integral transform method for explicit solutions to fractional differential equations

$Page 1: integral transform method for explicit solutions to fractional differential equations$

THEORY AND APPLICATIONS OFFRACTIONAL DIFFERENTIAL EQUATIONSTHEORY AND APPLICATIONS OFFRACTIONAL DIFFERENTIAL EQUATIONS

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THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS

ANATOLY A. KILBASBelarusian State UniversityMinsk, Belarus

HARI M. SRIVASTAVAUniversity of VictoriaVictoria, British Columbia, Canada

JUAN J. TRUJILLOUniversidad de La LagunaLa Laguna (Tenerife)Canary Islands, Spain

Amsterdam – Boston – Heidelberg – London – New York – Oxford

Paris – San Diego – San Francisco – Singapore – Sydney – Tokyo

THEORY AND APPLICATIONSOF FRACTIONAL DIFFERENTIALEQUATIONS

ANATOLYA. KILBASBelarusian State UniversityMinsk, Belarus

HARI M. SRIVASTAVAUniversity of VictoriaVictoria, British Columbia, Canada

JUAN J. TRUJILLOUniversidad de La LagunaLa Laguna (Tenerife)Canary Islands, Spain

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Preface

The subject of fractional calculus (that is, calculus of integrals and derivativesof any arbitrary real or complex order) has gained considerable popularity andimportance during the past three decades or so, due mainly to its demonstratedapplications in numerous seemingly diverse and widespread fields of science andengineering. It does indeed provide several potentially useful tools for solvingdifferential and integral equations, and various other problems involving specialfunctions of mathematical physics as well as their extensions and generalizationsin one and more variables.

The concept of fractional calculus is popularly believed to have stemmed from aquestion raised in the year 1695 by Marquis de L'Hopital (1661-1704) to GottfriedWilhelm Leibniz (1646-1716), which sought the meaning of Leibniz's (currentlypopular) notation ^f- for the derivative of order n £ No := } whenn = | (What if n = |? ). In his reply, dated 30 September 1695, Leibniz wrote toL'Hopital as follows: "... This is an apparent paradox from which, one day, usefulconsequences will be drawn. ..."

Subsequent mention of fractional derivatives was made, in some context orthe other, by (for example) Euler in 1730, Lagrange in 1772, Laplace in 1812,Lacroix in 1819, Fourier in 1822, Liouvill e in 1832, Riemann in 1847, Greer in 1859,Holmgren in 1865, Griinwald in 1867, Letnikov in 1868, Sonin in 1869, Laurent in1884, Nekrassov in 1888, Krug in 1890, and Weyl in 1917. In fact, in his 700-pagetextbook, entitled "Traite du Calcul Differentiel et du Calcul Integral" (Secondedition; Courcier, Paris, 1819), S. F. Lacroix devoted two pages (pp. 409-410) tofractional calculus, showing eventually that

y —

In addition, of course, to the theories of differential, integral, and integro-differen-tial equations, and special functions of mathematical physics as well as their exten-sions and generalizations in one and more variables, some of the areas of present-day applications of fractional calculus include Fluid Flow, Rheology, DynamicalProcesses in Self-Similar and Porous Structures, Diffusive Transport Akin to Diffu-sion, Electrical Networks, Probability and Statistics, Control Theory of DynamicalSystems, Viscoelasticity, Electrochemistry of Corrosion, Chemical Physics, Opticsand Signal Processing, and so on.

vii

vii i Theory and Applications of Fractional Differential Equations

The first work, devoted exclusively to the subject of fractional calculus, is thebook by Oldham and Spanier [643] published in 1974. One of the most recentworks on the subject of fractional calculus is the book of Podlubny [682] pub-lished in 1999, which deals principally with fractional differential equations. Someof the latest (but certainly not the last) works especially on fractional models ofanomalous kinetics of complex processes are the volumes edited by Carpinteri andMainardi [132] in 1997 and by Hilfer [340] in 2000, and the book by Zaslavsky [915]published in 2005. Indeed, in the meantime, numerous other works (books, editedvolumes, and conference proceedings) have also appeared. These include (for ex-ample) the remarkably comprehensive encyclopedic-type monograph by Samko,Kilbas and Marichev [729], which was published in Russian in 1987 and in Englishin 1993, and the book devoted substantially to fractional differential equationsby Miller and Ross [603], which was published in 1993. And today there exist atleast two international journals which are devoted almost entirely to the subject offractional calculus: (i) Journal of Fractional Calculus and (ii) Fractional Calculusand Applied Analysis.

The main objective of this book is to complement the contents of the otherbooks mentioned above. Many new results, obtained recently in the theory of or-dinary and partial differential equations, are not specifically reflected in the book.We aim at presenting, in a systematic manner, results including the existenceand uniqueness of solutions for the Cauchy Type and Cauchy problems involv-ing nonlinear ordinary fractional differential equations, explicit solutions of lin-ear differential equations and of the corresponding initial-value problems by theirreduction to Volterra integral equations and by using operational and composi-tional methods, applications of the one- and multi-dimensional Laplace, Mellin,and Fourier integral transforms in deriving closed-form solutions of ordinary andpartial differential equations, and a theory of the so-called sequential linear frac-tional differential equations including a generalization of the classical Frobeniusmethod.

This book consists of a total of eight chapters. Chapter 1 (Preliminaries)provides some basic definitions and properties from such topics of MathematicalAnalysis as functional spaces, special functions, integral transforms, generalizedfunctions, and so on. The extensive modern-day usages of such special functionsas the classical Mittag-Leffler functions and its various extensions, the Wright(or, more precisely, the Fox-Wright) generalization of the relatively more familiarhypergeometric pFq function, and the Fox H-function in the solutions of ordinaryand partial fractional differential equations have indeed motivated a major part ofChapter 1. Chapter 2 (Fractional Integrals and Fractional Derivatives) containsthe definitions and some potentially useful properties of several different families offractional integrals and fractional derivatives. Chapter 1 and Chapter 2, together,are meant to prepare the reader for the understanding of the various mathematicaltools and techniques which are developed in the later chapters of this book.

The fundamental existence and uniqueness theorems for ordinary fractionaldifferential equations are presented in Chapter 3 with special reference to theCauchy Type problems. Here, in Chapter 3, we also consider nonlinear and linearfractional differential equations in one-dimensional and vectorial cases. Chapter

Preface ix

4 is devoted to explicit and numerical solutions of fractional differential equa-tions and boundary-value problems associated with them. Our approaches in thischapter are based mainly upon the reduction to Volterra integral equations, uponcompositional relations, and upon operational calculus.

In Chapter 5, we investigate the applications of the Laplace, Mellin, and Fourierintegral transforms with a view to constructing explicit solutions of linear differen-tial equations involving the Liouville, Caputo, and Riesz fractional derivatives withconstant coefficients. Chapter 6 is devoted to a survey of the developments and re-sults in the fields of partial fractional differential equations and to the applicationsof the Laplace and Fourier integral transforms in order to obtain closed-form so-lutions of the Cauchy Type and Cauchy problems for the fractional diffusion-waveand evolution equations.

Linear differential equations of sequential and non-sequential fractional order,as well as systems of linear fractional differential equations associated with theRiemann-Liouville and Caputo derivatives, are investigated in Chapter 7, whichincidentally also develops an interesting generalization of the classical FrobeniusMethod for solving fractional differential equations with variable coefficients and adirect way to obtain explicit solutions of such types of differential equations withconstant coefficients. And, while a survey of a variety of applications of fractionaldifferential equations are treated briefly in many chapters of this book (especiallyin Chapter 7), a review of some important applications involving fractional modelsis presented systematically in the last chapter of the book (Chapter 8).

At the end of this book, for the convenience of the readers interested in furtherinvestigations on these and other closely-related topics, we include a rather largeand up-to-date Bibliography. We also include a Subject Index.

Operators of fractional integrals and fractional derivatives, which are based es-sentially upon the familiar Cauchy-Goursat Integral Formula, were considered by(among others) Sonin in 1869, Letnikov in 1868 onwards, and Laurent in 1884. Inrecent years, many authors have demonstrated the usefulness of such types of frac-tional calculus operators in obtaining particular solutions of numerous families ofhomogeneous (as well as nonhomogeneous) linear ordinary and partial differentialequations which are associated, for example, with many of the celebrated equa-tions of mathematical physics such as (among others) the Gauss hypergeometricequation:

l)z}-£ -a/3w=0

and the relatively more familiar Bessel equation:

9d2w dw , o o.

z 7 T + Z 1 ~ + (z ~ v ) w = °-dzz dz

In the cases of (ordinary as well as partial) differential equations of higher or-ders, which have stemmed naturally from the Gauss hypergeometric equation, theBessel equation, and their many relatives and extensions, there have been severalseemingly independent attempts to present a remarkably large number of scat-tered results in a unified manner. For developments dealing extensively with such

x Theory and Applications of Fractional Differential Equations

applications of fractional calculus operators in the solution of ordinary and par-tial differential equations, the interested reader is referred to the numerous recentworks cited in the Bibliography.

This book is written primarily for the graduate students and researchers inmany different disciplines in the mathematical, physical, and engineering sciences,who are interested not only in learning about the various mathematical tools andtechniques used in the theory and widespread applications of fractional differentialequations, but also in further investigations which emerge naturally from (or whichare motivated substantially by) the physical situations modelled mathematicallyin the book.

Right from the conceptualization of this book project until the finalization ofthe typescript in this present form, the authors have gratefully received invaluablesuggestions and comments from colleagues at many different academic institutionsand research centers around the world. Special mention ought to be made of thehelp and assistance so generously and meticulously provided by Dr. MargaritaRivero, Dr. Blanca Bonilla, Dr. Jose R. Franco, Ldo. Luis Rodriguez-Germa,all at the Universidad de La Laguna. Special mention should be made also ofEngr. Sergio Ortiz-Villajos Eirfn and of Dr. Luis Vazquez at the UniversidadComplutense de Madrid.

The first- and the second-named authors would like to express their apprecia-tion for the kind hospitality and support by the Universidad de La Laguna duringtheir many short-term visits to the Departamento de Analisis Matematico (Univer-sidad de La Laguna) throughout the tenure of this book project. All three authors,and particularly the second-named author, are especially thankful to the second-named author's wife and colleague, Professor Rekha Srivastava, for her constantsupport, encouragement, and understanding. The first-named author would alsolike to thank his wife, Mrs. Tamara Sapova, for her support and encouragement.

We are immensely grateful to Professor Nacere Hayek Calil for his constantinspiration and encouragement throughout the tenure of this book project.

Finally, the authors (particularly the third-named author) would like to thank-fully acknowledge the financial grants and support for this book project, whichwere awarded by the Ministry of Education and Sciences (MTM2004/00327),FEDER and the Autonomous Government of Canary Islands (PI 2003/133). Thisbook was prepared in the frame of joint collaboration of the first- and the third-named authors supported by Belarusian Fundamental Research Fund under Pro-jects F03MC-008 and F05MC-050. This work was also supported, in part, by theUniversidad de La Laguna, the Academia Canaria de Ciencias, and the NaturalSciences and Engineering Research Council of Canada.

September 2005 Anatoly A. KilbasBelarusian State University

Hari M. SrivastavaUniversity of Victoria

Juan J. Trujill oUniversidad de La Laguna

Contents

1 PRELIMINARIE S 1

1.1 Spaces of Integrable, Absolutely Continuous, and Continuous Func-tions 1

1.2 Generalized Functions 61.3 Fourier Transforms 101.4 Laplace and Mellin Transforms 181.5 The Gamma Function and Related Special Functions 241.6 Hypergeometric Functions 271.7 Bessel Functions 321.8 Classical Mittag-Leffler Functions 401.9 Generalized Mittag-Leffler Functions 451.10 Functions of the Mittag-Leffler Type 491.11 Wright Functions 541.12 The ^-Function 581.13 Fixed Point Theorems 67

2 FRACTIONA L INTEGRAL S AN D FRACTIONA LDERIVATIVE S 69

2.1 Riemann-Liouville Fractional Integrals and Fractional Deri-vatives 69

2.2 Liouvill e Fractional Integrals and Fractional Derivatives on the Half-Axis 79

2.3 Liouvill e Fractional Integrals and Fractional Derivatives on the RealAxis 87

2.4 Caputo Fractional Derivatives 902.5 Fractional Integrals and Fractional Derivatives of a Function with

Respect to Another Function 992.6 Erdelyi-Kober Type Fractional Integrals and Fractional Deriva-

tives 1052.7 Hadamard Type Fractional Integrals and Fractional Derivatives . . 1102.8 Griinwald-Letnikov Fractional Derivatives 1212.9 Partial and Mixed Fractional Integrals and Fractional

Derivatives 1232.10 Riesz Fractional Integro-Differentiation 1272.11 Comments and Observations 132

xi i Theory and Applications of Fractional Differential Equations

3 ORDINAR Y FRACTIONA L DIFFERENTIA L EQUATIONS.EXISTENC E AND UNIQUENESS THEOREM S 135

3.1 Introduction and a Brief Overview of Results 1353.2 Equations with the Riemann-Liouville Fractional Derivative in the

Space of Summable Functions 1443.2.1 Equivalence of the Cauchy Type Problem and the Volterra

Integral Equation 1453.2.2 Existence and Uniqueness of the Solution to the Cauchy Type

Problem 1483.2.3 The Weighted Cauchy Type Problem 1513.2.4 Generalized Cauchy Type Problems 1533.2.5 Cauchy Type Problems for Linear Equations 1573.2.6 Miscellaneous Examples 160

3.3 Equations with the Riemann-Liouville Fractional Derivative in theSpace of Continuous Functions. Global Solution 1623.3.1 Equivalence of the Cauchy Type Problem and the Volterra

Integral Equation 1633.3.2 Existence and Uniqueness of the Global Solution to the

Cauchy Type Problem 1643.3.3 The Weighted Cauchy Type Problem 1673.3.4 Generalized Cauchy Type Problems 1683.3.5 Cauchy Type Problems for Linear Equations 1703.3.6 More Exact Spaces 1713.3.7 Further Examples 177

3.4 Equations with the Riemann-Liouville Fractional Derivative in theSpace of Continuous Functions. Semi-Global and Local Solutions . 1823.4.1 The Cauchy Type Problem with Initial Conditions at the

Endpoint of the Interval. Semi-Global Solution 1833.4.2 The Cauchy Problem with Initial Conditions at the Inner

Point of the Interval. Preliminaries 1863.4.3 Equivalence of the Cauchy Problem and the Volterra Integral

Equation . . 1893.4.4 The Cauchy Problem with Initial Conditions at the Inner

Point of the Interval. The Uniqueness of Semi-Global andLocal Solutions 191

3.4.5 A Set of Examples 1963.5 Equations with the Caputo Derivative in the Space of Continuously

Differentiable Functions 1983.5.1 The Cauchy Problem with Initial Conditions at the Endpoint

of the Interval. Global Solution 1993.5.2 The Cauchy Problems with Initial Conditions at the End and

Inner Points of the Interval. Semi-Global and LocalSolutions 205

3.5.3 Illustrative Examples 209

Contents xii i

3.6 Equations with the Hadamaxd Fractional Derivative in the Space ofContinuous Functions 212

4 METHOD S FOR EXPLICITL Y SOLVING FRACTIONA LDIFFERENTIA L EQUATION S 221

4.1 Method of Reduction to Volterra Integral Equations 2214.1.1 The Cauchy Type Problems for Differential Equations with

the Riemann-Liouville Fractional Derivatives 2224.1.2 The Cauchy Problems for Ordinary Differential Equa-

tions 2284.1.3 The Cauchy Problems for Differential Equations with the

Caputo Fractional Derivatives 2304.1.4 The Cauchy Type Problems for Differential Equations with

Hadamard Fractional Derivatives 2344.2 Compositional Method 238

4.2.1 Preliminaries 2384.2.2 Compositional Relations 2394.2.3 Homogeneous Differential Equations of Fractional Order with

Riemann-Liouville Fractional Derivatives 2424.2.4 Nonhomogeneous Differential Equations of Fractional Or-

der with Riemann-Liouville and Liouville Fractional Deriva-tives with a Quasi-Polynomial Free Term 245

4.2.5 Differential Equations of Order 1/2 2484.2.6 Cauchy Type Problem for Nonhomogeneous Differential

Equations with Riemann-Liouville Fractional Derivatives andwith a Quasi-Polynomial Free Term 251

4.2.7 Solutions to Homogeneous Fractional Differential Equationswith Liouville Fractional Derivatives in Terms of Bessel-Type Functions 257

4.3 Operational Method 2604.3.1 Liouville Fractional Integration and Differentiation Opera-

tors in Special Function Spaces on the Half-Axis 2614.3.2 Operational Calculus for the Liouville Fractional Calculus

Operators 2634.3.3 Solutions to Cauchy Type Problems for Fractional Differen-

tial Equations with Liouville Fractional Derivatives 2664.3.4 Other Results 270

4.4 Numerical Treatment 272

5 INTEGRA L TRANSFORM METHO D FOR EXPLICI TSOLUTION S TO FRACTIONA L DIFFERENTIA LEQUATION S 279

5.1 Introduction and a Brief Survey of Results 2795.2 Laplace Transform Method for Solving Ordinary Differential Equa-

tions with Liouvill e Fractional Derivatives 283

xiv Theory and Applications of Fractional Differential Equations

5.2.1 Homogeneous Equations with Constant Coefficients 2835.2.2 Nonhomogeneous Equations with Constant Coefficients . . . 2955.2.3 Equations with Nonconstant Coefficients 3035.2.4 Cauchy Type for Fractional Differential Equations 309

5.3 Laplace Transform Method for Solving Ordinary Differential Equa-tions with Caputo Fractional Derivatives 3125.3.1 Homogeneous Equations with Constant Coefficients 3125.3.2 Nonhomogeneous Equations with Constant Coefficients . . . 3225.3.3 Cauchy Problems for Fractional Differential Equations . . . 326

5.4 Mellin Transform Method for Solving Nonhomogeneous FractionalDifferential Equations with Liouvill e Derivatives 3295.4.1 General Approach to the Problems 3295.4.2 Equations with Left-Sided Fractional Derivatives 3315.4.3 Equations with Right-Sided Fractional Derivatives 336

5.5 Fourier Transform Method for Solving Nonhomogeneous Differen-tial Equations with Riesz Fractional Derivatives 3415.5.1 Multi-Dimensional Equations 3415.5.2 One-Dimensional Equations 344

6 PARTIA L FRACTIONA L DIFFERENTIA L EQUATION S 347

6.1 Overview of Results 3476.1.1 Partial Differential Equations of Fractional Order 3476.1.2 Fractional Partial Differential Diffusion Equations 3516.1.3 Abstract Differential Equations of Fractional Order 359

6.2 Solution of Cauchy Type Problems for Fractional Diffusion-WaveEquations 3626.2.1 Cauchy Type Problems for Two-Dimensional Equations . . 3626.2.2 Cauchy Type Problems for Multi-Dimensional Equations . . 366

6.3 Solution of Cauchy Problems for Fractional Diffusion-Wave Equa-tions . 3736.3.1 Cauchy Problems for Two-Dimensional Equations 3746.3.2 Cauchy Problems for Multi-Dimensional Equations 377

6.4 Solution of Cauchy Problems for Fractional Evolution Equations . 3806.4.1 Solution of the Simplest Problem 3806.4.2 Solution to the General Problem 3846.4.3 Solutions of Cauchy Problems via the H-Functions 388

7 SEQUENTIA L LINEA R DIFFERENTIA L EQUATION S OFFRACTIONA L ORDER 393

7.1 Sequential Linear Differential Equations of Fractional Order . . . . 3947.2 Solution of Linear Differential Equations with Constant Coef-

ficients 4007.2.1 General Solution in the Homogeneous Case 4007.2.2 General Solution in the Non-Homogeneous Case. Fractional

Green Function 403

Contents xv

7.3 Non-Sequential Linear Differential Equations with Constant Co-efficients 407

7.4 Systems of Equations Associated with Riemann-Liouville and Ca-puto Derivatives 4097.4.1 General Theory 4097.4.2 General Solution for the Case of Constant Coefficients. Frac-

tional Green Vectorial Function 4127.5 Solution of Fractional Differential Equations with Variable Coef-

ficients. Generalized Method of Probenius 4157.5.1 Introduction 4157.5.2 Solutions Around an Ordinary Point of a Fractional Differ-

ential Equation of Order a 4187.5.3 Solutions Around an Ordinary Point of a Fractional Differ-

ential Equation of Order 2a 4217.5.4 Solution Around an a-Singular Point of a Fractional Differ-

ential Equation of Order a 4247.5.5 Solution Around an a-Singular Point of a Fractional Differ-

ential Equation of Order 2a 4277.6 Some Applications of Linear Ordinary Fractional Differential

Equations 4337.6.1 Dynamics of a Sphere Immersed in an Incompressible Vis-

cous Fluid. Basset's Problem 4347.6.2 Oscillatory Processes with Fractional Damping 4367.6.3 Study of the Tension-Deformation Relationship of Viscoelas-

tic Materials 439

8 FURTHER APPLICATION S OF FRACTIONA L MODEL S 449

8.1 Preliminary Review 4498.1.1 Historical Overview 4508.1.2 Complex Systems 4528.1.3 Fractional Integral and Fractional Derivative Operators . . 456

8.2 Fractional Model for the Super-Diffusion Processes 4588.3 Dirac Equations for the Ordinary Diffusion Equation 4628.4 Applications Describing Carrier Transport in Amorphous Semicon-

ductors with Multiple Trapping 463

Bibliography 469

Subject Index 521


Chapter 1

PRELIMINARIE S

This chapter is preliminary in character and contains definitions and propertiesfrom such topics of Analysis as functional spaces, special functions, and integraltransforms.

1.1 Spaces of Integrable, Absolutely Continuous,and Continuous Functions

In this section we present definitions of spaces of p-integrable, absolutely contin-uous, and continuous functions and their weighted modifications. We also givecharacterizations of those modified spaces which will be used later.

Let fi = [a, b] (—oo a < b oo) be a finite or infinite interval of the realaxis R = (-co, oo). We denote by Lp(a, b) (1 p ^ oo) the set of those Lebesguecomplex-valued measurable functions / on fi for which || / | |p < oo, where

\\f\\P= / \f(t)\Pdt) (l^p<oo) (1.1.1)\Ja )

and

= ess supo<a.<6|/(a;)|. (1-1-2)

Here ess sup|/(a;)| is the essential maximum of the function |/(a:)| [see, for example,Nikol'skii [628], pp. 12-13)].

We also need the weighted Lp-space with the power weight. Such a space,which we denote by X^{a,b) (c £ R; 1 ^ p ^ oo), consists of those complex-valued Lebesgue measurable functions / on (a,b) for which ||/||xf < °°> with

(1.1.3)

2 Theory and Applications of Fractional Differential Equations

andII/HJCC. = ess supa^ 6 [ z c | / ( z ) | ]. (1.1.4)

In particular, when c = 1/p, the space X^(a,b) coincides with the Lp(a, 6)-space:X»/p(a,b) = Lp{a,b).

Let now [a, b] (—00 < a < b < 00) be a finite interval and let AC[a, b] be thespace of functions / which are absolutely continuous on [a,b]. It is known [seeKolmogorov and Fomin ([434], p. 338)] that j4C[a, 6] coincides with the space ofprimitives of Lebesgue summable functions:

f{x) G AC[a,b] & f(x) =c+ f tp{t)dt (tp(t) G L(a,b)), (1.1.5)J a

and therefore an absolutely continuous function f(x) has a summable derivativef'{x) = ip(x) almost everywhere on [a,b\. Thus (1.1.5) yields

<p{t) = f'(t) and c = f(a). (1.1.6)

For n G N := {1,2,3, } we denote by ACn[a, b] the space of complex-valuedfunctions f(x) which have continuous derivatives up to order n — 1 on [a, b] suchthat /("-1)(x) £AC[a,b]:

b] (D = -£ )j , (1.1.7)

C being the set of complex numbers. In particular, ^ C1 ^ , b] = ^4C[a, b].

This space is characterized by the following assertion [see Samko et al. ([729],Lemma 2.4)].

Lemma 1.1 The space ACn[a, b] consists of those and only those functions f(x)which can be represented in the form

) \f{x) = m+<p)(x) + J2 ck(x - a)\ (1.1.8)k=0

where (p(t) G L(a, b), Ck (k = 0,1, , n — 1) are arbitrary constants, and

It follows from (1.1.8) that

<P(t) = f ( n ) ( t ) , ck = ^ ^ - (k = 0,l,---,n-l). (1.1.10)

We also use a weighted modification of the space ACn[a, b] (n G N), in whichthe usual derivative D = d/dx is replaced by the so-called ^-derivative, defined by

S = xD (D = 4~)- (1.1.11)ax

Chapter 1 (§ 1.1)

Such a modification, which we denote by AC£ [a, b] (n £ N; \i £ E), involves thecomplex-valued Lebesgue measurable functions g on (a, b) such that xfig(x) has^-derivatives up to order n — 1 on [a,b] and Sn~1[x tig(x)] is absolutely continuouson [a,b\:

£Ja,b} = {<? : [a,b] -> C : ^ "VsO* ) ] £ AC[a,6], /i 6 K, <5 = z ^ j .

(1.1.12)In particular, when fi = 0, the space .AC" [a, 6] := ACfo[a,b] is defined by

AC?[a,b] = L : [a,b] -> C : 5""1 [ G i4C[a,6], <5 = z j U . (1.1.13)

If fi = 0 and n = 1, the space AC^[a, b] coincides with vlC[a, 6].

When a > 0, the space AC/fâ, b] is characterized by the following result [seeKilbas ([370], Theorem 3.1)].

Lemma 1.2 Let 0 < a < b < oo, n £ N and /U £ E. The space AC% [a, b] consistsof those and only those functions g(x) which are represented in the form

g{x) - x"

w/iere <p(t) £ L(a, 6) and dk (k = 0,1, , n - 1) are arbitrary constants.

In particular, g(x) £ ^Cjâ, b] if, and only if,

( ^ ) (1.1.15)k=0

We note that <p(t) and dk in (1.1.14) are given by

<fi(t) = 9'n-S), dk = 9-^- (fc = 0 , l , . . - , n - l ) (1.1.16)

with

ffJk(a;) = (5fc[^5(a;)] (k = 1, , n - 1), ^ ( s) = ^^(a;), (1.1.17)

while in (1.1.15) they take the more simple forms

Let Q. = [a, b] (-00 ^ a < b ^ oo) and m £ No = {0,1, . We denote byCm(fi ) a space of functions / which are m times continuously differentiable on $7with the norm

fc=o k=o


In particular, for m = 0, C°(Q.) = C(fi) is the space of continuous functions / onQ with the norm

||c=max|/(z)|. (1.1.20)

When H = [a,b] is a finite interval and 7 £ C (0 ^ (7) < 1), we introducethe weighted space C7[a,b] of functions / given on (a,b], such that the function(x-ajtfix) £C[a, b], and

\\f\\c,=\\(x-a)y(x)\\c, C0[a,b] = C[a,b]. (1.1.21)

For n G N we denote by C"[a, b] the Banach space of functions f(x) which arecontinuously differentiable on [a, b] up to order n—1 and have the derivative /(") (x)of order n on (a, b] such that f^n\x) G Cy[a,b]:

C;[a,b] = \f: \\f\\c»=J2\\fik)\\c + \\f{n)\\cX C°[a,b]=C7[a,b}.{ k=o )

(1.1.22)From this definition we have the following characterization of the space C™[a, b]

[see Kilbas et al. ([375], Lemma 1)].

Lemma 1.3 Let n G No = {0,1, } and 7 G C (0 ^ ^(7) < 1). The spaceC"[a, b] consists of those and only those functions f which are represented in theform

where <p(t) G C^[a, b] and Ck {k = 0,1, , n — 1) are arbitrary constants.

Moreover,

<p(t) = / W ( t ) , Ck=f-^T- (* = 0 , l , - - -, n - l ) . (1.1.24)

/n particular, when 7 = 0, i/ie space Cn[a,b] consists of those and only thosefunctions f which are represented in the form (1.1.23), where (p(t) G C[a, b] and Ck(k = 0 ,1, - -- , n — 1) are arbitrary constants. Moreover, the relations in (1.1.24)hold.

We also introduce two subspaces C°[a, b] and C®[a, b] of the space C[a, b] definedby

C°a[a,b] = {f{x) G C°a[a,b}; / (a) = 0, | | / | |c. = | | / | |c} (1.1.25)

andC°b[a,b] = {f{x) G C°a[a,b]; f(b) = 0, | | / | |Ca = | | / | | c }. (1.1.26)

When Q, = [a, b] (0 < a < b < 00) is a finite interval and 7 G C (0 % 9t(7) < 1),we introduce the weighted space C1:\og[a, b] of functions g, given on (a, b] and suchthat [\og{x/a)]ig{x) G C[a,b}:

7 / ^ ) | c , C0,los[a,b] = C[a,b]. (1.1.27)

Chapter 1 (§ 1.1)

For n € N we denote by C^â^b} the Banach space of functions g(x) which havecontinuous ^-derivatives on [a, b] up to order n — 1 and derivative (Sng) (x) on (a, b]of order n such that (6ng)(x) G C7ii 0gK b]:

CUa, b] = \g: \\g\\c^ = £ \\Skg\\c + \\Sng\\c } , Câ, b] = C7,log[a, 6].

(1.1.28)From this definition we have the following characterization of the space C$ [a, b].

Lemma 1.4 Let 0 < a < b < oo, n £ No and 7 6 C (0 ^ $K(7) < 1). The spaceC|*7[a, 6] consists of those and only those functions g which are represented in theform

where ip(t) G C-ytiog[a, b] and du (fe = 0 ,1, , n — 1) are arbitrary constants.

Moreover,

<p(t) = (6ng)(t), c4 = ^ f ^ (* = 0 , 1, . (1.1.30)

/n particular, when 7 = 0, i/ie space Q ' J a, 6] = C^fa, b] consists of thoseand only those functions g which are represented in the form (1.1.29), where<p{t) e C[a, b] and <4 (fc = 0,1, , n — 1) are arbitrary constants. Moreover,the relations in (1.1.30) hold.

For Banach spaces X and Y, we denote by X — Y the continuous embedding:

(a) if / G X, then / G Y; (b) \\f\\Y £ K\\f\\x, (1.1.31)

where the constant K > 0 does not depend on / .

From the definitions (1.1.22) and (1.1.28) we derive the property of the spacesG > , 6] and <?£>,&].

Propert y 1.1 Let n E No, and let 71 and 72 be complex numbers such that

0^*(7i)^*(72)<l . (1-1.32)

The following embedding hold:

Cn[a,b] -+ C£[a,6] -» G?2[a,6], (1.1.33)

with

c-a ^ ^ll/llc^ , K = min [l , (6 - a)*^-^)] ; (1.1.34)

C?[a, b] -> C^71 [a, 6] ^ C4%[a, 6], (1.1.35)

Theory and Applications of Fractional Differential Equations

with

Ik" ^ Ks\\f\\c« , K5= min l , ( log-J . (1.1.36)

In particular,

C[a,b] ->C7l[a,6] -»C72[a,&] , wii/ i | | / | |c72 ^ (6 - a)m ( 7 2~7 l ) | | / | |c7 i ; (1.1.37)

C[a, b] -» C7l,iog[o, 6] -> C72,log[a, 6], mfft ||/||Cj;72 g (log ^ | | / | |c, ; 7 1.

(1.1.38)

1.2 Generalized FunctionsIn this section we present definitions and some properties of certain spaces of testand generalized functions. More detailed information for such classical spaces canbe found in the books by Gel'fand and Shilov [276], Vladimirov [859] and Zemanian[921], and for other ones in the books by Samko et al. [729] and McBride [566].

Let En be the n-dimensional Euclidean space, let x = (a;i,--- , xn) andt = (ii , - , in) be points in M™, x t = Xiti -I h xntn be their scalar product,

a nd | t| = ( i f + + t 2n ) 1 / 2 , dt = dt1---dtn. L et

N " = { k = , k n ) , i t , £ N ; j = l , - - - , n } , ( 1 -2 .1)

Nn — / t — ( h , . . . b \ b - C N « . i — 1 . . . n 1 (~\ 9 9 ^

The elements k £ NQ are called multi-indices. For x £ E™ and k £ NQ we shalluse the notations

x k = xkl xkn Ikl = k\ + + k (12 3)

and/Ft A \ . 1 1

(1.2.4)

We denote by C™ (n € N) the n-dimensional space of n complex numbersz = (zi, , zn) (ZJ e C; j = 1, , n).

We shall consider generalized functions over ft, where ft is a domain in E". Wechoose test functions on fi as the infinitely differentiable functions at the interiorpoints of Cl with prescribed behavior at the boundary points of fi. We denoteby < / , V5 > t n e value of the generalized function / as a functional on the testfunction (p. A generalized function / is called regular if it is a locally integrablefunction such that fn f (x)ip(x)dx exists for each test function y(x) and

< / , ¥ > >= / " / ( xMx)dx . (1.2.5)

Chapter 1 (§ 1.2)

I t is assumed that the bilinear form < /, is chosen in such a way that itcoincides with (1.2.5) in the case of a regular generalized function.

We assume that the space of test functions X = X(Cl) is a topological spaceand denote by X' = X'(Q.) the space of continuous linear functional space on Xwhich is called the topological dual of X.

First of all, we recall the notion of a generalized function concentrated ata point. A generalized function / G X' is said to be zero on open set G, if< /, = 0 for each test function <p £ X which is zero beyond G. The union Ofof all open sets where / = 0 is called a null set of the function f. The complementof the null set with respect to ft is called the support of the generalized functionand such a complement is denoted by supp(/) = Cl\ Of. The generalized functionis said to be concentrated at the point to, if supp(/) is this point to-

The classical Dirac function 6(t - t0) , t0 £ ft, defined by

{S(t-to),<p)=<p{to) (1.2.6)

and its derivatives defined as

| k | V (keif), (1-2-7)

provide simple examples of generalized functions concentrated at the point to.

Any function with domain ft in E" or <C™ and range in E = E1 or C = C1

is called an ordinary function. An ordinary function is called smooth (or C°°) ifit is infinitely differentiate. An ordinary function <p on En is said to be of rapiddescent if <p(t) = o (|t|~ra) for every integer m € Z = {0, , , . An ordinaryfunction <p is said to be of slow growth if there exists an integer m £ Z such that(p(t) = o( | t |m) . If an ordinary function <p(t) is defined and continuous on someopen set Cl in R", then the support of = 0 for every ip E X(€l) withsupp(y) C fi. If Clf is the largest open set where / = 0, then the set 0 \ Q/is called the support of the generalized function f and is denoted by supp(/). Ifsupp(/) c G c f l , then / is said to be concentrated on G. If G is bounded, then/ is called a generalized function with compact support.

When 17 = E, the support of an ordinary or generalized function / is said tobe bounded on the left or on the right if there exists a real number Tf € E suchthat f(t) = 0 for t < Tf or t > Tf, respectively.

Now we give definitions and certain properties of some spaces of test and gen-eralized functions. First we present the Schwartz test-function space S = <S(K"),defined on the whole space (fi = R"), and its dual S' = <S'(R"). <S is the linearspace of all complex-valued smooth functions cp(t) on En such that

7m,k = sup [(1 + |t|2ro] |D*V(t) | < oo (1.2.8)tGK"

for all nonnegative integers m E No and k € NQ . The topology of S is generatedby the countable collection of seminorms {7 m,k} ^ ik|=o- ^ follows from (1.2.8)


that every ip(t) £ <S is a function of rapid descent, and therefore S is called thespace of smooth functions of rapid descent.

The Schwartz space of generalized functions S' is the dual space of S. Everygeneralized function f £ S1 can be represented in the form

/( t ) = DmF(t ), (1.2.9)

where m £ NQ and F(t) is a continuous function of slow growth as |t| -> oo.Therefore, <S' is called the space of generalized functions of slow growth or thespace of tempered distributions.

When n = 1 and Cl = [0, oo), the space <S+ = <S+([0, oo)) is defined as thetest-function space consisting of all smooth functions <p(t) on [0, oo) which are ofrapid descent as t —> +oo. The corresponding space of generalized functions S'+ isthe dual space of <S+.

Next we consider the test-spaces VK, "D and their dual spaces. Let Q, = Kbe a compact (i.e. bounded and closed) set in E". We denote by T>K = T>K(M.n)the linear space of all complex-valued smooth functions </?(t) that are equal tozero outside K. The topology of T>K is generated by the countable collection ofseminorms {7k(v)} Ck=o> where

7 k M = sup |D"V(t)|, <p(t) e VK, (1.2.10)t£R™

and k £ NJ. The space of generalized functions V'K is the dual space of T>K-Every function / £ V'K has a representation of the form

/( t ) = DmF(t ), (1.2.11)

where F(t) is some ordinary function on K and m £ N J.

Let {Km}m=i be a sequence of compact sets Km in W™ such that

K1cK2^---cKmc---, Rn = U 1Km. (1.2.12)

The test space V = V(M.n) is denned as the countable-union space of all VKm T> =U^=1Z>icm. This space consists of all smooth functions with compact supports.The space of generalized functions V is the dual space of V. Every generalizedfunction / 6 V can be represented in the form

/(t) = Y, DP"^m(t), (1.2.13)|m|=l

where m, p m 6 NQ, and Fm(t) are continuous functions with compact supportssuch that the supports of Fm are separated from the origin as |m| —> oo and arecontained in an arbitrary neighborhood of supp(/).

The elements of the spaces T>'K and D' are called distributions.

Chapter 1 (§ 1.2)

The notation of (1.2.13) for every generalized function f £ V with compactsupport can be simplified to

M

fit) = J2 D m ^ ( t ) (M G No), (1.2.14)|m|=0

where m e NJ and Fm(t) are continuous functions concentrated in an arbitraryneighborhood of supp(/). In particular, every generalized function / € V con-centrated at the point t0 is expressed in terms of the Dirac delta function (1.2.6)and its derivatives (1.2.7) by

M

f{t) = Y, «mDm(S(t - t0), (1.2.15)|m|=0

where m £ NQ and am are given constants.When n = 1 and fi = [0,oo), we denote by T>+ = T>+([0, oo)) the subspace of

V(M) consisting of all (p(t) £ V(M) concentrated on [0, oo), and the correspondingspace of generalized functions T>'+ is the dual space of T>+. We denote by V'R andV'L the subspaces of X>'(R) consisting of all generalized functions with supportsbounded on the left and on the right, respectively.

The test-function space £ = £ (Rn) is the linear space of all complex-valuedsmooth functions on Mn. Its dual £' = £'{M.n) consists of all generalized functionswith compact supports. When n = 1 and fi = [0, oo), £+ = £+([0,oo)) is definedas the space of all complex-valued smooth functions on [0, oo), and £'+ is the dualspace of £+.

The test-function space £ = £(IRn) is the linear space of all entire functions(p(t) such that

supteC"

< oo, (1.2.16)

where k, m £ NQ and the constants a,j (j = 1, , n) depend on <^(t). Thetopology of £ is generated by the countable set of seminorms {£k,m(¥0}i£| |m|=0-The space of generalized functions £' = £'(R") dual to £ is called the space ofultradistributions.

Other properties of the above spaces of test and generalized functions may befound in the books by Gel'fand and Shilov [276], Vladimirov [859], and Zemanian[921]. Now we indicate some spaces of test and generalized functions suitable forfractional differentiation and integration operators.

First we present the Lizorkin spaces \&, $ and St', $' of test and generalizedfunctions on Kn; see Section 25.1 in the book by Samko et al. [729]. The formertest-function space *& is a linear space of all complex-valued infinitely differentiablefunctions (p(t) whose derivatives vanish at the origin:

, (D"V)(0)=0 (|k|£No). (1.2.17)


Lastly, we introduce the test-function space $ which is a subspace of the Schwartzspace (S(Mn). It consists of those Schwartzian functions <p 6 S which are orthogonalto polynomials, that is,

JWtk<p(t)dt = 0 (k e NJ). (1.2.18)

The spaces of generalized functions \P' and $' are the spaces dual to S& and $,respectively.

Finally, we mention McBride spaces Tp and J-'p of test and generalizedfunctions denned for 1 p ^ oo and ^ e t [see McBride [566]]. The test-functionspace Tv is a linear space of all complex-valued functions <p(x) on R+ such that

T H ? ) = II '*[ '~M')]IIL ' < oo, (1.2.19)

where k £ No- The topology of TPtil is generated by the countable collection ofseminorms {/y^'fi(lp)}'kLo- The space of generalized functions T'p M is the dual spacef

1.3 Fourier TransformsIn this section we present definitions and some properties of one- and multi-dimensional Fourier transforms in spaces of p-summable and generalized functions.More detailed information may be found in the books by Sneddon [773] and Titch-marsh [819] (one-dimensional case), Kolmogorov and Fomin [435], Nikol'skii [628]and Stein and Weiss [802] (multidimensional case), and Brychkov and Prudnikov[108] (generalized functions).

We begin with the one-dimensional case. The Fourier transform of a functionip(t) of real variable t £ M. = (—oo, oo) is defined by

(T<p){x) = F[<p(t)](x) = $(x) := f eixtip(t)dt ( i £ l ) . (1.3.1)

The inverse Fourier transform is given by the formula

(x e R). (1.3.2)

The integrals in (1.3.1) and (1.3.2) converge absolutely for functions ip,g £ .LÊ)and in the norm of the space L2(R) for ip,g € L2(R). The Z^-theory and L2-theory of the above Fourier integrals is described in the books by Sneddon [773]and Titchmarsh [819].

In particular, each of these transforms is inverse to the other one for "suffi-ciently good" functions ip, g:

g = g, (1.3.3)

Chapter 1 (§ 1.3) 11

and the following simple relation holds:

(??<p)(x) = <p{-x). (1.3.4)

The next properties of the Fourier transform give the connection between it andthe translation Th and dilation Ii\ operators defined by

(Th<p)(x) = <p(x - h) (x, l i e l ) (1.3.5)

and{TL\(p){x) = ip(\x) ( i £ l ; A > 0), (1.3.6)

respectively. The following relations are valid for tp e lJ(R) (j = 1,2):

{FTh<p){x)=e-i* h(?<p)(x) ( i ,A£ i ) , (1.3.7)

j (j) (1.3.8)

T[eiat<p{t)]{x) = (T-aF)[<p]{x) = T(x + a) (x,a e E). (1.3.9)

If ip(t) G Z/i(M), then (^^(x) is a bounded continuous function and, in accor-dance with the Riemann-Liouville theorem, (!F(p)(x) tends to zero as |a;| —> oo:

lim / eixt<p{t)dt = O, ¥>(i)6ii(E). (1.3.10)

The rate of decrease of (Jr<f)(x) at infinity is connected with the smoothness ofthe function <p(t). This connection is given by the following relations:

(fceN) (1.3.11)

and

Dk(r<p){x) = {it)k?[<P(t)](*) ( ^ N ) . (1.3.12)

These equations are valid for "sufficiently good" functions <p; for example, forfunctions ip{t) £ Ck{M) such that tpW(t) € L\R) (j = 0,1, , k).

The Fourier convolution operator of two functions h and (p is defined by theintegral

O

h*(p:=(h*tp)(x)= h(x - t)(p(t)dt ( i £ l ) , (1.3.13)J — OO

which has the commutative property

h*(p = f*h. (1.3.14)

The boundedness of the convolution operator in the space LP(M) is given bythe Young theorem.


Theorem 1.1 If h(t) G Li(R) and ip(t) £ LP(E), then (h * (p)(x) £ LP{R)(1 5: p 5= oo) and ifte following inequality holds:

In particular, if h{t) G Li(K ) and (p(t) G L2(K) , #ien (/i * ip){x) G L2(R) and

(1-3.16)

When h(t) G L2{M) and <£>(£) G £2(K) , then the convolution (1.3.13) has thefollowing property: the function (h * <fi)(x) is continuous, bounded, and vanishesat infinity.

The Fourier transform of the convolution (1.3.13) is given by the Fourier con-volution theorem.

Theorem 1.2 Let either h(t) £ L i ( l ) and tp(t) G Li(M) , or h{t) G Li(K ) andtp{t) £ L2(E), or h{t) £ L 2( l ) and <p(t) £ L2{M).

Then the Fourier transform of the convolution h* ip is given by

{T{h * <p)) (x) = {Fh){x){TV){x). (1.3.17)

We also indicate that the cosine- and sine-Fourier transforms of a function<p(t) (t £ M+) are defined by

O

= / cos{xt)ip(t)dt (x £ K+) (1.3.18)

andO

{?s<p){x) = / sin{xt)ip{t)dt {x £ K+), (1.3.19)Jo

respectively, while the corresponding inverse transforms have the forms

2(T~1g)(x) = - cos(xt)g{t)dt (x G E+) (1.3.20)

^* t/o

and

(a; G R+). (1.3.21)

Next we consider the multidimensional case. We shall use the notation givenin Section 1.2.

The n-dimensional Fourier transform of a function (p(t) of t £ E™ is dennedby

/ (XGE" ), (1.3.22)

Chapter 1 (§ 1.3) 13

while the corresponding inverse Fourier transform is given by the formula

(x G R").

(1.3.23)The integrals in (1.3.22) and (1.3.23) have the same properties as those of the one-dimensional ones in (1.3.1) and (1.3.2). They converge absolutely for functionsip,g e Li{Rn) and in the norm of the space L2(K

n) for <p,g £ L2{M.n). TheL\-theory and the L -theory of the above Fourier integrals can be found in thebooks by Kolmogorov and Fomin [434], Nikol'skii [628], and Stein and Weiss [802].

All of the above properties of the one-dimensional Fourier transforms are ex-tended to the multidimensional Fourier transforms. In particular, the relations(1.3.7)-(1.3.9) take the following forms:

(x, h£l"), (1.3.24)

x e f ; A > 0 ), (1.3.25)

+ a) (x, a G Rn); (1.3.26)

Th and H.\ being defined by

(rh^)(x) = y>(x - h), (IIA¥>)(x) = *{**) (x, h e E"; A e R+); (1.3.27)

while (1.3.10)-(l.3.12) are extended as follows:

lim / e" -V( t )d t=0, ^( t)GLi(K n) , (1.3.28)

(x £ 1 " ; ke Nn) (1.3.29)

andD k ( ^ ) ( x ) = Jf[(it)k^(t)](x) (x e »"; k e N"), (1.3.30)

respectively. If A is the n-dimensional Laplace operator

then (1.3.29) yields

(xGKn) . (1.3.32)

Analogous to (1.3.13), the Fourier convolution operator of two functions h and(p is defined by

h*ip:= (h*ip)(x) = [ /i(x - t)<p(t)dt ( x £ l " ) , (1.3.33)

which has the commutative property (1.3.14). For such a convolution operator,the Young theorem (Theorem 1.1) and the Fourier convolution theorem (Theorem1.2) still apply.


Theorem 1.3 If h(t) G Li(E.) and y{t) 6 Lp(Rn) with 1 ^ p ^ oo, then (h *

<p)(x) G Lp(M.n) and the inequality (1.3.15) holds.

In particular, if h(t) G Li(E" ) and <p(t) G L2{Wl), then {h * ip)(x) G L2(M.n)

and the formula (1.3.16) is valid.

Theorem 1.4 Let either h(t) G Li(En) and </?(£) G Li(E") , or ft(f) G Li(En)and y>(*) G 2(K"), or h(i) G Z,2(IR") anrf <p{t) G L2(R

n)-

T/ien i/ie Fourier transform of the convolution h * <p is given by the formula(1.3.17).

Now we present Fourier transforms of generalized functions defined in Section1.2; see Section 2.2 in the book by Brychkov and Prudnikov [108]. First we presentthe definitions by L. Schwartz of the Fourier transform in the space of generalizedfunctions <S'(Kn). Such a direct Fourier transform is defined by

(<peS), (1.3.34)

while the inverse Fourier transform has the form

1 ^ ) ] (feS'). (1.3.35)

When /(t) G -Li (Mn)), it is easily seen that T[f] and J7" 1^] coincide with (1.3.22)and (1.3.23), respectively.

T and T~x specify an automorphism of the generalized function space S' andthey are inverse to each other:

T~l [T[f]} = f, T [F-1^]] = f (f G S'). (1.3.36)

The formulas (1.3.24), (1.3.26), (1.3.29), and (1.3.30) are valid for / G S':

jr[r h/(t)](x ) = e--hjT[/](x ) (x,h G R"), (1.3.37)

Q) (xe i "; AGE+), (1.3.38)

= ^(x + a) (x,a G E"); (1.3.39)

(xef; kenn) (1.3.40)

(xGE"; k€N"). (1.3.41)

Here rh/ , IW and D k / are defined for / G 5' by

< rbf, f > = < f, T-hv > {<p E S), (1.3.42)

(V GS) (1.3.43)

and^ (ve S). (1.3.44)

Chapter 1 (§ 1.3) 15

To obtain a convolution formula of the form (1.3.33), we need to introduce theFourier transform in the space of generalized functions £'. For / G £', the Fouriertransform J-[f] can be represented in the form

^[/](x ) = </(t),77(t)eix t>, (1.3.45)

where 77 (t) is any function in the space V of test functions that is equal to unityin the neighborhood of supp/. Moreover, .?"[/] (x) is an entire function of x, andfor any keNg, there exist numbers A^^.0 and m ^ 0 such that

| Dk ^ [ / ] ( x ) | ^ k ( l + |x|2)m. (1.3.46)

Now the convolution / * g of two generalized functions / G S' and g £ £' isdenned as follows:

< f*9,f > = (/(t),< g(u),y(t + u)>) (<p€S). (1.3.47)

Here the right-hand side makes sense because < g(u),ip(t + u) > G S. Such aconvolution / * g is a member of S' and the formula of the form (1.3.17) holds:

F[f*9]=F[f\F\9] ( / £ < * ' ; 9e£'). (1.3.48)

The Fourier transform (1.3.22) is an isomorphism from the test space V ontothe other one C A function ip(t) € T> has the property that <>(t) is equal to zerooutside the domain

G = {t , i j G K " : \tj\âh aj > 0 (j = l,---n)} , (1.3.49)

dj > 0 (j = 1, -n) being positive numbers, if, and only if, its Fourier transform^[^{z) is an entire function of z — x + iy satisfying the inequalities

\z*T[<p](z)\ Bkea-X (a = (oi, , an) e R

n) (1.3.50)

for every nonnegative integer k G NQ and for some Bu 0.

The above properties of functions were used by Gelfand and Shilov[276] to define the direct and inverse Fourier transforms in the space of generalizedfunctions V by

<r-D\f\, = <f,?[<p\> {<p€C) (1.3.51)

and<Tv1[f], = <f,T-1[ip\> (veC), (1.3.52)

respectively. Analogous to (1.3.51)-(1.3.52), the Fourier transforms are defined inthe space V as follows:

(<peV) (1.3.53)

and< ^ 1 [ / ] , V > = < / , ^ -1 M > (<peV). (1.3.54)


These Fourier transforms have the properties (1.3.37)-(1.3.41) in the spaces Vand CJ. The convolution / * g is defined by (1.3.47) for / G 23' and g G £', andthe formula (1.3.48) holds for such a convolution.

To conclude this section, we present formulas for the Fourier transforms (1.3.1)and (1.3.22) of some elementary and generalized functions. First we present theone-dimensional case. The Fourier transform (1.3.1) of power functions |i|A and|i|Asgn(i) are given for x G E (x ^ 0) by

r (-£)'" "= -2 sin ( ^ ) F(A + l ) ^ ! " * - 1 (A G C; A # 0; A - 1 - 2k; ke No), (1.3.55)

)] (fc 6 No), (1.3.56)

and

.F [|t|Asgn(t)] (a;) = 2icos (^ j T(A + l)|a;|-A-1sgn(a;) (A -2ife; k G N),

(1.3.57)

^ [ r 2 f csgn ( t ) ] ( ^= i | ^ ^ -a;2 f c- 1 [ ^2 f c ) - l og ( | a; | )] (k G N), (1.3.58)

VK-2) being the Euler psi function (1.5.17) [see Brychkov and Prudnikov ([108], for-mulas 10.1.11 and 10.1.12)]. The Fourier transform of generalized power functionstk (k G No) and of the polynomial Pm(t) of degree m G N are expressed via theDirac delta function (1.2.6) and its derivatives (1.2.7):

f[l]{x) = 2TTS(X), T [tk] (x) = 2n(-i)k8^{x) (k G N), (1.3.59)

and

( d \ m

-i—U(x) (x G E; Pm(x) = 5 > M am / 0; m G N),X ' 3=0

(1.3.60)with aj G C (j — 0,1, , m; am ^ 0) [see Brychkov and Prudnikov ([108],formulas 10.1.1 and 10.1.5) and Gel'fand and Shilov ([276], formula (4) in Table1)]. The following relations hold for the one-dimensional Fourier transform (1.3.1)of the Dirac delta function (1.2.6) and of its derivatives (1.2.7):

7[5(t)](x) = 1 {x G R), (1.3.61)

T[5(t - a)]{x) = e~iax (x, a G E), (1.3.62)

f[8(k\t)](x) = {-ix)k ( i £ l ; k G N), (1.3.63)

F [ 6 k ( t - a)](x) = e ~ i a x ( - i x ) k ( i . a e l ; k e N ) . (1.3.64)

Chapter 1 (§ 1.3) 17

The next formulas are extensions of the relations given in (1.3.55) and (1.3.59)-(1.3.62) to the multidimensional Fourier transform (1.3.22) in K" (n G N):

T [|t|A] (X) =

J 3

(A G C; A 0; A - 1 - 2k, k G No; x £ l " ; n G N; x ^ 0 := (0, , 0)),

?[l, 1, , l](x ) = 2TT5(X) (X e l " , n e N ), (1.3.66)

T [tk tkn] (x) = (2ir)n(-i)kl+-+k»S^(xx) 6{k»\xn) (1.3.67)

(x e l " , t i e N; kj e N, j = 1, , n),

(x e l ", neN),

P|TOj (x) being a polynomial of degree |m| = mi H \-mn (rrij G N; j = 1, , n)

mi mn

P{xu , xn) =j 1 =0

with constant coefficients a,j1...jm G C,

i , - - - , tn) ] (x ) = l : = ( l , - - - ,

(1.3.69)

(1.3.70)

- " ) / 2 J( n_2 ) / 2(a|x|); (1.3.71)

(x e l " , n e N\{1} ; a G M\{0} )

J(n-2)/2(z) being the Bessel function of the first kind (1.7.1); in particular, whenn = 3,

( X £ l 3 ; a e l ; a / O ). (1.3.72)

[see Brychkov and Prudnikov ([108], formulas 707-709, 711, 713, 714)].

We also note that the relations in (1.3.55) and (1.3.65) for A = k G N take thefollowing forms:

T [|i|2fc] (x) = 0 ( x e l ; k G N), (1.3.73)

F[\t\ 2k+1]{x)=2(-l) k+1{2k (1.3.74)


andT [|t|2* ] (x) = 0 (x G En; n, k G N), (1.3.75)

^[|t|2fc+1](x)=2"(-l)*+V" - k]

(1.3.76)(x G Rn; n G N; k G No),

respectively.

1.4 Laplace and Mellin Transforms

In this section we present definitions and some properties of one- and multi-dimensional Laplace and Mellin transforms. More detailed information may befound in the books by Ditkin and Prudnikov [195], Doetsch [198], [199],Sneddon[773] and Titchmarsh [819] (for the one-dimensional case), and Brychkov et al.[107] (for the multidimensional case).

The Laplace transform of a function <p(t) of a real variable t G M+ = (0, oo) isdefined by

O

{£<p)(s) = £[<p(t)](s) = <p(s) := e-st<f{t)dt (s G C). (1.4.1)Jo

If the integral (1.4.1) is convergent at the point so G C, then it converges absolutelyfor s G C such that 9t(s) > 9t(so). The infimum av of values s for which theLaplace integral (1.4.1) converges is called the abscissa of convergence. Thus theLaplace integral (1.4.1) converges for 9t(s) > av and diverges for 9K(s) < av.

The inverse Laplace transform is given for x G IR+ by the formula

v y /v / — YJv /JV / * — c\ / y\ J*-"" v i — v / v /* i i >^ iu i

The direct and inverse Laplace transforms are inverse to each other for "sufficientlygood" functions ip and g:

£-1£<p = tp and ££~1g = g. (1.4.3)

First we present some simple properties of the Laplace transform analogous tothose given in Section 1.3 by (1.3.7)-(1.3.9) and (1.3.11)-(1.3.12) for the Fouriertransform:

{£Th<p)(p)=e-*h(£<p)(p) {heR), (1-4-4)

(£UxV)(p) = U (?) (A G E+), (1.4.5)

= (r-aC)(p) = C(p + a) (a G C), (1.4.6)k (*eN) (1.4.7)

Chapter 1 (§ 1.4) 19

andDk(C<p)(s) = (-l)*£[*VW](a ) ( i e N ). (1.4.8)

These equations are valid for "sufficiently good" functions (p.

When (p £ Ck(R+), the Laplace transforms {C<p)(s) and C[Dk <p{t)](s) existand lim(_>._|_oo(i)y))(-?)(i) = 0 for j = 0,1, , k — 1, then the relation in (1.4.7) isreplaced by the more general one

fc-ik<p{t)](s) = skC[Dk<p{t)](s) = sk(&p)(s) - ^ V - ^ p V H O ) (& € N). (1.4.9)

The Laplace convolution operator of two functions /i(i ) and <£>(£), given on K+,is defined for x G IR+ by the integral

(x) := h(x - t)<f{t)dt, (1.4.10)Jo


h* (p — <f * h. (1.4.11)

The Convolution Theorem 1.2 applied to (1.4.10) yields the form

(C(h * V)(p) = {Ch){p){Cip){p), (1.4.12)

which holds for "sufficiently good" functions h and ip.

We denote by K!f....+ the region in M.n with positive coordinates:

»+...+ = { t = , * „ ) € » " : t j > 0 (j = l , - - - , n ) } . (1.4.13)

The n-dimensional Laplace transform of a function </?(t) of t € K"...+ is definedby

(Cip)(s)=jC[ip(t)}(s)=ip(s):= f e-s-^(t)dt (s e C"), (1.4.14)

while the inverse Laplace transform is given for x £ JR+...+ by the formula

1 7+iOO 7+J0O

( £ - 1 ) ( ) £ - 1 [ ( ) ] ( ) ex s5(s)ds (1.4.15)

where 7 = 9i{sj) > cr , (cr j e l ; j = 1, , n). The theory of these multidi-mensional Laplace transforms is found in the book by Brychkov et al. [107].

The integrals in (1.4.14) and (1.4.15) have some properties analogous to thosefor the one-dimensional Laplace transforms given by (1.4.1) and (1.4.2). In partic-ular, they satisfy the mutually invertible relations in (1.4.3) for "sufficiently good"functions ip and g, and for such functions the above properties (1.4.4)-(1.4.8) ofthe one-dimensional Laplace transforms are extended to the multidimensional caseas follows:

(h G R"), (1.4.16)


(CUxlp)(p) = (j) (A £ R+), (1.4.17)

) (a G R»); (1.4.18)

(*GN" ) (1.4.19)

andDk(£<^)(x) = (-l) 'k l£[tV(t)](p ) (k e N"), (1.4.20)

respectively.

Analogous to (1.4.10), the Laplace convolution operator of two functions h andip is defined by

h*(p = (h*tp)(x) = / h(x - t)(p(t)dt (xel!f...+), (1.4.21)Jo

where

f := f / and (/i(x - t) := h(Xl - tu , xn - tn)). (1.4.22)Jo Jo Jo

The operation in (1.4.21) has the commutative property (1.4.11), and for such aconvolution the Laplace convolution theorem (1.4.12) also holds.

The Mellin transform of a function ip(t) of a real variable t e l + = (0, oo) isdefined by

(M<p)(p) = M[<p(t)](p) = ¥>'(*) := f ts~\{t)dt (s € C) (1.4.23)

and the inverse Mellin transform is given for x G R+ by the formula

p+ioox-sg(s)ds (7 = m(s)). (1.4.24)

1 p+i:= — /

2m J~(-ic

These relations can be derived from (1.3.1) and (1.3.2) if we replace <p(t) by y(e*)and ix by s. Thus the conditions for the existence of the integrals in (1.4.23)and (1.4.24) can be derived from the corresponding conditions for the direct andinverse Fourier transforms.

The direct and inverse Mellin transforms are inverse to each other for "suffi-ciently good" functions ip and g:

M~lM<p = <p and MM~1g = g. (1.4.25)

Two simple properties of the Mellin transform are connected with the elemen-tary operators Ma and Na defined by

(Matp)(x) =xaV(x) ( i £ l ;oeC) (1.4.26)

and(Na<p)(x)=tp{xa) ( i £ l ; ael\{0}) , (1.4.27)

Chapter 1 (§ 1.4) 21

respectively. The following relations hold for "sufficiently good" functions <p:

(MMa<p){s)=M-a{s):=M{s + a) (a G C) (1.4.28)

and

(MNa<p)(s) = ^-M (-) (a € R\{0} ) (1.4.29)\a\ \a/

In particular,(MN-!ip)(s) = M{-s). (1.4.30)

We also indicate some other properties for the Mellin transform which are validfor "sufficiently good" functions p:

(MUX(p)(s) = \~sM(s) (A G M+) (1.4.31)

with the n,\-dilation operator (1.3.6);

M[Dmip{t)]{s) = (-l)m{s - 1) (s - m)(Mv){s - m) (1.4.32)

= r ( r ( t - J ) a ) ( A f y ) ( a ~ m) (L4-33)VIA

- m ) , (1.4.34)

M[Smv(t)}(s) = (sriMîs), (1.4.35)and

Dm{Mtp){s) = M[(logt)mv(t)}(s), (1.4.36)

with m eN and the -derivative (1.1.11).

When <p £ Cm(K+), the Mellin transforms (M[<p(t)](s-m) and M[Dmip{t)]{s)

exist, and hmtô+[ts~k~1V{m~k~1){ t)] a nd lim t_H.0O[i s- fc-V m~fc-1)( i)] a re finite

for k = 0,1, , m — 1, then the relations in (1.4.33) and (1.4.34) are replaced bythe following more general ones:

M[Dm<p{t)]{s) = ^ " ^

E r (rn*"/ )[»'~''~V( ro- fc-1)(»)]80 (1-4-37)

and

= (-1)" r ( 8 ) (%) (, - m)J. ( S lib I

with raeN, respectively.


The Mellin convolution operator of two functions h(t) and <p(t), given on B.+,is defined for x € M.+ by the integral

h * </> = (h * ( z) := | X h ( | ) p ( i ) y, (1.4.39)


h*tp = tp*h. (1.4.40)

The Convolution Theorem 1.2 with respect to (1.4.39) yields the form

{M{h * ¥>)) (a) = CM/i)(s)(.M<p)(s), (1.4.41)

which holds for "sufficiently good" functions h and </?.

The n-dimensional Mellin transform of a function (p(t) of t G B&+...+ is definedby

(Mtp){8)=M[p(t)]( 8) = <p*{s):= [ e—V(t)dt (s G C"), (1.4.42)

while the inverse Mellin transform is given for x € 1K+...+ by the formula

-i />7i+ioo /-7n+joo(M- 1

5)(x)=At- 1[ 5(p)](x) = — — / / x-sff (s)ds (1.4.43)

with 7j = ^(sj) (j = I,--- , n). The theory for these multidimensional Mellintransforms appears in the book by Brychkov et al. [107].

The integrals in (1.4.42) and (1.4.43) have some properties analogous to thoseof the one-dimensional Mellin transforms given by (1.4.23) and (1.4.24). In partic-ular, the mutually invertible relations in (1.4.25) are valid for "sufficiently good"functions <p and g, and for such functions the above properties (1.4.28)-(1.4.36) ofthe one-dimensional Mellin transforms are extended to the multidimensional caseas follows:

(MMa<p){s) = M-a{s) = M{s + a) (a e Cn); (1.4.44)

(MNa<p)(s) = (-) (a e i " , a / 0); (1.4.45)\a\ \ a/

s); (1.4.46)

(MIl x<f)(s)=\~WM(s) (XER+); (1.4.47)

M[D"V(t)](s ) = (-l)l ml( s - 1) (s - m ) ( % ) (s - m)

= r ( p (+

im ~ S ) ( X ^ ) ( s - m) (1.4.48)

= (-1)1"1' F ( S) AM<p)(s - m) (1.4.49)1 (s — m)

Chapter 1 (§ 1.4) 23

(m = (mi, , mn) G Nn := N x x N; |m| = mi H h mn);

M[8m<p(t)](s) = {-s)m{M<p)(s) (m € Nn) (1.4.50)

andT>m(M<p)(s) = M[(logt)mp{t)](s) (m€Nn) . (1.4.51)

Here the operators Ma and Na are defined by

(Ma¥>)(x) = x V ( x ) ( x e I " ; a £ C ") (1.4.52)

and(AraVj)(x) = ip (xa) (x e l " , a e l n ; a ^ 0), (1.4.53)

respectively, while

(1.4.54)

(1.4.55)

Analogous to (1.4.39), the Mellin convolution operator of two functions h andis defined by

h*<p=(h*<p)(x):= j ( i e R J . . .+ ) , (1.4.56)

where

(1.4.57)The commutative property (1.4.40) holds for such a convolution, as does the Mellinconvolution theorem (1.4.41).

To conclude this section, we present formulas for the Laplace transform (1.4.1)and the Mellin transform (1.4.23) of some elementary and generalized functions.First we present the former. The Laplace transform (1.4.1) of power function tx

is given by

[** ] (P) = -TT

£ [ r 1- * ] (p) = (-l) fe+1^[log(p) - ^(fc + 1)] (k e No; W(p) > 0), (1.4.60)

where ^(2) is the Euler psi function (1.5.17) [see Brychkov and Prudnikov ([108],formulas 739 and 742)]. The Laplace transform (1.4.1) of the Dirac delta function(1.2.6) and its derivatives (1-2.7) have the following forms:

£{6{t)]{s) = 1 ( s e C ), (1.4.61)


C [<5(fc)(i)j (s) =sk (s € C; k G N), (1.4.62)

£ \5{k)(t - a)] (s) = e~assk (a e C; s e C; A; € N); (1.4.63)

[see Brychkov and Prudnikov ([108], formulas 732 and 733)].

The Mellin transform of the exponential function e~xt and the power function(1 + t)~a is expressed in terms of the Euler Gamma function (1.5.1) by

M [e~Xi] (s) = ^ (9t(A) > 0; <R(8) > 0) (1.4.64)

A

and

M[(t+l)-']( 8) = T{8)^~8) ( 0 < <K(s) < 9l(a)), (1.4.65)respectively. In particular, when A = 0, (1.4.64) yields

M [e"*] (s) = T{s) (<H(a) > 0). (1.4.66)

1.5 The Gamma Function and Related SpecialFunctions

In this section we present the definitions and some properties of the Euler gammafunction and of some special functions connected with this function. More detailedinformation may be found in the book by Erdelyi et al. ([249], Vol. 1, Chapter I).

The Euler gamma function T(z) is defined by the so-called Euler integral ofthe second kind:

T(z) = [ t'-ie-'dt («(z) > 0), (1.5.1)

where tz~1 = e^"1)1 0 8^. This integral is convergent for all complex z £ C{Ol(z) > 0). It follows from (1.4.66) that the gamma function is the Mellin trans-form of the exponential function:

M[e-t](z)=T(z) (K(z)>0). (1.5.2)

For this function the reduction formula

r(z + 1) = zT(z) (9\{z) > 0) (1.5.3)

holds; it is obtained from (1.5.1) by integration by parts. Using this relation, theEuler gamma function is extended to the half-plane 9K(z) 0 by

r(f|n) >-n; n £ N; z$ Zo~ := {0, - 1 , - 2, . (1.5.4)

H e re (z)n i s t he Pochhammer symbol, de f ined for c o m p l ex z e C a nd n o n - n e g a t i vei n t e g er n € No by

(2)0 = 1 and (z)n = z{z + l)---{z + n-l) (n G N). (1.5.5)

Chapter 1 (§ 1.5) 25

Equations (1.5.3) and (1.5.5) yield

r(n + 1) = (l)n = n! (neNo) (1.5.6)

with (as usual) 0! = 1.It follows from (1.5.4) that the gamma function is analytic everywhere in the

complex plane C except at z = 0, — 1, —2, , where T(z) has simple poles and isrepresented by the asymptotic formula

r(z) =Z i K

O(z + k)] (1.5.7)

We also indicate some other properties of the gamma function such as thefunctional equation:

Y{z)Y{\ -z) =SlSlllt

(z i Zo; 0

the Legendre duplication formula:

(z£C);

and the more general Gauss-Legendre multiplication theorem:

omz —1 m~}' ( k\

{ ™> fc=o ^ m '

the Stirling asymptotic formula:

T(z) = / V - ^ e " * [l + O Q ) ] (| arg(z)| < TT;

and its corollary for |F(a; + iy)\ (x,y G ffi):

i) ; r - =V5F; (1.5.8)

(1.5.9)

(1.5.10)

oo); (1.5.11)

l + O (-\\ (x -) oo). (1.5.12)

In particular,

and

(n G N; n -> oo) (1.5.13)

+ O (y -» oo); (1.5.14)

the quotient expansion of two gamma functions at infinity:

+ O Q ) ] |(arg(z + a)| < TT; |2| ^ oo); (1.5.15)


and the equality:

{2n2"1)V'^' ( 2 « -1 ) ! ! : = 1 - 3 - - - ( 2 " - 1 ) ( «eN), (1.5.16)

which follows from (1.5.3) and the second relation in (1.5.8).

The Euler psi function is defined as the logarithmic derivative of the gamma-function:

£ = y (*eC). (1.5.17)

This function has the following property:

m — 1

(z G C; m G N), (1.5.18)

which, for m = 1, yields

tp(z + 1) = tp{z) + - (zeC). (1.5.19)

The beta function is defined by the Euler integral of the first kind:

B(z,w) = f tîl - tr^dt (m{z) > 0; m(w) > 0), (1.5.20)Jo

This function is connected with the gamma functions by the relation

B(z,w) = ———-— (z,w ^ ô")- (1.5.21)

The binomial coefficients are defined for a G C and n G No := {0,1, } bythe formula

n! n!(1.5.22)

In particular, when a — m (m G No), we have

m ) = „ m ! ,, (m,n e No; m > n) (1.5.23)« / n ! (m-n)! ~

and/ ™ \

= 0 (m,nG No; 0 ^ m < n) (1.5.24)

If a $ Z~ := } =: ZQ\{0}, (1.5.22) is represented via thegamma function by

a l — - . -~ ^ - . n G N o ) . (1.5.25)n 7 n!F(a-n + 1)

Chapter 1 (§ 1.6) 27

Such a relation can be extended from n G No to arbitrary complex (3 G C by

D ^ e Ci ° * z">- <L5'26)

The incomplete gamma functions -y(z, w) and r(z, w) are defined for 2,u; Gby

-y{z,w) = tz-xe-ldt (m(z) > 0) (1.5.27)7o

and

r(2,w) = / tz-xe-ldt, (1.5.28)

respectively. The following relation is evident:

I(z,oo) = r(z,0) = r(z)=j(z,w) + T(z,w) (9t(z) > 0). (1.5.29)

1.6 Hypergeometric Functions

In this section we present the definitions and some properties of Gauss, Kum-mer, and generalized hypergeometric functions. More detailed information maybe found in the book by Erdelyi et al. ([249] Vol. 1, Chapters II , IV, and VI) .

The Gauss hypergeometric function 2-F1 (a, b; c; z) is defined in the unit disk asthe sum of the hypergeometric series

2F1{atbiciz) = Z ^ , (1.6.1)

where \z\ < 1; a,b G C; c G C\Zjj", and (a)k is the Pochhammer symbol (1.5.5).The series in (1.6.1) is absolutely convergent for \z\ < 1 and for \z\ = 1, when9l(c — a — b) > 0, while it is conditionally convergent for \z\ = 1 (z 1) if— 1 < 9l(c — a — b) ^ 0. For other values of z, the Gauss hypergeometric functionis defined as an analytic continuation of the series (1.6.1). One such analyticcontinuation is given by the Euler integral representation:

2F1(a,b;c;z)= J^] f ^ ( 1 - ty-b~\l - zt)~adt (1.6.2)r(6)T(c - 6) Jo

(0 < 51(6) < 3t(c); | a r g ( l - z )| < TT).

If c ZQ~, then 2F\(a,b;c; z) has another integral representation in terms of theMellin-Barnes contour integral.

Here | arg(—z)\ < IT and the path of integration starts at the point 7 — ioo (7 G M)and terminates at the point 7+ioo, separating all the poles s = — k (k = 0,1,2 )


to the left and all the poles s = a + n (n € No) and s = b + m (m G No) to theright.

The asymptotic behavior of 2F\ (a, fc; c; z) at infinity is given by

2F1(a,b;c;z) = A(-zya Fl + O (|)1 + B(-z)-» f l+of t j ] (1.6.4)

(H -> oo; |axg(-z)| < n)

when a - 6 £ Z := {0, , , } , and by

2F1(a,fe;C;z) = C(-^)-alog(z) Fl + o Q ^ l (1.6.5)

(\z\ -> oo; |arg(-z)| < TT)

when a — b G Z, where A,B,C £ C are constants.

The Gauss hypergeometric function has the following simple properties:

2Fi(b,a;c;z) = 2Fi(a,b;c;z), (1.6.6)

{a, b; c; 0) = 2FX (0,6; c; z) = 1, (1.6.7)

1(a,6;6;2) = ( l - z ) - < \ (1.6.8)

the Euler transformation formula:

2F1(a,b;c;z) = (l-z)c-a-b 2F1(c-a,c-b;c;z), (1.6.10)

and the following differentiation relations:

a + n , b + n;c+;z) (n e N) (1.6.11)6 ; c ; z ) =(c)n

and

12f i ( °> i : c i 2) ] =(a)nza~l iFâ + n^b-, c;z) (n G N). (1.6.12)

We also note that, if c is not an integer (c ^ Z), then the Gauss hypergeometricfunction

ui(z) =2F1(a,b;c;z)

and the function

U 2 ( z) = z 1 ~ c 2 F 1 ( a - c + l,b - c + 1;2 - c; z)

are two linearly independent solutions of the hypergeometric differential equation

d ii dti

z{l-z)- + [c-(a + b+ \)z]— - abu(z) = 0. (1.6.13)

Chapter 1 (§ 1 . 6 ) 2 9

The confluent hypergeometric Kummer function is also defined by the series

0 0 / \ k

$(a;c;z) = .F^c; z) := £ g |p (1.6.14)

where z, a G C;, c G C\ZQ~; but, in contrast to the hypergeometric series in (1.6.1),this series is convergent for any z £ C. It has an integral representations similarto (1.6.2) and (1.6.3):

(a;c-,z) = ^(C) f ta~'{l -r(a)r(c - a) Jo

i i ? ?2 (1.6.16)

*(a;c-,z) = ^(C) f ta~'{l - tf-ê^dt (0 < K{a) < 9t(c)) (1.6.15)r(a)r(c a) J

and

( z ) r ( a ) d S ,7 o o T(c - s)

where | arg(—z)\ < w and the path of integration separates all the poles s = —k(k = 0,1,2 ) to the left and all the poles s = a + n (n G No) to the right.

The asymptotic behavior of $(a; c; z) at infinity has the form

when |z| —) oo and | arg(z)| < TT, and where e = 1 for 3(z) > 0, while e = —1 for3{z) < 0.

In particular,

^ y [ O Q ) ] (K(z) -)- oo) (1.6.18)

andr ( C ^ ^ [ Q ) ] -> -oo). (1.6.19)

The following differentiation relations hold for Kummer's confluent hypergeo-metric function:

) $(a;c;z) = j^$(a + n;c + n;z) (n G N), (1.6.20)(C)n

;c; )] = (- ir( l-c) n^-"- 1$(a;c-n; (n G N), (1.6.21)

a;c;^) ]=(- i r^Tê-^(a;c + n;Z) (n G N), (1.6.22)


and

( A \ n

j - z J [e-'zc-a+n-1*{a; c; z)] = (c - o)ne-^c-°-1*( a - n; c; 2) (n € N).

(1.6.23)

The Kummer hypergeometric function Ui(z) = $(a;c; 2:) is the solution to theconfluent hypergeometric equation

z ^ + (c-z)^-au(z)=0. (1.6.24)

The other solution to this equation is given by Tricomi 's confluent hypergeometricfunction ^(a; c; z) defined by

-I /-OO

V(a;c;z) = —— ta-1{l + t)c-°-1e-ztdt {m(z) > 0; 9t(a) > 0). (1.6.25)r ( a) ./o

Using the (^-derivative defined in (1.1.11), equations (1.6.13) and (1.6.24) canbe rewritten as

[8{5 + c-l)-z(S + a){8 + b)]u{z) = 0 {5 = z^-) (1.6.26)

and

[S{5 + c-l)-z{5 + a))]u(z) = 0 {6 = z-^-), (1.6.27)

respectively.

The Gauss hypergeometric series (1.6.1) and the Kummer hypergeometric se-ries (1.6.14) are extended to the generalized hypergeometric series defined by

(1.6.28)^ (Oi)k---(Oq)k

where ai,bj £ C, bj 0, — 1, — 2, (I = 1, , p; j = 1, , q). This seriesis absolutely convergent for all values of z £ C if p ^ q. When p = q + 1,the series in (1.3.28) is absolutely convergent for \z\ < 1 and for \z\ = 1 when

91 (Z)j=i bj - X)f=i ai) > 0> while it is conditionally convergent for \z\ = 1 (z 1)

( )

If 6j ZQ (i = 1, , 9), this function has an integral representation in termsof the Mellin-Barnes contour integral, generalizing those in (1.6.3) and (1.6.16),given by

pFq(ai, , ap; 61, , bq; z)

»). z)sr(s]ds ( 1 6 2 9 )

s){-z) r{s)ds- ( L 6-2 9)

Here | arg(—z)\ < TT and the path of integration separates all the poles s = —k(k = 0,1,2, ) to the left and all the poles s = a,j + n (n e No; j = 1, , p) tothe right.

Chapter 1 (§ 1.6) 31

If dj — a,[ ^ Z are not integers for all j ,Z = 1, , p, then the asymptoticbehavior of (1.6.28) at infinity , generalizing the one in (1.6.4), is given by

0Fq(au , a p ; b u - O - (1.6.30)

(|z| ->oo; |arg(—z)| < TT),

where A,- G C (j = 1, , p) are constants. When a,j — ai G Z for some ofj,l = 1,- , p, then, analogous to (1.6.5), the corresponding (-z)~a;» has to bemultiplied by log(^).

For pFq(ai, , ap; b\, , 69; 2) the following differentiation relation holds,generalizing those in (1.6.11) and (1.6.20):

— ) pFq(ai,- , bg\Z)

(ai)n1

{bq)n- pFq{ai+n,- + n;z) (n £ N). (1.6.31)

The function u{z) = pFg(ai, , ap; 61, , bq; z) is a solution to the generalizedhypergeometric equation

3=1 1=1

u(z) = (1 .6 .32)

W e n o te t h at ( 1 . 6 . 3 2) y i e l ds ( 1 .6 .26) a nd (1 .6 .27) i n t he c a s es w h e re p = 2,q=l,a\ = a, a,2 = b, bi = c and p = 1, q = 1, a\ = a, 61 — c, respectively.

To conclude this section, we present formulas for the Laplace transform (1.4.1)and the Mellin transform (1.4.23) of the functions considered here. First we con-sider the former. Applying the relation (1.4.59) to (1.6.1), (1.6.14), (1.6.25), and(1.6.28), direct evaluation leads to the following relations for the Laplace transformof the above hypergeometric functions:

X,a,b;c; -s

(1.6.33)

(a, b e C; c G C\Z^; Sft(s) > 0; SR(A) > 0),

where G2'3 [s | . .. ] denotes the Meijer G-function given by (1.12.51);

£[tx-1$(a;c;t)](s) = A, a; c; -s

(o G C; c G C\Z(7; <R(s) > 0; <R(A) > 0)

(1.6.34)

; l - i ) (1.6.35)


(9t(A) > 0; SK(c) < 1 + «(A); |1 - s\ < 1);

C[tx~1pFq[ai,- , ap;bi, , bq;t)](s) = —— p+iFq A , a i , - - - , ap;bi,- , bq; -s L S J

(1.6.36)

(Oj 6 C, (j = 1, , p); 6,- G C\Z~, (j = 1, , g); p q)

(*H(A) > 0; !K(s) > 0 when p < q; SK(s) > 1 when p = g).According to (1.4.23) and (1.4.24), from relations (1.6.3), (1.6.16), and (1.6.29)we derive the following formulas for the Mellin transform (1.4.23) of the Gauss,Kummer, and generalized hypergeometric functions (1.6.1), (1.6.14), and (1.6.28):

(a,b,c,seC, c £ ZQ ; 0 < 9t(s) < min[9t(o),9i{b)] ) ,

(a, 6, c, s G C; c Z^"; 0 < <H(s) < 9l(a)) ,

(1.6.39)

(ah bjGC; bj$Zo; (I = 1, , p; j = 1, , q); 0 < 9t(a) < min [fH(a;)]) .V 1=1=P /

1.7 Bessel Functions

In this section we present the definitions and some properties of classical Besselfunctions and some of their modifications. More detailed information about theBessel functions may be found in the book by Erdelyi et al. ([249], Vol. 2).

The Bessel function of the first kind Jv{z) is defined by

where z G C\(—oo, 0] and i / f C . This function can be represented in terms of thehypergeometric function

_ (1.7.2)fc=o K^lk a :

by

(1.7.3)

Chapter 1 (§ 1.7) 33

and so the series in (1.7.1) is convergent for all z € C. Thus Jv(z) is analytic in z.In particular, when v — —1/2 and v = 1/2, we have

/ 2 x i / 2 / 2 \ 1 / 2

J_i/2(2) = I — 1 cos(z) and J1/2{z) = — sin(z). (1.7.4)

The Bessel function J^( ) is given by the Poisson integral representation:

The following lemma yields the other integral representation of Jv{z) by theMellin-Barnes contour integral.

Lemma 1.5 For z £ C (| arg(z)| < there holds the relation

2-1-1/ p+iao T(u + s)Mz) = T /

where the path of integration separates all the poles s — —v — n (n 6 No) to theleft.

Proof. The usual technique for evaluating the Mellin-Barnes contour integral in(1.7.6) and thus prove Lemma 1.5 can be found (for example) in Erdelyi et al.([249], Vol. 1, Section 1.19).

The main terms of the asymptotic behavior of Jv (z) near zero and infinity aregiven by

Jv{z) = r vT I ) ( I ) " [ 1 + 0{z)] {z "*"0;

and

respectively.

The function u(z) = Jv{z) is a solution to the Bessel differential equation

z2 + z^ + (z2-^)u(z)=0. (1.7.9)az dz

The Bessel function of the first kind satisfies the differential recurrence relationswhich, in terms of the (^-operator, have the forms

5n [z"Ju(z)] = zv~nJv-n{z) and <5" \z~v Jv{z)\ = (-l)nz-v-nJv+n{z), (1.7.10)

where 8 = z- and n £ N.


The Bessel function of the second kind, or the Neumann function Yv{z), isdefined via the Bessel function of the first kind by

Yv(z) = . ] ACOS{U-K)JV{Z) - J-V{z)} {p G C\Z). (1.7.11)sin(j/7r)

The integral representation of Yv{z) in terms of the Mellin-Barnes contourintegral is given by the following lemma.

Lemma 1.6 For z G C (|arg(z)| < IT) the following relation holds:

Y (Z) = ^ - 1 f T { u + 8 ) T ( [ V 8 ] / 2 ) zVK) iV* y r ( [ u + l + s ] / 2 ) r ( [ V + l 8 ] / 2 ) T ( l + [ V 8 ] / 2 ) '

(1.7.12)where the path of integration separates all the poles s = v + 2m (m € No) to theright and all the poles s = — v — n (n G No) to the left.

Proof. We can easily prove Lemma 1.6 by using the method based on the directand inverse Mellin transforms (see Section 1.4).

The Neumann function Yu(z) has the following asymptotic behavior near zeroand infinity:

v Q E1 + 0W] (M->o) (1-7.13)

and1 /*?

The functions Jv{z) and Yv{z) are linearly independent solutions to the Besseldifferential equation (1.7.9). The Neumann function satisfies the relations similarto those in (1.7.10):

Sn[zvYv{z)] =zv-nY n(z) and 5n [z-"Yv(zj\ = (-l)nz-v-nYv+n(z), (1.7.15)

where 6 = z- and n G N.

The modified Bessel function Iv (z) is defined by

g l ^ oc,0]). (1.7.16)

Such a function is expressed via the Bessel function of the first kind as follows:

Iv(z) = e~vvi/2Jv (e^z) . (1.7.17)

Analogous to (1.7.5), Iv{z) has integral representations of the Poisson type:

h{z) = ^r^Ti/2) / ( 1 " i2y~1'2cosh^dt (*(" ) > ~\) ' (L7-18)

Chapter 1 (§ 1.7) 35

and by the contour integral

h ^ t ( C; | argz| < J) . (1.7.19)2)

7T + OO

/ e d t (v e C; | argz| < J

The main terms of the asymptotic behavior of Iv(z) near zero and infinity aregiven by

h{z) = i>TT) © " [ 1 + 0(* )]' (z 0; " G c v n (1-7'20)

and

(1.7.22)

The function u(z) = Iv{z) is a solution to the modified Bessel differentialequation

^ + « | - C + . (1.T.23)

The modified Bessel function satisfies the differential recurrence relations of theform (1.7.10):

Sn [z"Iv(z)] = z"-nIv-n(z) and Sn [z-"Iv(z)] = z-"-nIv+n(z), (1.7.24)

where S = z£ and n G N.The modified Bessel function of the third kind, or the Macdonald function

Kv{z), is defined via the modified Bessel function by

This function has the following integral representations:

Kv{z)=\J0°° i~v~x exp H d+0]dt' mz) > o)' (L7'26)

but, when 0 < %\.{z) < \, then

The following lemma presents the integral representation of Kv{z) by meansof the Mellin-Barnes contour integral.


Lemma 1.7 For z e C (| arg(z)| < it) the following relation holds:

- , , ( 1.7.2 8)

where the path of integration separates all the poles s = —v — n (n £ No) ands = v - 2m (m G No) io £fte left.

Proof. This lemma can be proved just as Lemma 1.6 by using the known Mellintransform of Kv(x), properties of the gamma function, and the inverse Mellintransform.

The asymptotic representations of Kl/(z) near zero and infinity are given by

^ ( ) [1 + O(z)] (z-> 0; *(./) >0) (1.7.29)

and

KAz) = (^ )1 /2 e~' [l + O Q )] (\z\ -» oo; | arg(Z)| < y ) , (1.7.30)

respectively.

The functions Iv{z) and Kv{z) are linearly independent solutions to the Besseldifferential equation (1.7.23). The Macdonald function Kv(z) satisfies the relationssimilar to those in (1.7.24):

Sn [zvKv(z)\ = {-l)nzv-nKv.n{z) (5 = zj~z;ne N) (1.7.31)

and

Sn [z-"Kv(z)] = {-l)nz-v-nKv+n(z) (S = z^;ne N). (1.7.32)

We also note that the functions Yn(z) and Kv(z) have the symmetric propertywith respect to indices:

Y-n{z) = Yn(z) (n G N) and K^(z) = Kv{z) {v £ R). (1.7.33)

Now we present formulas for the Laplace transform (1.4.1) and the Mellin trans-form (1.4.23) of the functions considered in this section. The Laplace transformsof Bessel functions (1.7.1), (1.7.11), (1.7.16), and (1.7.25) are given by

£[Mt)](s) = (s2 + I)-1'2 [s + (s2 + I ) 1/2]" " , (1.7.34)

when 9t(j/) > - 1 and *K(s) > 0.

Chapter 1 (§ 1.7) 37

when |9<t(i>)| < 1, v 0, and 9t(s) > 0.

£[I»(t)](s) = (s2 - 1)"1/2 [s + (s2 - 1)1 /2]" " (1.7.36)

when 9K{v) > - 1 , and SH(s) > 1, and

C[Kv(t)](s) = 7rcsc(^7r)(S2 - ly1'2 | [a + (s2 + 1)1/2] " - [a - (s2 - 1)1/2] ~" j

(1.7.37)when |<R(z/)| < 1, v 0, and SK(s) > - 1 .

The Mellin transforms of the Bessel functions are given by

when -<R(i/) < <R(s) < §.

when |9l(i/)| < «R(s) < |, and

= 2 -^ r ( ; ^ f = 2-when |9l(i/)| < <R(s).

We note that the formulas (1.7.38), (1.7.40), and (1.7.41) are well known; [see,for example, Prudnikov et al. ([689], Vol. 2, formulas 2.12.8.2, 2.13.2.2, and2.16.6.1)].

Next we consider a function Zvp{z) denned by

/ 7\Zv

p(z)= I t"-1 exp (-tp - - ) dt (z/GC; p G R), (1.7.42)Jo ^ t)

with an additional condition 3\(v) ^ 0 for p < 0. This function was introducedby Kratzel [451] for p ^ 1 and investigated (for any real p G R) by Kilbas et al.[401]. When p = 1 and z is replaced by z2/4, then in accordance with (1.7.26), itcoincides with the Macdonald function K_v{z) with the exactness of the multiplier2(z/2)":

T ) = 2 © / f - " ( z ) (sn(z) >0 ) (L7-43)The function Z"p{z) is analytic with respect to z G C (9t(z) > 0). It has

different asymptotic estimates at zero and infinity for p > 0 and p < 0. Thus wehave

Zvp[z) = O ^™»[°.*(")] j (p > 0; z -» 0), (1.7.44)


if the poles of the gamma functions T(s) and T([v + s]/p) do not coincide, whileif these poles coincide, the logarithmic multiplier log(,z) must be included in theasymptotic estimate (1.7.36);

'exp \-/3zp^p+1>\ (1.7.45)

where p > 0, \z\ —> oo, | arg(z)| < ^p+2 '*' — e, and

7 =p) 2p

(1.7.46)When p < 0 and |arg(z)| < [(p — l)?r/(2p)], then the asymptotic estimates ofZvAz) at zero and infinity are given by

; (p<0; 2^0) (1.7.47)

andZv

p(z) = O (z*W) ( p < 0; |* | ->oo), (1.7.48)

respectively. The results in (1.7.44)-(1.7.48) were established in Kilbas et al. [401].We only note that the relations in (1.7.44) and (1.7.45) for p > 1 and x > 0 weregiven by Kratzel [451].

The integral representation of Zvp(z) in terms of the Mellin-Barnes contour

integral is given by the following lemma.

Lemma 1.8 For z G C (| arg(z)| < TT) the following relation holds:

"ds' ( 1 7 4 9)

where the path of integration separates all the poles at sn = — n (n G No) and atsm = —mp — v (m £ No), to the left when p > 0, while for p < 0 it separates allthe poles at sn = —n (n £ No) to the left and at sm = —mp — v (m G No) to theright.

Proof. This lemma is proved just as Lemma 1.6 by using the following knownMellm transform M of Zv

p(x):

;) (s) = ^ r (S) r (*(* ) > min[0, -*(«/)]). (1.7.50)

Such a relation for p > 0 was proved by Kilbas and Shlapakov [403], and for thegeneral case when p G E (p ^ 0) by Kilbas et al. [401].

Finally, we consider a function Xy,l (z) defined by

Pi

(v+ 1-1/0) L

Chapter 1 (§ 1.8) 39

with /3 > 0, m{p) > i - 1, a G C, and 9t(z) > 0.This function was introduced by Glaeske et al. [283]. It is an analytic function

with respect to z £ C. When z = x > 0, the asymptotic behavior of Xi,J(x) atzero and infinity is given by

and

A^(z) ~ pv+i-We-XxWP)-*-1 (m( ) > l - I ; a; -^ oo). (1.7.53)

The following lemma gives the integral representation of \J]l(z) as a Mellin-Barnes contour integral.

Lemma 1.9 For z G C (| arg(z)| < TT) the following relation holds:

where the path of integration separates all the poles at s = — n (n € No) and ats = -m/3 + vfi + a (m E No) to the left.

Proof. This lemma is proved just as Lemma 1.6 by using the following knownMellin transform M. of \\y^{x):

which was established in Glaeske et al. [283].

To conclude this section, we indicate that the function \\^l (z) is a generaliza-tion of the function A (z) introduced by Kratzel [447] and defined by

(n G N; <R(7) > - - 1; 9t(z) > 0).n

When n = 2, this function reduces to the Macdonald function K~7(z) as follows:

> 0; 1« (7)>_ I ) . (1.7.57)

Therefore, in accordance with (1.7.43) and (1.7.57), functions such as Z"p{x) and

Av,a{x) are called Bessel-type functions.


1.8 Classical Mittag-Leffler Functions

In this section we present the definitions and some properties of two classicalMittag-Leffler functions. More detailed information may be found in the books byErdelyi et al. ([249], Vol. 3, Section 18.1) and Dzhrbashyan ([205], Chapter IIIand in [208]).

The function Ea(z) defined by

x.—^ ZI; <K(a)>0), (1.8.1)

I I f ¥ KT - t - I I

was introduced by Mittag-Leffler [604] and is, therefore, known as the Mittag-Leffler function. The basic properties of this function were studied by Mittag-Leffler ([604], [605] and [606]) and by Wiman ([890]). We present some propertiesof this function. It is an entire function of z with order [9^(a)]-1 and type 1. Inparticular, when a = 1 and a = 2, we have

Ex(z) = ez and E2(z) = cosh(y/z). (1.8.2)

When a = n G N, the following differentiation formulas hold for the functionEn(Xzn):

( — ) En {Xzn) = XEn (Xzn) (n G N; A G C) (1.8.3)

and

A

z'0; n G N; A G C). (1.8.4)

When a = 1/n (n £ N\{1}) , the function Ei/n{z) has the following represen-tation:

E1/n(z) = ez 1+n

In particular, for n = 2, we have

E1/2(z) = e'2 [l + -^ e~t2dtj , (1.8.6)

which yields the asymptotic estimate

^ ) (1.8.7)

The asymptotic behavior of Ea (z) for any a is more complicated. It is basedon the integral representation of Ea(z) in the form

Chapter 1 (§ 1.8) 41

Here the path of integration C is a loop which starts and ends at — oo and encirclesthe circular disk |i| ^ Izl1/" in the positive sense: | arg(i)| ^ w on C. The integrandin (1.8.8) has a branch point at t = 0. The complex t-plane is cut along the negativereal axis, and in the cut plane the integrand is single-valued: the principal branchof ta is taken in the cut plane.

When a > 0, Ea(z) has different asymptotic behavior at infinity for 0 < a < 2and a ^ 2. If 0 < a < 2 and is a real number such that

TTOt

— < n < min[7r,7ra], (1.8.9)

then, for TV G N\{1} , the following asymptotic expansions are valid:

1 N / \

£>a{Z) — —Z'- e xp \Z' I — 2_ ^jr. T T ~ "T" U 1 jv+1 / (.1.8.1UJ

with |jsr| —\ oo, | arg(z)| ^ / ; and

fc=i

with |z| —> oo, // ^ | arg(z)| ^ TT.

When a ^ 2, then (for N G N\{1} ) the following asymptotic estimate holds:

/ znm \ , /„ |exp

—' 1 (1 — ak) zK

(1.8.12)with \z\ —> oo, | arg(z)| ^ sj-, and where the first sum is taken over all integers nsuch that

| arg(z) + 27rn| ^ ^ . (1.8.13)

When a > 0, the following lemma yields the other integral representation ofEa(z) as a Mellin-Barnes contour integral.

Lemma 1.10 For a > 0 and z G C (| arg(z)| < TT), the following relation holds:

-y-ioc r ( ! a 5)

where the path of integration separates all the poles at s = — k (k G No) io ifte Ze/£and all the poles at s = n + 1 (n £ No) to the right.

Proof. This lemma is proved just as Lemma 1.5.

According to (1.4.23) and (1.4.24), from (1.8.14) we derive the Mellin transformof the Mittag-Leffier function (1.8.1):

M[Ea{-t)](s) = rff11^1 '} (0 < 9t(a) < 1). (1.8.15)

J- I J- CxS}


The Laplace transform (1.4.1) of Ea(t) has a more complicated form. It is ex-pressed in terms of the Wright function (1.11.1) with s = 2 and q = 1 by

1C[Ea(t)](8) = - 2*1

s

(1,1), (1,1)> 0). (1.8.16)

The Mittag-Leffler function Ea p{z), generalizing the one in (1.8.1), is definedby

-r r (z,/3eC; <R(a)>0). (1.8.17)

This function, sometimes called a Mittag-Leffler type function, first appeared ina paper by Wiman [890], and its basic properties were investigated (almost fivedecades later) by Agarwal [5], Humbert [353], Humbert and Agarwal [354], andDhrbashyan[205]. When /? = 1, Ea p(z) coincides with the Mittag-Leffler function(1.8.1):

EaA{z) = Ea{z) (z&C; 9t(a) >0). (1.8.18)

We also recall two other particular cases of (1.8.17):

pz — 1£:12(^) = , and £2,2(2) = ^ - . (1.8.19)

z \Jz

Like the Mittag-Leffler function Ea(z), Ea}p(z) is an entire function of z withorder [fH(a)]"1 and type 1, and satisfies the following differentiation formulasgeneralizing those in (1.8.3) and (1.8.4):

(^-\ [zÊnf (Xzn)] = z?-n-lEnS-n (Xzn) (n G N; A G C) (1.8.20)

and

(1.8.21)It may be directly proved that the usual derivatives of Ea^(z) are expressed interms of the generalized Mittag-Leffler functions (1.9.1) by

f %\an{z) (nGN). (1.8.22)

In particular, we have

£) [Ea{z)]=n\Enaf+an{z) (n£N). (1.8.23)

When Q = 1/n (n £ N), the function Ei/n^{z) has a more general representa-tion than the one in (1.8.5):

[^-1)ne-*°E1/nS{z0) (1.8.24)

Chapter 1 (§ 1.8) 43

t0n-k-k-1

ir(p-[k/n}) /

for any z0 £ C\{0} .

The function Ea^{z) has the integral representation

(1.8.25)

(1.8.26)

which generalizes the one in (1.8.8) with the same path C.

The representation (1.8.26) can be applied to obtain the asymptotic behaviorof Ea>p{z) at infinity, which is different for the cases 0 < a < 2 and a ^ 2. When0 < a < 2 and for a real number fi satisfying (1.8.9), then (for N £ N)

fe — 1

with \z\ —¥ oo, ^ /i, and

JV

(1.8.28)

with \z\ -t oo and ix | arg(z)| ^ ir

If a ^ 2, then

i-t

S - ) , (1.8.29)N

k=l

where the first sum is taken over all integers n satisfying the condition (1.8.13).

In particular, when a — 2, we have

1= 2;

N

-E-k=i

- 2k) zK

1(\z\ -> o o; g TT), (1.8.30)

and, for z = x > 0, we have

cos

iV

-Efc=i

(-1)* 1,-iV+l

(x > 0 ; a: —> oo ) . (1.8.31)

The following result, generalizing Lemma 1.10, provides the integral represen-tation for Ea,$(z) (a > 0) in terms of a Mellin-Barnes contour integral.


Lemma 1.11 For a > 0 and z £ C (| arg(—z)\ < n) the following relation holds:

~-{-z)-sds, (1.8.32)

where the path of integration separates all the poles at s = —k (k £ No) to the leftand all the poles at s = n + 1 (n £ No) to the right.

From (1.8.32) we arrive at the Mellin transform of the Mittag-LefHer functionin the form

M[EaiP(-t)](s) = (1.8.33)

It can also be directly shown that the Laplace transform of Ea^(t) is given interms of the Wright function (1.11.1) by

C[EaJI{t)](s) = -s

(1,1), (1,1)0). (1.8.34)

When /? = 1, the relations (1.8.33) and (1.8.34) coincide with the ones in (1.8.15)and (1.8.16).

To conclude this section, we consider the Fourier transform (1.3.1) of theMittag-Leffler function Ea^{\t\) in (1.8.17) with a > 1. Making a term-by-termintegration of the series and using the formula (1.3.55), we have

= E D* r +fc ]

2TT6{X) I—2fc—2

k=0 T{2ak + a + /?)

2w5(x)Y(/3) kV

(1.8.35)

where S(x) is the Dirac delta function (1.2.6). Hence, in accordance with (1.11.10),the result is expressed in terms of the generalized Wright function 2*1(2) by

(2,2), (1,1)

(a + (3,2a)

1(Q > 1; p £ C).

(1.8.36)According to Theorem 1.5 in Section 1.11, the Wright function is defined for a > 1.In particular, when /3 = 1, in accordance with (1.8.18), we have

T[Ea(\t\)}(x) = 2TT8(X) -(2,2), (1,1)

(a + 1,2a)(a > 1). (1.8.37)

Chapter 1 (§ 1.9) 45

Since

E«A\t\) - t\Ea,a+p(\t\),

if a > 1, we find from (1.8.36) and (1.8.37) that

2x2

(2,2), (1,1)

(a +/3,2a)

and

respectively.

(2,2), (1,1)

(a + 1,2a)

(1.8.38)

(/? e C), (1.8.39)

(1.8.40)

1.9 Generalized Mittag-Leffler FunctionsIn this section we present definitions and properties of some functions which gen-eralize Mittag-Leffler functions (1.8.1) and (1.8.17).

First we consider the generalized Mittag-Leffler function defined for complexzeC,a,/3,peC, and 9t(a) > 0 by

(Ph 1

k=0T(ak k\ T(p)

(p,

w,1)

a)z (1.9.1)

where (p)k is the Pochhammer symbol (1.5.5). This function, introduced by Prab-hakar [687], is an entire function of z of order [D^(a)]""1. Its properties wereinvestigated by Prabhakar [687] and Kilbas et al. [400].

In particular, when p = 1, it coincides with the Mittag-Leffler function (1.8.17):

= EaS{z) (zeC). (1.9.2)

When a = 1, E Az) coincides with the Kummer confluent hypergeometric func-tion <&{p;P;z) (see (1.6.14)), apart from a constant factor [r(/?)]-1:

[z = (1.9.3)

When a = m € N is a positive integer, Epm (z) coincides with the generalized

hypergeometric function (1.6.28) with p = 1 and q = m, apart from a constantmultiplier factor:

[z = F ( &tm p; —,m m m m


For the generalized Mittag-Leffler function (1.9.1) the following differentiationformulas hold:

(n G N) (1.9.5)

and

T) \^~lK,e(â)\=ze-n-1Epap_n{\z

a) (AGC; n G N), (1.9.6)

When p = 1, (1.9.5) coincides with (1.8.22); while for p = 1 and for a = 1, p = a,P = c, (1.9.6) coincides with (1.8.20) and (1.6.21), respectively.

The following relation is valid for the -derivative (1.1.11):

When p = 1, (1.9.7) yields the corresponding formulas for the Mittag-Leffler func-tion (1.8.17):

71 / \ 0 — Tl — 1 Tp 1 -\ I~\ C\ Q\Ja,p\z) = Z Hiafi — n\Z)] ^l.y.oj

and, in particular,

z1-' Ea(z) = z-nEa^n(z); (1.9.9)

where 6 = 2 ^ and n G N.

For a > 0, the integral representation for .E o(z) can be expressed in terms ofa Mellin-Barnes contour integral. Thus, just as we proved Lemma 1.5, we arriveat the following result.

Lemma 1.12 For a > 0 and z G C (| arg(—z)\ < TT), the following relation holds:

where the path of integration separates all the poles at s = —k (k G No) to the leftand all the poles at s = n + p (n G No) to the right.

In accordance with (1.4.23) and (1.4.24), (1.9.10) yields the Mellin transformof the generalized Mittag-Leffler function (1.9.1) as follows:

'> (0 < <K(s) < W(p)). (1.9.11)

Chapter 1 (§ 1.9) 47

I t may be directly verified that the Laplace transform (1.4.1) of Epa At) is expressed

in terms of the Wright function (1.11.1) with p = 2 and q = 1 as follows:

CM), (U)

(a, 13)(1.9.12)

When p = 1, (1.9.11) and (1.9.12) coincide with (1.8.33) and (1.8.34), respectively.

The following formula also holds for the Laplace transform of the function (Xta):

n] (s) =where <H(s) > 0, <K(/3) > 0, A G C, and |As"a| < 1.

The next function generalizing the one in (1.9.1) is defined by

(1.9.13)

(1.9.14),(Pm,Otn

where z,p,/3j e C, 9t(a,-) > 0, j = 1, , n, and m e N.

Whenm = 1, (1.9.14) coincides with (1.9.1), that is, Ep ((a1; ft); z) =-E£i/3(z).

For this function the differentiation formula, generalizing the one in (1.9.6), isgiven by

(1.9.15)

The following lemma, generalizing Lemma 1.12, presents the integral represen-tation of Ep ((a, /3)n; z) with ctj > 0 (j = 1, , m) via the Mellin-Barnes contourintegral.

Lemma 1.13 If aj > 0 (j = I,--- , m), then, for z € C (|arg(—z)| < TT), thefollowing relation holds:

Ep((aj,Pj)1,m;z) =2mT{p) y7^ i

( L 9 . 1 6)

where the path of integration separates all the poles at s = —k (k £ No) to the leftand all the poles at s = n + p (n G No) to the right.


From (1.9.16) we derive the Mellin transform of the generalized Mittag-LefHerfunction (1.9.14):

Ai[ ((a it^-)i,m;-*)](* ) = J m " ? " ' 1 ^ (0 < m{s) < K ^) -1 \P) llj= i i \Pj ajs))

(1.9.17)The Laplace transform of Ep ((ctj, f3j) i ,m; i) is given in terms of the Wright function(1.11.1) with p = 2 and q = mby

s

1

s(<R(s) > 0). (1.9.18)

When m = 1, (1.9.18) yields (1.9.12).

The above functions (1.9.1) and (1.9.14), generalizing the classical Mittag-Leffler function Ea{z), extend such properties of Ea,p(z) as the differentiationformula of the form (1.8.20) and the integral representation (1.8.32) via Mellin-Barnes contour integrals. Now we consider the generalized Mittag-Leffler functionwhich extends the differentiation properties (1.8.20) and (1.8.21). Such a functionEa,m,i(z), introduced by Kilbas and Saigo [392], is defined by the following series:

cfcz* (1.9.19)k=0

with

^ : r + 1 ; : v (i-9-2o)Here an empty product is taken to be one, and a, / G C are complex numbers,and m £ l such that

9t(a) > 0, m > 0, and a{jm + I) <£ Z~ {j G No). (1.9.21)

Ea,m,i(z) is an entire function of z of order [^(a)]" 1 and type m [see Gorenfio etal.'[294], [295]].

In particular, if m = 1, the conditions in (1.9.21) take the form

m(a) > 0 and a(J + I) £ Z~ (j G No), (1.9.22)

and (1.9.19) is reduced to the Mittag-Leffler type function given in (1.8.17), apartfrom a constant factor T(al + 1):

Ea,iAz) = vial + l)Ea,ai+1(z). (1.9.23)

When a = n G N, (1.9.18) takes the form

oo I k — X n

(ss

Chapter 1 (§ 1.10) 49

(n G N; m > 0; n(qm + I) <£ Z~(q G No))

The last function has the following differentiation properties:

and

[ ( ' ) ( ^ ) ] ^ ( ^ ) (z * 0), (1.9.26)

with n G N and A G C. When m = 1 and nl + 1 = /?, then, in accordance with(1.9.23), the relations (1.9.25) and (1.9.26) coincide with (1.8.20) and (1.8.21),respectively.

The so-called multivariate Mittag-Leffler function E(ait...t an),b{zii > zn) ofn complex variables Zi, , zn G C with complex parameters ai, , an, b G C isdefined by

+ ln=k ( k \ FT" 'i1 1 j = l J

j(1.9.27)

in terms of the multinomial coefficients:

k v k\

'1) ) ' n /

(k, h,---,lne No). (1.9.28)

1.10 Functions of the Mittag-Leffler Type

In this section we present the definitions and properties of special functions definedin terms of the Mittag-Leffler function Ea,a{z) and the power function za~l andits matrix analog. These functions will play the main role in investigating theso-called subsequential differential equations of fractional order.

First we consider a function defined for z £ C\{0} and a, A G C in terms ofthe Mittag-Leffler function (1.8.1) by

Ea (Xza) {z 6 C\{0} ; A G C; <R(a) > 0). (1.10.1)

The following differentiation formulas hold for this function with respect to z:

J£) (Xza) (1.10.2)

and with respect to A:

) n +\1 (Xza), (1.10.3)


where Ea (Xza) is the generalized Mittag-Leffler function (1.9.1) and n G N.

Putting p = P — 1 in (1.9.13) and taking (1.9.2) and (1.8.18) into account, weobtain the Laplace transform of the function (1.10.1):

^ >0; AeC; |A«"a| < 1), (1.10.4)S /\

and differentiating (1.10.4) n times with respect to A leads to the relation

s)=(^9^ (n e N)- (Lio-5)Next we consider a function, more general than that in (1.10.1), defined by

z0-1Ea,p (Xza) (z € C\{0} ; a, /3, A G C; <R(a) > 0). (1.10.6)

The following relations, analogous to those in (1.10.2)-(1.10.5), are valid for thefunction in (1.10.6) (n € N):

Ea^(Xza)]=z^-n-1Ea^n(Xza), (1.10.7)

{\za)] = nh^-'E^+e (Xza), (1.10.8)

^ >0; A e C; |AS"Q| < 1), (1.10.9)S

and

i (jLyEaJl (At-)] (,) = ^ T T (|A*-" | < 1)- (1-10.10)

Now we consider the special case of the function (1.10.6) when /3 = a. Thisspecial function, which we denote by e^2, and it is called a-Exponential functions,is defined by

e** := za-lEa,a {\za) (1.10.11)

where z G C\{0} , d\{a) > 0, and A e C.According to (1.8.17), e^z is a series of the form

f *' >0). (1.10.12)T[(k + l)a]

Hence e^f is analytic with respect to z G C\{0} and

lim [s1- 0^ ] = - ^y (9l(a)>0). (1.10.13)

Chapter 1 (§ 1.10) 51


hm+ [(x - a )1 - ^ " " ) ] = - L (5R(a) > 0). (1.10.14)

When a = 1, then in accordance with (1.8.18) and (1.8.2), eXz coincides withthe exponential function eXz:

eXz = eXz (z .AeC), (1.10.15)

Therefore, the function in (1.10.11) can be considered as a generalization of theexponential function, and as such we can call e*z the a-exponential function. Thesemigroup property for the exponential functions is well known:

e\zefz = e(\+u,)z ( 2 ) A ) M € Q. (1.10.16)

But the a-exponential function does not possess such a property:

(a # 1). (1.10.17)

For example, if a = 2 and z = 1, then in accordance with the second relation in(1.8.19), we have

ex = v/Asinh(v/A), (1.10.18)

but[v /Asinh(VA)][VMsinh(v^)] ^ y/[\ + p) sinh(y(A + fj). (1.10.19)

From (1.8.27)-(1.8.29) we derive the asymptotic estimates for (1.10.11) at in-finity . If 0 < a < 2 and for a real number fi satisfying (1.8.9), then

where \z\ -* oo, N £ N\{1} , and | arg (A2:a) | £ /x; and

°az=- y^

where \z\ -> oo, TV € N\{1} , and n ^ | arg (Xza) \ ^ ?r;

If a ^ 2, then

k = l

(1.10.22)


where \z\ —> oo, N G N\{1} , | arg(Aza)| ^ gf-, and the first sum is taken over allintegers n satisfying the condition

| arg (\za) + 2im\ ^ ^-. (1.10.23)

Relations (1.10.7)-(l. 10.10) yield the corresponding formulas for the functionin (1.10.11) as follows:

{£) baZ]=Z]=za~n~1 Ea^n{\za) (nGN; A G C),

sa-and

(n G N).

(1.10.24)

(1.10.26)

(1.10.27)

Formulas (1.10.25) and (1.10.27) above allow us to generalize the function in(1.10.11) to the form

h (I)" wi (110-28)

(1.10.29)

(1.10.30)

where z G C - {0} , 9\(a) > 0, A G C, and n G No.

When n = 0, (1.10.28) coincides with function (1.10.11):

^\z \z

By (1.10.25), the following relation holds:

za) (n G N),

and hence, in accordance with (1.9.1), e^zn has the series representation

fc!

(n + 1,1)

in+l 1\

(1.10.31)Formulas (1.10.26) and (1.10.27) yield the Laplace transform of tane£n:

n\(sa -

(n G No; m(s) > 0; |As~a| < l ) . (1.10.32)

If x > 0 and A = b + ic (b, c G M), then it directly follows that the real andimaginary parts of e^xn are given by

p

bxe

j = 0

(1.10.33)

Chapter 1 (§ 1.11) 53

and

ea,n\ -!

bx" C (2? + 1)! a>

respectively.

We also need a matrix analog of the function e*z. Let Mn(R) (n G N) be a setof all matrices A = [a,jk] of order n x n with CLJU G M. (j = 1, , n). By analogywith (1.10.12) for z G C\{0} (9t(a) > 0), and A G Mn(ffi) , here we introduce amatrix a-exponential function defined by

,ak——v (1.10.35)

When a = 1, then ef z coincides with the matrix exponential function eAz:

exz = eAz e Q_ (1.10.36)

In general, the semigroup property of the form

eAzeBz = e(A+B)z (z £ c . A B g M n(R) ) i (1.10.37)

does not hold for the matrix exponential functions eAz and eBz. So neither doessuch a property hold for the matrix a-exponential functions eAz and e^z:

eAze*z 4A + B ) 2 (*, a G C; A, B e Mn(R)). (1.10.38)

Similarly, the inversion formula

(e-42)"1 =e~Az, (1.10.39)

valid for the matrix exponential function eAz, is not true for matrix a-exponentialfunction eAz:

( e ^ r1 # e-Az. (1.10.40)

We define the norm ||A|| of the matrix A with elements a,jk G K(i, & = 1, , n) by

||A|| = max|aife|. (1.10.41)

Then, from (1.10.35), we derive the estimate for the norm of eAz. For any fixedz € C, the following relation holds:

When z = x > 0 and a > 0, the above formula takes the more simple form

0 0 ctk

eAx\\ < X^


1.11 Wright Functions

In this section we present definitions and properties of the so-called Wright (or,more appropriately, the Fox-Wright) functions given in Erdelyi et al. ([249], Vol.1, Section 4.1), ([249], Vol. 3, Section 18.1), Kilbas et al. [402], and in Srivastavaand Karlsson [790].

The simplest Wright function <p(a, 0; z) is defined (for z,a,(3 £ C) by the series

4>(a,/3;z) =03, a)

i

k=0

If a > —1, the series in (1.11.1) is absolutely convergent for all z € C, whilefor a = — 1 this series is absolutely convergent for \z\ < 1 and for \z\ = 1 andm(/3) > - 1 [see Kilbas et al. ([402], Corollary 1.2)]. Moreover, for a > - 1 ,<j>(a, P; z) is an entire function of z. The function <p(a, /?; z) was introduced byWright [891], who in [892] and [896] investigated its asymptotic behavior at infinityprovided that a > — 1.

When a = 1 and /3 = v + 1, the function (f>{\,v + 1; 2/i) is expressed interms of the Bessel functions Jv{z) and Iv{z), given by (1.7.1) and (1.7.16), asfollows:

4> (l, v + 1; -Z-pj = 0 ) Jv{z), <t>(l,v+ 1; "-pj = 0 ) !„(*) . (1.11.2)

The integral representation of (1.11.1) in terms of the Mellin-Barnes contourintegral is given by

L [ f(s) -z)-sds, (1.11.3)4>{a,(3;z) = [ r , f2m Jc F(P - as)

where the path of integration C separates all the poles at s = -fc (fc G No) to theleft. If C = (7 - 200,7 + 200) (7 e R), then the representation (1.11.3) is valid ifeither of the following conditions holds:

0 < a < 1, I axg{-z)\ < ( 1~2Q ) 7 r, z jt 0 (1.11.4)

ora = 1, 9t(/J) > 1 + 27, arg(-z) = 0, z ^ 0. (1.11.5)

This result was established by Kilbas et al. ([402], Corollary 4.1). In their paper,other conditions for the representation (1.11.3) were also given for the case whenC = £-00 (£00) is a loop situated in a horizontal strip starting at the point—00 4- i<pi (00 + i<pi) and terminating at the point —00 + iy>2 (°° + i'-Pi) with— OO < (fii < (fi2 < OO.

In accordance with (1.4.23) and (1.4.24), the relation (1.11.3) means that theMellin transform of (1.11.1) is given by the formula

^ >0). (1.11.6)

Chapter 1 (§ 1.11) 55

The Laplace transform (1.4.1) of (1.11.1) is expressed in terms of the Mittag-LefHerfunction (1.8.17):

(a > - 1 ; P & C; 0); (1.11.7)

[see Erdelyi et al. ([249], Vol. 3, Section 18.2)].If z € C and | arg(z)| ^TT — e ( 0 < e < 7 r ), then the asymptotic behavior of

4>{a, (3; z) at infinity is given by

0 ((

(1.11.8)where a0 = [2?r(a + 1)]~1/2 (z -> oo).

The following differentiation formula for cf>(a,/3;z) is a direct consequence ofthe definition (1.11.1):

jd \— j 0(a,/3;z) = >(a, a + n/3; z) (n G N). (1.11.9)

When a = n, p = i/ + 1, and z is replaced by —z, the function (f>(a,P;z) isdenoted by J^{z):

and this function is known as the Bessel- Wright function, or the Wright generalizedBessel function [see Prudnikov et al. ([689], Vol.3), and Kiryakova([417], p. 352)].When (i = 1, the Bessel function of the first kind (1.7.1) is connected with (1.11.10)by

Mz) = (I) " 31 ( £ ) . (1.11.11)

Using this definition, one may derive from (1.11.3), (1.11.6)-(1.11.8), and (1.11.9)the representation of Jv{z) in terms of a Mellin-Barnes contour integral, its Mellinand Laplace transforms, and the differentiation formula. We only note that iffj, = p/q is a rational expression, the function u(z) = JZ(z) is the solution to thedifferential equation of order p + q:

u{z) = 0p

where 8 = z-^. This equation can be rewritten in the alternative form as follows:

P / j \ «n

dz,= 0

(1.11.12)is follows:

(1.11.13)


[see Kiryakova ([417], formula (E.37))].

The more general function p9q(z) is defined for z G C, complex ai, bj G C, andreal a/, @j G M. (I = 1, , p; j = 1, , q) by the series

(aj,af)i,j

(frz,/3z)i,q

where z, a;, 6j € C, a;, j3j G M, / = 1, , p, and j = 1, , q. This general Wright(or, more appropriately, Fox-Wright) function was investigated by Fox ([263] and[264]) and Wright ([893], [894] and [895]), who presented its asymptotic expansionfor large values of the argument z under the condition

Q P

\ a. _ \ ^ n,, ~» _i n i i ic \/ HJ / (-*l ^ —I * ^

If these conditions are satisfied, the series in (1.11.14) is convergent for any z G C.This result follows from the following assertion, which yields the conditions for theconvergence of the series (1.11.14) [see Kilbas et al. ([402], Theorem 1)].

Theorem 1.5 Let o;, bj G C and cti, (3j G M. (I = 1, -p; j = 1, , q) and let

A =

3 = 1

and

p-q

(1.11.17)

(1.11.18)

(a) If A > —1, then the series in (1.11.14) is absolutely convergent for allz G C.

(b) / / A = —1, then the series in (1.11.14) is absolutely convergent for \z\ < Sand for \z\ = 5 and 9%) > 1/2.

Proof. (1.11.14) is a power series of the form

We investigate the asymptotic behavior of \ck\ as k —> oo. In accordance with(1.5.12), we have

(at +ctik)\ ~ At I - \ A, = (27r

Chapter 1 (§ 1.11) 57

w i t h k —> oo, a nd I — 1,- , p; a nd

\r(bj+(3jk)\ ~5

with k —> oo and j = 1, , q, while, by (1.5.13), we have

k\ ~ (27T)1/2 (-) k1'2 (fc-»oo).

Using these estimates and taking (1.11.16)-(1.11.18) into account, we obtainan estimate for \ck\, as k —> oo, of the form

where

(1.11.19)

(1.11.20)

Now the results in (a) and (b) follow from the known convergence principles forpower series, which completes the proof of Theorem 1.5.

When ai,0j G R+, (I = 1, , p; j = 1, , q), the generalized Wright func-tion p q(z) has the following integral representation as a Mellin-Barnes contourintegral.

where the path of integration C separates all the poles at s = -k (k G No) to theleft and all the poles at s — (a; + ni)/ai (I — 1, , p; ni € N) to the right. IfC = (7 — 200,7 + *°°) (7 £ K), then the representation (1.11.21) is valid if eitherof the following conditions holds:

(1.11.22)

orA = 1, (A + 1)7 + 1 < JR(/i), arg(-z) =0, 0. (1.11.23)

This result was established by Kilbas et al. ([402], Theorem 4). In their paper,other conditions for the representation (1.11.21) were also given for the case whereC — £-oo (£00) is a loop situated in a horizontal strip starting at the point—00 + i<fi (00 + i<pi) and terminating at the point —00 + i(f2 (00 + ^ 2) with- 0 0 < cpi < ip2 < OO-

By (1.4.23) and (1.4.24), (1.11.21) yields the Mellin transform of the generalizedWright function (1.11.14):

M *q

(ai,(

-t- ais)

(1.11.24)


with ; I = 1, , p; j = 1, , q; 0 < 9t(a) < min^,

It can be directly verified by using (1.11.14) and (1.4.59) that the Laplacetransform of the generalized Wright function (1.11.14) is given by

P * 9

' (ai

\bj

,ati)

! Pj,

1,P

1,9

-t1S

P+l*9

(1,1), I> 0).

(1.11.25)To conclude this section, we note that when a/ = f3j = 1, with / = 1, , p;

j = 1, , q, and bj (fc ZQ , with j = 1, , q; then, in accordance with (1.5.4),the generalized Wright function p q(z) in (1.11.10) coincides with the generalizedhypergeometric function pFq(z) in (1.6.28) as follows:

,;z] (1.11.26)

In particular, the representation (1.11.21) yields the one in (1.6.29).

1.12 The ^-Function

In this section we present the definitions and properties of the so-called //-functionof Fox [264]. More detailed information may be found in the books by Mathai andSaxena ([560] Chapter 1), Srivastava et al. ([789], Chapter 1), Prudnikov et al.([689], Vol. 3, Section 8.2), and Kilbas and Saigo [398].

For integers m,n,p,q such that 0 ^ m ^ q and 0 ^ n ^ p, for ai,bj € C, andfor ai,(3j £ l + (/ — 1, , p; j = 1, , q), the //-function H™"(z) is definedvia a Mellin-Barnes contour integral of the form

Hp,q KZ) =

9 .

) , - - - , (ap,ap)

) , - - - , (bq,f3q)

nr=1

with

An empty product in (1.12.2), if it occurs, is taken to be one, and the poles

bji = -^P- U = 1, , "i ; ' e No) (1.12.3)Pi

Chapter 1 (§ 1.12) 59

and the poles

^ (r = , n; fcG No) (1.12.4)

do not coincide:

o). (1.12.5)

C in (1.12.1) is an infinite contour which separates all the poles at s = bji in(1.12.3) to the left and all the poles at s = aru in (1.12.4) to the right of C, and Chas one of the following forms:

(a) C = C-oo is a left loop situated in a horizontal strip starting at the point—oo + itpi and terminating at the point — oo + iipi with — oo < ipi < if2 < oo;

(b) C = £+oo is a left loop situated in a horizontal strip starting at the point+oo + iipi and terminating at the point +oo 4- i</52 with —oo < <fi < ip2 < oo;

(c)) C = (7 — ioo, 7 + zoo) (7 £ E) is a contour starting at the point 7 — zooand terminating at the point 7 + ioo.

The properties of the i7-function (1.12.1) depend on the numbers A, 5, and fi,w h i ch a re e x p r e s s ed v i a m,n,p,q, ai,ai (I = I,--- , p) a nd bj,/3j (j — 1, , q)by the relations (1.11.12)-(1.11.14), and on the number a*:

n P m qa* = Ea< - E a< + £ & - E ft- (L12-6)

1=1 l=n+l j=l j=m+l

The conditions for the existence of the ff-function are given by the following result[see Kilbas and Saigo ([398]; Theorem 1.1)].

Theorem 1.6 The H-function H™^n(z) defined by (1.12.1) makes sense in thefollowing cases:

C = C.OC, A > 0 , z / 0; (1.12.7)

£ = £^00, A = 0, 0 < \z\ < 5; (1.12.8)

C = £-oc, A = 0, |z| = <J, <R(/x) < - 1 ; (1.12.9)

£ = C+oo, A<0, zÔ; (1.12.10)

C = C+ao, A = 0, \z\>8; (1.12.11)

£ = £+ o o, A = 0, |z|=<J, fR( /u)<- l; (1.12.12)

£ = (7 - i oo, 7 +zoo), o *>0, | a r g ( z ) | < , ^ / 0; (1.12.13)

£ = (7-400,7+100), a* = 0, 7A+«R(/x) < - 1 , arg(z) = 0, z 7 0. (1.12.14)

Let the conditions in (1.12.5) be satisfied. If all poles at s = bji in (1.12.3) aresimple, i.e., if

p j ( b r + k) p r ( b r + I) ( r ^ j ; r , j = , m ; * , J G N 0 ) , (1.12.15)

#£«"(*) = E E fyzlW*, (1.12.16)j=l 1=0


and either A > 0, and z / 0 or A = 0 and 0 < \z\ < S, then the ff-function(1.12.1) has the following power series expansion:

where

h, (-i)' IKU,

(1.12.17)If all poles at s = arfc in (1.12.4) are simple, i.e., if

aj{l-a r + k)?ar(l-aJ+l) (r j ; r,j = 1 ,- , n; k,l e N o ) , (1 .12.18)

and either A < 0 and z 0 or A = 0 and \z\ > S, then the F-function (1.12.1)has a power series expansion of the form

where

Kk =

hrkz(ar~1~k)l<Xri (1.12.19)r = l ife=0

(a,- - [1 - ar + k}%) n?= m +i r (l - bj - [1 - a(1.12.20)

Remark 1.1 When the conditions in (1.12.5) are satisfied, but some of the polesat s = bji in (1.12.3) or at s = ark in (1.12.4) coincide, the power series expansions(1.12.16) and (1.12.19) are replaced by expansions containing two terms, one ofwhich has power series expansions of the form (1.12.17) and (1.12.19), while theother has power-logarithmic expansions [see Kilbas and Saigo ([398], Section 1.4)].

The relation (1.12.16) yields the power asymptotic expansion of the .ff-function(1.12.1) near zero. If the conditions in (1.12.5) and (1.12.15) are satisfied and eitherA ^ 0 or A < 0 and a* > 0, then the principal terms of the asymptotic estimateof Hgf(z) have the form

H™f{z) = ] T \h*zb>IP> + o ( z 6 ^ ) ] {z -> 0) (1.12.21)3 = 1

with the additional condition | arg(x)| < a*n/2 when A < 0 and a* > 0, where

^ ^ A (1-12.22)-A -^ —, Âr-r (ar -bft) UUm+1 r (I - fcr + bj I )

Chapter 1 (§ 1.12) 61

The main term of this asymptotic estimate is given by

( U 2 ' 2 3>

For the case where A < 0 and a* = 0, the .ff-function (1.12.1) has exponentialasymptotic behavior near zero. If the conditions in (1.12.5) and (1.12.15) aresatisfied, the main part of such an expansion has the form

m

tfpTM = E [h

+ A* 2-(,+l/2)/\A\ (c,

+olz-^-|-1 /^ / | i i | l (z-»0; |arg(z)| fê*). (1.12.24)

Here hj (1 j ^ m) are given by (1.12.22), e* is a constant such that

0 < e * <x min far,/3j], (1.12.25)

(1.12.26)-( E * , -$>

= (27ri)n+m-« exp9 n

E ^ " E ar7TI (1.12.27)

r = l j = l

In particular, when

m i n

(1.12.28)

(1.12.29)

(1.12.30)

the main term of the asymptotic estimate (1.12.24) has the form

Formula (1.12.19) yields the power asymptotic expansion of the iJ-function(1.12.1) near infinity. If the conditions in (1.12.5) and (1.12.18) are satisfied and


either A 0 or A > 0 and a* > 0, then the principal terms of the asymptoticestimate of H™^n(z) have the form

(1.12.32)

with the additional condition | arg(z)| < a*ir/2 when A > 0 and a* > 0, where

ar

The main term of this asymptotic estimate is given by(1.12.33)

-> oo). (1.12.34)

For the case where A > 0 and a* = 0, the if-function (1.12.1) has exponentialasymptotic behavior near infinity. If the conditions in (1.12.5) and (1.12.18) aresatisfied, the main part of such an expansion has the form

+ Az(p+l/2)/A j _+o

Here hr (1 r ^ n) are given by (1.12.33), e is a constant such that

7T

0 < e< -

p

kr=n+l

P

ir=n+l

(1.12.35)

(1.12.36)

(1.12.37)

(1.12.38)

(27r)(p"?+1)/2 A"" n " +i/2 n " 1/2 ( xr = l

(1.12.39)

(1.12.40)

Chapter 1 (§ 1.12) 63

In particular, when

m i n9t(ar) - 11 _ *(/* ) + 1/2

ar

(1.12.41)

the main term of the asymptotic estimate (1.12.35) has the form

\m(ar) - 1Q := mm

l<r<n aT

(\z\

(1.12.42)

Remark 1.2 When conditions in (1.12.5) are satisfied, but some of the poles ats = bji in (1.12.3) or at s = aru in (1.12.4) coincide, the logarithmic multipliers ofthe form [log(z)]n (n € N) must be included in the asymptotic estimates (1.12.21),(1.12.23), (1.12.24), (1.12.31) and (1.12.32), (1.12.34), (1.12.35), (1.12.42) [seeKilbas and Saigo ([398], Sections 1.8, 1.9 and 1.5, 1.6)].

Now we present some simple properties of the i7-function (1.12.1).

The HpnqJl(z) in (1.12.1) is symmetric in the set of pairs (en, a i ), , (an, an) ,

in ( an + i , an + i ) , , (ap,ap), in , (bm,Pm), and in (& m+i,/3m+i), ,(bq,Pq).

If one of (a,[,ai) (I = 1, , n) is equal to one of (bj,/3j) (I = m + 1, , q),or if one of (ai,ai) (I = n + 1, , p) is equal to one of (bj,(3j) (j = 1, , m),then the if-function reduces to a lower order //-function, that is, p, q, and n (orm) decrease by one. For example, the following two reduction formulas hold:

jm,np,q

Tjm,n — 1

- hp-l,q-l (1.12.43)

and

HP,q (1.12.44)

The following relations are valid:

z°H.p,q

(ai,ai)itP

(bj^jKq

(a; + aai,ai)itP{z € C) (1.12.45)

and

rj-m,n+lnp+l,q+l

1p+l,q+l (1.12.46)

9 .


where cG C; a G R+; k G Z.

The following translation formula holds:

H:p .

(1.12.47)

Next we present the differentiation formulas for the if-function (1.12.1):

= zu~kH.m,n+1 (w,ceC; O-GE+); (1.12.48)

- zUHp+k,q+k cz1- (1.12.49)

where ui, c, Cj G C; j = 1, , k; u G

dz

if^_L i ,

{cz

(cz + d)c(0,cr),(a;,a;)i>p

(1.12.50)

where (c, d G C; <r G

Many special functions are a particular case of the H-function (1.12.1). When

ai = Pj = 1 (' = l i ' - ' P ; i = 1, i ?)i ^ reduces to the Meijer G-function

HmnP,Q

zMi*

= <%? z, dp

Zm JC Yil=r(1.12.51)

Chapter 1 (§ 1.12) 65

Most of the functions presented in Sections 6-9 and 11 can be expressed interms of the ^-function (1.12.1). It follows from (1.6.3) and (1.6.16) that theGauss and Kummer hypergeometric functions are represented by

2Fi(a,b;c;z) =

—z

( l - o , l ) , 1-6,1

1 - a, 1 - b

0,1 - c(1.12.52)

and

T(c)—z —z

1 - 0

0 ,1 -c(1.12.53)

respectively.From (1.6.29) we have the following representation for the generalized hyper-

geometric function:

op;

0,(1 -. (1.12.54)

Formulas (1.7.6), (1.7.12), and (1.7.28) yield the following results for the Besselfunction of the first kind, the modified Bessel function, and the Macdonald func-tion:

Jv{z) =2-"y/^H\l

= 2 v Trie'

1,0

(1 - 1//2,1/2), ([1 + ^]/2,1/2)

and

Kv{z) =2-" 2,01,2

([1 + i/]/2,1/2)

{v, I), {-v 12,1/2) _

(1.12.55)

(1.12.56)

(1.12.57)

respectively.


Remark 1.3 There are other known representations for the Bessel function ofthe first kind, the modified Bessel function, and the Macdonald function given by

( 9{a E C) (1.12.58)

( 9\ a

- ) #l '°([a + v\l% 1), ([a - v]/2,1), ([a - v - l]/2,1) _

(oeC)

"(1.12.59)and

(a e C), (1.12.60)

[see Kilbas and Saigo ([398], (2.9.18)-(2.9.20))].

The following relations for the Bessel-type functions follow from (1.7.49) and(1.7.54):

1 Tj2fip-"0,2

T1,1

(0,1)

(P > 0),

(p < 0; 1R(I/) g 0)

(1.12.61)

and(1 - [a + l]//3,

(1.12.62)

From (1.8.14) and (1.8.32) we obtain the following representations for theMittag-Leffler functions (1.8.1) and (1.8.17):

—z

and

- TT1'1 —z

(0,1)

(0,1)

(0,1), (1-/3, a)

(1.12.63)

(1.12.64)

Chapter 1 (§ 1.13) 67

while (1.9.10) and (1.9.16) yield the corresponding formulas for the generalizedMittag-Leffler functions (1.9.1) and (1.9.14):

and

— Z

(0,1), ( 1 - / 3, a) _

( 1 - p . l)

(1.12.65)

(1.12.66)

Finally, from (1.11.3) we derive the representation for the Wright function<p{a,P;z) defined by (1.11.1):

(0,1), ( 1 - / 3, a)(1.12.67)

and (1.11.21) gives us the corresponding result for the generalized Wright function(1.11.14):

(1.12.68)(ai,ai)itP (1 - o;,a;)i,p

z — H —z(bj,Pj)l,q \ P'9 [ (0,l),(l-^,^)l,9

1.13 Fixed Point Theorems

In this section we present various existence and uniqueness theorems based onclassical theorems which assert the existence and uniqueness of fixed points ofcertain operators. We shall use known definitions and notions from functionalanalysis which can be found, for example, in the book by Kolmogorov and Fomin[434].

First we present Schauder's fixed point theorem which yields only the existenceof a fixed point without its uniqueness.

Theorem 1.7 Let (D,d) be a complete metric space, let U be a closed convexsubset of D, and let T : U —> U be the map such that the set Tu : u £ U isrelatively compact in D.

Then the operator T has at least one fixed point u* G U:

Tu* = u*. (1.13.1)

The necessary and sufficient conditions required for the set U to be relativelycompact in the space of continuous functions C[a, b] on a finite closed interval[a, b] of the real axis E are given by the Arzela-Ascoli theorem. To formulate this


theorem, we recall the definitions of equicontinuous and uniformly bounded sets. Aset G is called equicontinuous if, for every e > 0, there exists some S > 0 such that,for all g G G and all Xi,x2 G [a, b] with \x\ - x2\ < 5, we have \g(xi) - g(x2)\ < e.A set G is called uniformly bounded if there exists a constant M > 0 such thatHfflloo S M for every g £ G. The following Arzela-Ascoli theorem can now berecalled.

Theorem 1.8 Let G be a subset of C[a,b] equipped with the Chebyshev norm.Then G is relatively compact in C[a,b] if, and only if, G is equicontinuous anduniformly bounded.

Next we present the classical Banach fixed point theorem in a complete metricspace.

Theorem 1.9 Let {U,d) be a nonempty complete metric space, let 0 ^ to < 1,and let T : U -» U be the map such that, for every u, v G U, the relation

d{Tu,Tv)ûjd(u,v) (0 1 w < 1) (1.13.2)

holds. Then the operator T has a unique fixed point u* E U.

Furthermore, if Tk (k G N) is the sequence of operators defined by

T1 =T and Tk = TTk~l {k e N\{1}) , (1.13.3)

then, for any UQ € U, the sequence {TkUo}'£L1 converges to the above fixed pointu*.

We note that the map T : U —> U satisfying the condition (1.13.2) is called acontractive map or a contractive mapping.

We also indicate a generalization of the above Banach fixed point theoremgiven by Weissinger [see Diethelm ([173], Theorem C.5)].

Theorem 1.10 Let (U,d) be a nonempty complete metric space, and let Wu 0for any k G No be such that the series YlT=o wk converges. Further, let T : U —> Ube such a map that, for every k G N and for every u,v G U, the relation

d{Tku,Tkv)ûkd(u,v) (fceN) (1.13.4)

holds. Then the operator T has a unique fixed point u* G U. Furthermore, for anyUQ G U, the sequence {Tkuo}' =1 converges to this fixed point u*.

Chapter 2

FRACTIONALINTEGRALS ANDFRACTIONALDERIVATIVES

This chapter contains the definitions and some properties of fractional integralsand fractional derivatives of different types.

2.1 Riemann-Liouville Fractional Integrals andFractional Derivatives

In this section we give the definitions of the Riemann-Liouville fractional integralsand fractional derivatives on a finite interval of the real line and present some oftheir properties in spaces of summable and continuous functions. More detailedinformation may be found in the book by Samko et al. ([729], Section 2).

Let fi = [a, b] (-oo < a 0)are defined by

and

f{aj

respectively. Here F(a) is the Gamma function (1.5.1). These integrals are calledthe left-sided and the right-sided fractional integrals. When a = n G N, the

69


definitions (2.1.1) and (2.1.2) coincide with the nth integrals of the form

= r dh r

= T^TnT A * - O"" 1/^)* (n G N) (2.1.3)

and

(J6"_/)(z) = [ dh [ dt2--- [ f(tn)dtn

The Riemann-Liouville fractional derivatives D"+y and D"_y of order a G C(<H(a) ^ 0) are defined by

{Daa+y){x) := (£ {i:+ ay){x)

n = PKa)] + l ; « > a ) (2.1.5)

and

respectively, where [9t(a)] means the integral part of $H(Q). In particular, whena = n G No, then

and (D?_j/)(aO = ( - l ) y n )( * ) (n G N), (2.1.7)

where y^n\x) is the usual derivative of y(x) of order n. If 0 < 9i(a) < 1, then

1 d fx v(t)dt{D%+y)(x) = — -— / J_[m(a)] (0 < K(a) < 1; ^ > a), (2.1.8)

When a € M+, then (2.1.5) and (2.1.6) take the following forms:

ii *

Chapter 2 (§ 2.1) 71

and

while (2.1.8) and (2.1.9) are given by

and

respectively.If 9t(a) = 0 (a / 0), then (2.1.5) and (2.1.6) yield fractional derivatives of a

purely imaginary order:

i[0 *>a)- (2L14)

It can be directly verified that the Riemann-Liouville fractional integration andfractional differentiation operators (2.1.1), (2.1.5) and (2.1.2), (2.1.6) of the powerfunctions (a; — a)^~1 and (b — x)!3~1 yield power functions of the same form.

Property 2.1 // <R(a) ^ 0 and /? G C (3t(/?) > 0), fften

(j°+(t - af-i) (x) = j ^ ^ y ( a r - a ) ^ " 1 ( ^ H > 0), (2.1.16)

r ( ^ Q ) ^ 0) (2.1.17)

and

6a_(6 - tf-1) (x) = r ^ a ) ( b ~ ^)0+a~1 (Ha) > 0), (2.1.18)

(D%_(b - tf-1) (x) = r ( ^ ° Q ) (b - xf-"-1 (9t(a)^0). (2.1.19)

In particular, if 0 — 1 and *R(a) ^ 0, (ften i/ie Riemann-Liouville fractionalderivatives of a constant are, in general, not equal to zero:

On the other hand, for j = 1,2, , pH(a)] + 1,

(DZ+(t-a)a-i)(x)=0, {D?_(b-t)a-i)(x) = 0. (2.1.21)


From (2.1.21) we derive the following result.

Corollar y 2.1 Let 9t(a) > 0 and n = [<K(a)] + 1.

(a) The equality (D"+y)(x) = 0 is valid if, and only if,

where Cj £ M. (j = 1, , n) are arbitrary constants.In particular, when 0 < fH(a) ^ 1, the relation (D"+y)(x) = 0 fto/ds «/, and

only if, y(x) = c(x — a)a~1 with any c £ l .

(b) The equality (D"_y)(x) — 0 is valid if, and only if,

where dj € IK (j = 1, , n) are arbitrary constants.In particular, when 0 < SR(a) ^ 1, i/»e relation (D£_y)(x) = 0 fto/ds i/, and

only if, y(x) = d(b — re)""1 with any d G R.

We indicate some properties of the left- and right-sided fractional calculusoperators I"+, D%+ and 7^_, D "_ basically presented in Samko et al. ([729],Section 2). The first result yields the boundedness of the fractional integrationoperators I"+f and / "_/ from the space Lp(a, b) (1 p ^ oo) with the norm ||/||p,defined according to (1.1.1) and (1.1.2) with c = 1/p, by

b Y/P

\f(x)\pdx\ ( l g p < o o ), H / H p - e s s s u p ^ ^ l / l (p = oo).\\f\\P-=j ^ ^(2.1.22)

Lemma 2.1 (a)The fractional integration operators I°+ and /£_ with !R(a) > 0are bounded in Lp(a, b) (1 ^ p ^ oo);

(b) 7/0 < a < 1 and 1 < p < I/a, then the operators J"+ and /£_ are boundedfrom Lp(a,b) into Lq(a,b), where q = p / (l — otp).

Remark 2.1 Lemma 2.1(a) was proved in Samko et al. ([729] Theorem 2.6),while Lemma 2.1(b) is known as the Hardy-Littlewood theorem [see Samko et al.([729], Theorem 3.5)].

The next result characterizes the conditions for the existence of the fractionalderivatives D«+ and D%_ in the space ACn[a,b] defined in (1.1.7).

Chapter 2 (§ 2.1) 73

Lemma 2.2 Let 9t(a) ^ 0, and n = [fR(a)] + 1. 1/ y(z) G 4Cn[a,&], f ft era £ftefractional derivatives D"+y and Dg_y exist almost everywhere on [a, b] and can berepresented in the forms

r(n-a)7o (x-t)-(2 1 24)

{t _ x)a-n+i'1 25)

Proof. Relation (2.1.24) was established in Samko et al. ([729], Theorem 2.2) onthe basis of Definition (2.1.5) and Lemma 1.1. Formula (2.1.25) is proved similarlyby using Definition (2.1.6) and the representation for the function g(x) € ACn[a, b]of the form (1.1.8):

where

cf>(t) = and dfc = (2.1.27)

Corollar y 2.2 If 0 <R(a) < 1 (a ^ 0) and y(a;) € ,4C[a,&],

and

T ( l - a) ( 6 - a : ) -

Remark 2.2 Relations (2.1.28) and (2.1.29) were proved in Samko et al. ([729],Lemmas 2.2 and 2.3).

The semigroup property of the fractional integration operators I"+ and I^_ aregiven by the following result [see Samko et al. ([729], Sections 2.3 and 2.5)].

Lemma 2.3 // <K(a) > 0 and <ft(/?) > 0, then the equations

(7aVf+/)(* ) = (£+"/)(*) ' and (I?_l£_/)(i) = (2.1.30)

are satisfied at almost every point x G [a, b] for f(x) G Lp(a,b) (1 ^ p ^ oo). //a + /3 > 1, then the relations in (2.1.30) hold at any point of [a, b].

The following assertion shows that the fractional differentiation is an operationinverse to the fractional integration from the left.


Lemma 2.4 If y\{a) > 0 and f(x) £ Lp(a,b) (1 ^ p ^ oo), £/ien the followingequalities

{DZ+IZ+f)(x) = f(x) and (D?_I?_f){x) = f(x) (3t(a) > 0) (2.1.31)

hold almost everywhere on [a,b].

Remark 2.3 The first relation in (2.1.31) for f(x) € L(a,b) was established inSamko et al. ([729], Theorem 2.4). The second one can be proved similarly.

From Lemmas 2.2-2.4 we derive the following composition relations betweenfractional differentiation and fractional integration operators.

Property 2.2 // 9t(a) > 9t(/3) > 0, then, for f(x) G Lp(a,b) (1 ^ p ^ oo), therelations

(D%+IZ+f)(x) = i:+ 0f(x) and (fl£_J£_/)(s) = I^f(x) (2.1.32)

fto/d almost everywhere on [a,b].In particular, when /3 = k € N and SH(a) > fc, iften

(Z)fc/a"+/)(a:) = /aQ+fc/(^) and (DkI?_f)(x) = (-1)"l£kf(x). (2.1.33)

Property 2.3 Let 9l(a) ^ 0, m G N anrf D = d/dx.(a) If the fractional derivatives (D"+y)(x) and (D"+ my)(x) exist, then

{DmDaa+y){x) = (Da

a+my)(x). (2.1.34)

(b) If the fractional derivatives (D"_y)(x) and (D"^my)(x) exist, then

(DmDZ_y)(x) = (-l)m(D£my)(x). (2.1.35)

To present the next property, we use the spaces of functions I"+(LP) andIb-(Lp) denned for 9t(a) > 0 and 1 p ^ oo by

Ia+(LP) {f / = > V£l p(a,6)} (2.1.36)

andtf_(Lp):={/: / = /fe

Q_<A, </»GLp(a,b)}, (2.1.37)

respectively. The composition of the fractional integration operator I™+ with thefractional differentiation operator D"+ is given by the following result.

Lemma 2.5 Let <R{a) > 0, n = [9\{a)] + 1 and let fn-a{x) = (/"+af){x) be thefractional integral (2.1.1) of order n — a.

(a) If 1 p <; oo and /(a;) G IZ+(LP), then

{i: +Daa+f) (x) = f(x). (2.1.38)

Chapter 2 (§ 2.1) 75

(b) If f{x) G Li(a,b) and fn-a(x) G ACn[a,b], then the equality

{iZ+D«a+f){x) = fix) -

holds almost everywhere on [a, b].

In particular, if 0 < fH(a) < 1, then

[i: +Daa+f)ix) = fix) - f—^-ix - a)a~\ (2.1.40)

where fi-a{%) — (Ia+af)(x)> while for a = n G N, the following equality holds:

iI2+Dna+f)ix) = fix) - V ^P-ix - a)k. (2.1.41)

k=0

The following index rule Property is also easy to prove.

Propert y 2.4 Let a > 0 and /3 > 0 be such that n — 1 < a <n, m — 1 < j3 <min,m G N) and a + /3 < n, and let f G Li(a,b) and fm-a G ACmi[a,b]). Thenwe have the following index rule:

Proof. Since n > a + (3, then using (2.1.5) and the semigroup property (2.1.30),we have

(2.1.43)Since / G L\ia,b) and fm-a G ACmi[a,b}), then by Lemma 2.4 (with a replacedby /3) we have

(/ f +</)(t) = f(t) - £ ( a +r ( / l J + 1

(|* \* - «?->. (2-1.44)

According to (2.1.5), [il™+ pf)] (m~i] ix) = iD^~jf)ix), and thus substituting(2.1.44) into (2.1.44) using the relation (2.1.7), we arrive at (2.1.42).

Lemma 2.5 was proved in Samko et al. ([729], Theorem 2.4). The followingstatement, which can be proved just as Lemma 2.5, characterizes the composi-tion of the fractional integration operator /"_ with the fractional differentiationoperator D"_.

Lemma 2.6 Let «H(a) > 0 and n = [^(a)] + 1. Also let gn-aix) = il£~ag)ix)be the fractional integral (2.1.2) of order n — a.


(a) If 1 £ p g oo and 3(1) G I"-(LP)> then

(I?-D?_g) {x) = g(x). (2.1.45)

(b) If g(x) G Li(a, 6) and gn-a(x) G .AC™[a, 6], t/ien the formula

= g(x) - ] T ^ ^ ^ ^ (6 - *)<*-> (2.1.46)

holds almost everywhere on [a,b\.

In particular, if 0 < 9\(a) < 1, iften

(2.1.47)

where g\-a{x) = (I^Z"g){x), while for a = n G N, the following equality holds:

(IZ_DZ_g)(x) = g(x) - £ tlMM{b _ x)\ (2.1.48)fc=0

Now we present the rules for fractional integration by parts, which were provedin Samko et al. ([729], Corollary of Theorem 3.5 and Corollary 2 of Theorem 2.4).

Lemma 2.7 Let a > 0, p ^ 1, q ^ 1, and (1/p) + (1/g) ^ 1 + a (p ^ 1 and5 ^ 1 in £/ie case when (1/p) + (I/?) = 1 + a.)

(a) If ip(x) G Lp(a,b) and ip(x) G Lq(a,b), then

f <p{x) {iZ+rp) (x)dx = f ^(x) (/6a-

J a Ja(2-1.49)

(b) If f{x) G I1[LP) and g(x) G I^+{Lq), then

f f{x) {Daa+g) (x)dx = [ g(x) (IU) ix)dx- (2.1.50)

J a J a

Now we consider the properties of (2.1.1) and (2.1.2) and fractional deriva-tives (2.1.5) and (2.1.6) in the spaces Cy[a,b] and C™[a,b] defined in (1.1.21) and(1.1.22), respectively. The existence of the fractional integrals / " + / and / "_/in the space C7[a, 6] and the fractional derivatives D%+y and D%_y in the spaceC™[a,b] are given by the following lemma.

Lemma 2.8 Let 9t(a) ^ 0 and 7 G C.

(a) Let m(a) > 0 and 0 ^ 91(7) < 1-

If ^ (7) > 9i{a)> then the fractional integration operators I"+ and I"_ arebounded from C7[a,b] into C7_a[a, 6]:

Chapter 2 (§ 2.1) 77

_rpt(a)]|r(l-9t(7))|1 Inj

In particular I"+ and Ig_ are bounded in C7 [a, b].

If 9^(7) ^ 9^(a), then the fractional integration operators J"+ and /£_ arebounded from C-y[a,b] into C[a,b]:

\\IZ+f\\c fell/He,, and \\IU\\c fell/lc,, (2-1.52)

7n particular, J"+ and /"_ are bounded in C7 [a, 6].

(b) // <H(a) ^ 0, n = [9t(a)] + 1 and y(x) G C^[a,6], iften the fractionalderivatives D"+y and D"_y exist on (a,b\ and can be represented by (2.1.24) and(2.1.25), respectively. In particular, D"+y and D£_y are given by (2.1.28) and(2.1.29), respectively, when 0 ^ 9t(a) < 1 (a ^ 0) and y(x) G C7[a,b].

The following assertion for fractional calculus operators (2.1.1), and (2.1.2),and (2.1.5) and (2.1.6), is analogous to those in Lemmas 2.3-2.6 and Property 2.2.

Lemma 2.9 LetfR(a) > 0, «K(/3) > 0 andO ^ ^(7) < 1. The following assertionsare then true:

(a) If f(x) G (77[a, b], then the first and second relations in (2.1.30) hold at anypoint x G (0, b] and x G [a, b), respectively. When f(x) G C[a, b], these relationsare valid at any point x G [a,b\.

(b) If f(x) G C^[a,b], then the first and second equalities in (2.1.31) hold atany point x G (a,b] and x G [a, b), respectively. When f(x) G C[a,b], then theseequalities are valid at any point x G [a, b].

(c) Let 9t(a) > 9t(/3) > 0. // f(x) G C7[a,b], then the first and secondrelations in (2.1.32) hold at any point x G (a, 6] and x G [a, b), respectively. Whenf{x) G C[a, b], then these relations are valid at any point x G [a, 6]. In particular,when P = k G N andfH(a) > k, the relations in (2.1.33) are valid in their respectivecases.

(d) Let n = [9t(a)] + 1. Also let fn-a{x) = (Ia+af)ix) the fractional integral(2.1.1) and gn-a(x) = {IÎag)(x) the fractional integral (2.1.2) be, of order n — a.

If f(x) G C7[o,6] and fn-a(x) G C^[a,b], then the relation (2.1.39) holds atany point x G (0, b]. In particular, when 0 < 9l(a) < 1 and fi-a(x) G C^la, b], theequality (2.1.40) is valid.

If g{x) £ C7[a, b] and gn-a(x) G C"[a,b], then the equality (2.1.46) holds atany point x G [a, b). In particular, when 0 < 9l(a) < 1 and fi-a(x) £ Câ, b], theequality (2.1.47) is valid.

If f(x) G C[a,b] and fn-a(x) G Cn[a,b], then (2.1.39) and (2.1.46) hold atany point x G [a, b]. In particular, if f(x) G Cn[a,b], the relations (2.1.41) and(2.1.48) are valid at any point x G [a,b].


Remark 2.4 The assertions of Lemmas 2.8 and 2.9 for the left-sided fractionalcalculus operators / " + / and D"+y were given in Kilbas et al. [375]. The results ofLemmas 2.8 and 2.9 for the corresponding right-sided fractional calculus operators/£_/ and D%_y are proved similarly.

To conclude this section, we give the formula which shows that the Riemann-Liouvill e fractional integral (2.1.1) of the Mittag-Lefner function (1.8.17) withspecial parameters also yields a function of the same kind.

(i:+(t - af-'E [X(t - a)"]) (*) = (X- a)"^1 E^+p [X(x - a)"] (2.1.53)

with A G C, <K(a) > 0, 9t(/3) > 0, and 9%) ^ 0, [see, for example, ([729], Table9.1, formula 23)]. A similar relation for the Riemann-Liouville fractional derivative(2.1.5) directly follows:

(D«+(t - af~lE [X(t - a)"]) (x) = {x - af-Ê^-a [X(x - a)"} (2.1.54)

with A G C, 9t(a) > 0, «R(/3) > 0, and 9%) 0.

In particular, when j3 = n = a, we can apply the well-known limi t formula

Km - L = o (2.1.55)2^0 T(z)

to derive from (2.1.54) the following relation for the function e« defined in(1.10.12):

(l>o+ea(*~ o)) (x) = â{x~a) (9t(a) > 0; A G C). (2.1.56)

When p = 1 and fj, = a, then (2.1.54), in accordance with (1.8.17) and (1.8.18),yields the following formula for the Mittag-Leffler function (1.8.1):

(D%+Ea [\{t - a)a}) (x) = XEa [X{x - a)a] (<R(a) > 0; A e C). (2.1.57)

Finally, we give an analog of the well-known property from analysis

4- I K(x,t)dt= f -?-K{x,t)dt+ lim K(x,t), (2.1.58)dx Ja Ja ox t-Kc-o

which is valid provided that K(x, t) is continuous on [a, b] x [a, b] and that K(x, t)has a continuous partial derivative (d/dx)K(x, t) with respect to a; 6 [a, b] for anyfixed t G [a, b], for the Riemann-Liouville fractional derivative (D%+y)(x) of ordera (0 < a < 1) in the case when K(x,t) = k(x — t)f(t). For this we need theso-called partial Riemann-Liouville fractional derivative of order a (0 < a < 1)with respect to a; of a function y(x, t) of two variables (x, t) G [a, b] x [a, b] definedby

(x-u)

with 0 < a < 1; x > a; t G [a, 6], (see Section 2.9 in this regard). The followingnotation is also suitable for the Riemann-Liouville fractional derivative (D"+y)(x)

Chapter 2 (§ 2.2) 79

in (2.1.8) and for the Riemann-Liouville fractional integral (ll+ af)(x) defined by(2.1.1):

(DZ+y)(x) = D2+[y(t)](x) and (/^"/)(x) = /^[/(OK* ) (0 < a < 1).(2.1.60)

The following lemma can be proved fairly easily.

Lemma 2.10 Let o £ l and 0 < a < 1. Also let the function f(x) and k(x) bedefined on [a, b] such that

f{x) G C[a, b] and L{x) = f T~ak{x - r)dr e C1 [a, b]. (2.1.61)Jo

Then, for any x £ [a, b],

Daa+ \[ k(t- u)f(u)du] (x)

-fJ a

D%+[k(t-a)](u)f(x + a-u)du + f(x) lim ll+ a[k(t - a)](x). (2.1.62)

Remark 2.5 Lemma 2.10 is stated here for 0 < a < 1, provided that theconditions of (2.1.61) are satisfied. In fact, this lemma also holds for a £ C(0 < 9\(a) < 1) under conditions for functions f(x) and k(x) different from thosein (2.1.61). For example, that f(x) can be Lebesgue measurable on [a, b] and thatk(x) can have a measurable derivative k'(x) almost everywhere on [a, &].

Remark 2.6 For a = 0, the relation of the form (2.1.62) was formally proposedby Podlubny ([682], 2.213), together with its generalization of the form (2.1.58) inwhich the derivative d/dx is replaced by the Riemann-Liouville fractional deriva-tive D%+.

2.2 Liouvill e Fractional Integrals and FractionalDerivatives on the Half-Axis

In this section we present the definitions and some properties of the Liouvillefractional integrals and fractional derivatives on the half-axis R+. More detailedinformation may be found in the book by Samko et al. ([729], Section 5]).

The Riemann-Liouville fractional integrals (2.1.1) and (2.1.2) and fractionalderivative (2.1.5) and (2.1.6), defined on a finite interval [a, b] of the real line E, arenaturally extended to the half-axis M+. The fractional integration constructions,corresponding to the ones in (2.1.1) and (2.1.2), have the following forms:

and r


respectively, while the fractional differentiation constructions, corresponding tothose in (2.1.5) and (2.1.6), are defined by

and

/ d \ n 1 / d\n f{D-V){X) := \-li) {I-~ay){x)=TWâ)VTx) /

(2.2.4)with n = [91(a)] + 1; 91(a) ^ 0 ; x > 0.

The expressions for ig"+/ and /f / in (2.2.1) and (2.2.2), and D%+ y and D°_y in(2.2.3) and (2.2.4), are called the Liouville left- and right-sided fractional integralsand fractional derivatives on the half-axis R+. In particular, when a — n € No,then

\jyjry)\jJ) — yu _y jyd.) — y\^)i y^j^yjy^) — y \dj)i

{Dn_y){x) = (-l)ny^(x) (n € N), (2.2.5)

where y^n\x) is the usual derivative of y{x) of order n.

If 0 < <K(a) < 1 and x > 0, then

and

_a)dxJx

If <H(a) = 0 ( a^ 0), then the Liouvill e fractional derivatives (2.2.6) and (2.2.7)are of a purely imaginary order and have the following forms:

w^W)i[ O S F ('GEU0}; *> 0) (2"2-8)and

respectively.

The Liouvill e fractional calculus operators If?+ and DQ+ satisfy the relations(2.1.16) and (2.1.17) with a = 0, while the Liouvill e fractional calculus operators/ " and D°_ of the power function x$~x and the exponential function e~Xx yield apower function of the same form and the same exponential function, respectively,both apart from a constant multiplication factor.

Chapter 2 (§ 2 . 2 ) 8 1

Property 2.5 Let 9t(a) ^ 0.

(a) If <K(/?) > 0, iften

(b)

C/'

(c)

7//3GC,

* ^ ) (*)

: ^ " 1) (*)

J/W(A)>

1 res + a)a

T(8 a)'

then

T(l-a-/?)r(i-/3)

T(l + a-/3) ,

0, £/&en

(ê-At )(a;

;'3+Q-1 (9t(a):

r^-"" 1 (9t(a)

'3+°-1 («(a)5

>0;

^ 0 ;

>0;

)=AQe-A x (fR(a)

9l(/3)

9t(a4

>0),

>0).

>0),

>0).

(2.2

(2.2

(2.2

)<1)

(2.2

(2.2

(2.2

.10)

.11)

.12)

.13)

.14)

.15)

When 0 < a < 1 and 1 p < I /a, £/ie integrals I$+f and I f / are defined fora function f(x) G Lp(

Proof. Formulas (2.2.10) and (2.2.11) follow from (2.1.16) and (2.1.17) for /3 = 0.(2.2.12) is known [Samko et al. ([729], Table 9.3.1)]. (2.2.13) is proved by using thedefinitions (2.2.4) and (2.2.12) with a being replaced by n-a, where n = [*K(a)]+l .Using these formulas and differentiating the obtained relation n times, we have

dx t

T(l-n + a- (3)r(l3 + n-a) 0_a__1

~^~ ' r ( l - /?) r(^ - a) X

Also, by (1.5.8), we have

sin[(/? — a + n)ir] sin[(/8 — a)?i

and

1 F(l + a — (3) F(l + a — /?) sin[(,5 — a)?r]1 (p — a)l (1 + a — p) 7T

(2.2.17)

(2.2.18)

Substituting these relations in (2.2.16), we obtain (2.2.13). Properties (2.2.14) and(2.2.15) are well known [Samko et al. ([729],(5.20))].


Lemma 2.11 (Hardy-Littlewood Theorem). Let 1 fi p ^ oo, 1 q ^ oo,and a > 0. Then the operators IQ+ and I" are bounded from LP(M.+ ) into Lq(R

+)if, and only if,

1 v0 < a < 1, Kp< - andq= —^ (2.2.19)

a 1 — ap

The weighted analog of the assertion in the space XP(E+), defined in (1.1.3)and (1.1.4), is given by the following result.

Lemma 2.12 [Samko et al. ([729], Theorems 5.3 and 5.4)]. Let 1 g p < oo,/ i £ l and

0<a<m+-, 0 < m < a, q=- ; —, u = ( - - m) q, (2.2.20)p ~ ~ \-{a-m)p \p )

with m T 0 for p = 1.

(a) If n 0 does not depend on f.

bounded from X? +1y(b) If fi > ap — 1, then the operator I" is bounded from X? +1y (M.+ ) into

\ l/g / /.oo \ l/p

"\{lZf)(x)\"dxj ^*2^y ^ l / ^ l ^ j , (2.2.22)

where the constant k2 > 0 does noi depend on f.

Lemma 2.13 Lei 1 p < 00.(a) If 1 0, i/ien i/ie operator IQ+ is bounded from LP(M+)

into XP1/p_a(R+):

( + ) ) j^fcU ) (2-2.23)

(b) If 1 p < I / a and 0 < a < 1, tften i/ie operator I" is bounded from

^\ {II f) (x)\*dx) l " Z r(r(f /~)a) ( / " I/(^)IP^) ' ? (2-2.24)

The following assertion, similar to Lemma 2.3, also holds [see Samko et al.([729], Section 5.1)].

Chapter 2 (§ 2.2) 83

Property 2.6 Let a > 0, /3 > 0, p ^ 1 and a +/3 < l/p. If f(x) <E Lp(R+), thenthe semigroup properties

(#+!&./)(* ) = {l£Pf){x), and (Illif){x) = [I a_+f3f)(x) (2.2.25)

hold. The relations in (2.2.25) are also valid for "sufficiently good" functions f{x).

The Liouvill e fractional derivatives (DQ+V)(X) and (D°!_y)(x) exist for "suffi-ciently good" functions y(x); for example, for functions y(x) in the space CQ°(1R+)

of all infinitely differentiable functions on M.+ with a compact support. Therefore,the following properties, similar to those in Lemma 2.4 and Properties 2.2 and 2.3,are valid.

Property 2.7 If a > 0, then the relations

(D%+IZ+f)(x) = f{x), and (Da_Ilf)(x) = f(x) (2.2.26)

are true for "sufficiently good" functions f{x). In particular, these formulas holdforf{x)£L1{R+).

Property 2.8 If a > f3 > 0, then the formulas

(D0o+I£+f)(x) = (JoVVX*) and {DtHf)[x) = (IZ~0f){x) (2.2.27)

hold for "sufficiently good" functions f(x) such as (for example) f(x) £ Li(E+).In particular, when ft = k G N and !H(a) > k, then

(DkI£+f)(x) = (#+*/)(* ) and [DkIa_f){x) = (-l)k(I^kf)(x). (2.2.28)

Property 2.9 Let a > 0, m G N and D = d/dx.

(a) If the fractional derivatives (D([+y)(x) and (D^my)(x) exist, then

(DmD%+y)(x) = (D%+mf)(x). (2.2.29)

(b) If the fractional derivatives (D^_y)(x) and (D°i+my){x) exist, then

(DmD°Ly)(x) = {-l)m{D°L+mf)(x). (2.2.30)

The formulas for fractional integration by parts, analogous to those in Lemma2.7, for the Liouville fractional integrals (2.2.1) and (2.2.2) and fractional deriva-tives (2.2.3) and (2.2.4) are given by the following result.

Property 2.10 If a > 0, then the relations/»OO /»OO

/ V(x) (/Oa+V) (x)dx = / V(* ) {1-f) (*)dx (2.2.31)

Jo Jo

and

f{x) (D%+9) (x)dx = / g(x) (Dtf) (x)dx (2.2.32)

are valid for "sufficiently good" functions if, ip and f, g.

In particular, (2.2.31) holds for functions ip(x) G ip(IR+) and i>(x) G Lq(R

+),while (2.2.32) for f(x) G J (LP(M+)) and g(x) G I$+(Lg(R

+)), provided thatp>l,q>l, (l/p) + (I/?) = 1 + a.


Remark 2.7 Relation (2.2.31) was given in Samko et al. ([729], (5.16'))- Theexpression (2.2.32) follows from (2.2.31) by taking Ig+ip = g and F<p = f.

The next assertion yields the Laplace transform (1.4.1) of the Liouvill e frac-tional integrals and fractional derivatives Iff+f and D%+y in (2.2.1) and (2.2.3)[see Samko et al. ([729], Theorems 7.2 and 7.3)].

Lemma 2.14 Let fR(a) > 0 and f(x) G £i(0,6) for any b > 0. Also let theestimate

\f(x)\ Aepox (x>b>0) (2.2.33)

hold, for constants A > 0 and po > 0.

(a) If f(x) G Li(0,6) for any b > 0, then the relation

{CIZ+f){s)=s-a{Cf){p) (2.2.34)

is valid for £H(s) > po-

(b) If n = pK(a)] + 1, y{x) G ACn[0,b] for any b > 0, and the followingestimate of the form (2.2.33)

\y(x)\ Be90X {x>b>0) (2.2.35)

holds for constants B > 0 and q0 > 0, and if y (0) = 0 (k = 0,1, , n — 1),then the relation

(£DZ+y)(s)=sa(£y)(s). (2.2.36)

is valid for 9l(s) > qo-

Remark 2.8 If 9\{a) > 0, n = [Di(a)} + 1, y{x) G 4Cn[0,&] for any b > 0, thecondition in (2.2.35) is satisfied and there exist the finite limits

lim [Dkl£-ay(x)] and lim [DkI^~ay{x)] = 0 {D = d/dz; A; = 0,1, , n - 1),

then from (2.2.6) and (1.4.9) we derive a relation, more general than that in(2.2.36), of the form

{CDg+y)(8) = sa(Cy)(s) - £ 8n-"-1Dk(Iây)(0+) (5H(a) > «o). (2-2.37)k=0

In particular, when 0 < 9t(a) < 1 and y(a;) G 4C[0, 6] for any b > 0, then

% (2.2.38)

The Mellin transform (1.4.23) of the the Liouvill e fractional integrals IQ+

and I" 0, s € C and f(x) G X s1

+a(R+).

(a) Ifm(s) < l -SR(a), then

(b) IfSR(s) >0, tften

I^L + a) (<*(«) > 0). (2.2.40)(s + a)

Lemma 2.16 Let «H(a) > 0, n = [9t(a)] + 1, s G C and j/(z) 6

(a) If £H(s) < 1 + ?t(a) and the conditions

li m [a ;s- f c - 1(7^Qy)(a; )] = 0 (k = 0,1,--- , n - 1) (2.2.41)

andli m [ a : 8 - * - 1 ^ " ? / ) ^ ) ] = 0 (fe = 0 , l , - -- , n - 1) (2.2.42)

fto/d, tften

- a) («R(s - a) < 1). (2.2.43)S)

(b) //£H(s) > 0 and t/ie conditions:

l i m [ a ; s - * ; - 1 ( / ^ - Q y ) ( a ; )l = 0 (fe = 0 , l , - - - , n - 1 ) ( 2 . 2 . 4 4)x—S-0+

anrflim \x"-k-l(ri- ay)(xj\ = 0 (jfe = 0,1, , n - 1) (2.2.45)

Z-K3O L J

are ua/«d, then

(MD1y)(s) = r^Q)(My)(8 - a) (9t(s) > 0). (2.2.46)

Remark 2.9 If 9l(a) > 0, n = [9t(a)] + 1, and the conditions in (2.2.41), (2.2.42)and (2.2.44), (2.2.45) are not satisfied, then from (2.2.3), (1.4.37), (2.2.39) and(2.2.4), (1.4.38), (2.2.40) we derive the following formulas, more general than thosein (2.2.43) and (2.2.46):

and

(MD%+y)(s) = My)(s - a)


(MDa_y)(s) = r(s{S]a)(My)(s - a)

n—l

k=0 v '

respectively. In particular, when 0 < £H(a) < 1, then

(MDZ+y)(s) = T{1r + " s)

S) (My)(s - a) + [x'-Î^y^x)] (2.2.49)

and

(MDa_y)(s) = F ( S) (My)(s - a) + [ a ;8- ^ 1^ " " , / ) ^ ) ] ^ . (2.2.50)i (s a)

Formulas (2.1.53) and (2.1.54), involving the Mittag-Leffler function (1.8.17), arealso valid for the Liouvill e fractional integrals (2.2.1) and fractional derivatives(2.2.3):

{iZ+tP-iE^p (At")) (x) = xa+P-1Ell,a+fl (Az") (2.2.51)

and(DS+tP-1Ell,f) (At")) {x) = xP-Êâ (As") (2.2.52)

where A € C; 9\(a) > 0, <R(/3) > 0, and SR(/i) > 0.

When /j, — a, the formula (2.2.52) can be rewritten as follows:

(Dg+tP-iEcf) [\ta]) (x) = ^ a_ , + XxÊ (Xxa) (A€C). (2.2.53)

A relation similar to (2.2.53) is valid for the left-sided Liouvill e fractional derivative(2.2.4) with «R(a) > 0 and !K(/?) > [5H(a)] + 1:

atp [\r a]) (x) = r ^ _ + Xx-o-PEcp (Xx~a) (X e C). (2.2.54)

The results in (2.2.53) and (2.2.54) for a > 0 and /? > [a] + 1 were proved byKilbas and Saigo in [391] and presented in Kilbas and Saigo ([397] in 1997, [724]in 1998). They can be extended to a complex a and /? by analytic continuation.

In conclusion we indicate the composition properties of the Liouvill e fractionalintegration operators (2.2.1) and (2.2.2) with the operator of translation TT, and theoperator of dilation IIx, defined in (1.3.5) and (1.3.6), respectively. For "sufficientlygood" functions f(x), the following formulas hold:

hI1! = I1rU (a > 0; fcel), (2.2.55)

and

nA / 0V = Aa/oa+nA/, and UxIlf = XarUxf (a > 0; A > 0) (2.2.56)

[see Samko et al. ([729], (5.12)-(5.14))].

Chapter 2 (§ 2.3) 87

2.3 Liouvill e Fractional Integrals and FractionalDerivatives on the Real Axis

In this section we present the definitions and some properties of the Liouvillefractional integrals and fractional derivatives on the whole axis M. = (—00,00).More detailed information may be found in the book by Samko et al. ([729],Section 5).

The Liouvill e fractional integrals and fractional derivatives on M. are definedsimilarly to those on the half-axis in Section 2.2. The fractional integrals have theform ra nd

where i e l and 9l(a) > 0, while the fractional derivatives corresponding to thosein (2.2.3) and (2.2.4) are defined by

and

y{t)dt

{t - a;)a-«+1'(2.3.4)

where n = [fH(a)] + 1, 9t(a) 0 and i € l , respectively.The expressions for / " / and Iif in (2.3.1) and (2.3.2), and for D%y and Da_y

in (2.3.3) and (2.3.4), are called Liouville left- and right-sided fractional integralsand fractional derivatives on the whole axis M.. In particular, when a = n £ No,then

(rP+y){x) = (D0_y){x)=y(x);

(Dly)(x)=y^(x), [Dn_y){x) = {-l)ny^\x) (neN) (2.3.5)

where y^n\x) is the usual derivative of y(x) of order n.

If 0 < m{a) < 1 and x G K, then

and

When m(a) = 0 (a / 0), the Liouvill e fractional derivatives (2.2.38) and (2.3.7)of a purely imaginary order have the forms


and

frespectively.

Analogous to Property 2.5(b), the Liouvill e fractional calculus operators 7"and D+ of the exponential function eXx yield the same exponential function, bothapart from a constant multiplication factor.

Propert y 2.11 Let <K(A) > 0.

(a) Ifm{a) ^0, then(I%ext){x) = \-aeXx. (2.3.10)

(b) J/SK(a) ^ 0 , then(D%ext)(x)=\aeXx. (2.3.11)

When 0 < a < 1 and 1 p < I /a, the integrals / " / and / " / are defined for afunction / (a) £ LP(R). Indeed, the following Hardy-Littlewood theorem holds [seeSamko et al. ([729], Theorem 5.3)].

Lemma 2.17 Let 1 i£ p ^ oo, 1 ^ g ^ oo and a > 0. 77ie operators I" and I fare bounded from LP(R) into Lq(R

+) if, and only if, the conditions in (2.2.14) aresatisfied.

Let M. be the axis E supplemented by the one infinite point: R = K|J{oo} . Wedenote by L pw (1 5; p < oo) a space of functions /(a;) with an exponential weightu G 1 on 1 by defining the norm as follows:

LPM If II/I U = ( / " e-wi |/(t)|"dt) P < oo I , (2.3.12)

with 1 ^ p < oo, while by Z/oo,w = Cw the space of functions f(x) such thate-"xf{x) € C(R) with the norm

Ioo,,(R) = CU{R) := If : H/llc = maxe-"*|/(*)l < «>)I t€R J

(2.3.13)

Spaces (2.3.12) and (2.3.13) are invariant with respect to the Liouvill e fractionalintegration 1% and II [see Samko et al. ([729], Theorem 5.7)].

Lemma 2.18 Let 1 p ^ oo, a > 0 and OJ > 0.

(a) TTie operator J° «s bounded in the space LPtW:

(b) T/ie operator I" is bounded in the space -LPi_w:

_w, * = ( | ^ |) (1 g p < oo), fc = M Q (p = oo). (2.3.15)

Chapter 2 (§ 2.3) 89

The following assertions, similar to Property 2.6, hold [see Samko et al. ([729],Section 5.1)].

Lemma 2.19 Let a > 0, j3 > 0, and p ^ 1 be such that a + (3 < 1/p. Iff(x) G LP(M), then

and {Ia_lif){x) = (H+Pf)(x). (2.3.16)

The Liouville fractional derivatives(D"2/)(a;) and (D°!_y)(x) exist for "suffi-ciently good" functions y(x); for example, for functions y(x) in the space C£°(W.)of all infinitely differentiable functions on K with compact support. Therefore, thefollowing properties, similar to Properties 2.7 and 2.10, are valid.

Lemma 2.20 If a > 0, then, for "sufficiently good" functions f(x), the relations

(D«I»f)(x) = f{x), and {Da_IZf){x) = f(x) (2.3.17)

are true. In particular, these formulas hold for f(x) G Li(E).

Property 2.12 If a > /3 > 0, then the formulas

{DÎ%f){x) = ( /oWXz), and {D^_IZf){x) = {FT*f){x) (2.3.18)

hold for "sufficiently good" functions f(x). In particular, these relations hold forf(x) e Li(M) .

Furthermore, when j3 = k £ N and d\(a) > k, then

[D kI%f){x) = I$-"f{x), and (Dkl2f)(x) = (-l)klZ~kf(x). (2.3.19)

Property 2.13 Let a > 0, m G N and D = d/dx.(a) If the fractional derivatives (D+y)(x) and {D°£rmy){x) exist, then

[DmD%y){x) = [Da+

+mf){x). (2.3.20)

(b) If the fractional derivatives {Dty){x) and (D0^~my)(x) exist, then

{DmDa_y){x) = (-l)m{Da_+mf){x). (2.3.21)

Property 2.14 If a > 0, then the relations

/ <p(x) (7?V) (x)dx = / 1){x) (lZ(p) (x)dx (2.3.22)J— OO J—CO

and/»OO /»OO

/ f(x) (D%g) {x)dx = / g[x) (Da_f) (x)dx (2.3.23)J—OO J—OO

are valid for "sufficiently good" functions ip, tj) and f, g.In particular, (2.3.22) holds for functions ip{x) G Lp(M) and ip(x) G Lq(R),

while (2.3.23) holds for f(x) G 12 (LP(R+)) and g(x) G I2(Lq{M+)), providedthat p > 1, q > 1, and (1/p) + (l/q) = 1 + a.


Remark 2.10 Property 2.14 was given in Samko et al. ([729], (5.16) and (5.17)).

The Fourier transform (1.3.1) of the Liouville fractional integrals / " / and / " /is given by the following result [see Samko et al. ([729], Theorem 7.1)].

Property 2.15 LetO < 9i(a) < 1 and f(x) G Ia(M). Then the following relationshold:

and

Here (îx)a means{îx)a = \x\aeTani s g n ( l ) / 2. (2.3.26)

Remark 2.11 When 9l(a) > 0, then, for "sufficiently good" functions f{x), theequations (2.3.24) and (2.3.25) are valid as well as the following correspondingrelations for the Liouville fractional derivatives D^_f and Dtf:

f){x) = {-ix)a{Ff)(x) (2.3.27)

and{TDa_f){x) = {ix)a{Tf){x), (2.3.28)

where {îx)a is defined by (2.3.26).

In conclusion, we give the composition properties of the Liouville fractionalintegration operators (2.3.1) and (2.3.2), similar to those given in (2.2.55) and(2.2.56) for the Liouville fractional integration operators on R+. The formulas

( a > 0; hem), (2.3.29)

and

nxi$f = xaiîixf, nA /f / = A°/«nA / (a > o; A > o), (2.3.30)

hold for "sufficiently good" functions f{x). In particular, they are true for f(x) GLP(R), l^p< I/a, when 0 < a < 1 [see Samko et al. ([729], (5.12)-(5.13))].

2.4 Caputo Fractional Derivatives

In this section we present the definitions and some properties of the Caputofractional derivatives. Let [a, b] be a finite interval of the real line R, and let

Da+[y(t)](x) = {D%+y)(x) andDgL[j/(t)](ar) = {D%_y){x) be the Riemann-Liouvillefractional derivatives of order a £ C (<H(a) ^ 0) defined by (2.1.5) and (2.1.6),respectively. The fractional derivatives (cD%+y)(x) and (GD%__y)(x) of order a £ C

Chapter 2 (§ 2.4) 91

(a) ^ 0) on [a, b] are defined via the above Riemann-Liouville fractional deriva-tives by

and

[CD?_y){x) :=

respectively, where

n = W

n—l (i

' ( < - a )* (*)

(b-t)k

+ 1 /or a ^ No; n = a for a e No.

(2.4.1)

(2.4.2)

(2.4.3)

These derivatives are called left-sided and right-sided Caputo fractional derivativesof order a.

In particular, when 0 < £H(a) < 1, the relations (2.4.1) and (2.4.2) take thefollowing forms:

(cD°+y){x) = (x), (2.4.4)

(2.4.5)

If a $ No and y(x) is a function for which the Caputo fractional derivatives(cD%+y){x) and (cD£_y)(x) of order a € C (9t(a) ^ 0) exist together with theRiemann-Liouville fractional derivatives {D^+y){x) and (D£_y)(x), then, in accor-dance with (2.1.7) and (2.1.9), they are connected with each other by the followingrelations:

n—1

{CDaa+y){x) = (DZ+y){x) -

k=0T{k-a

and

n-l

(c£>6a_y)(:r) = ( ^_2 / ) ( a ;) -

In particular, when 0 < 9l(a) < 1, we have

{cDaa+y){x) =

( 6- x )k ~ a (n=[SR(a)] + l) . (2.4.7)

(2.4.8)

yip)r(l-a)

(2.4.9)

If a ^ No, then the Caputo fractional derivatives (2.4.1) and (2.4.2) coincidewith the Riemann-Liouville fractional derivatives (2.1.5) and (2.1.6) in the follow-ing cases:

(°DZ+y)(x) = {Daa+y){x), (2.4.10)


if y{a) = y'(a) = = y^n-^(a) = 0(n = [9t(a)] + 1); and

(CDZ_y)(x) = (D?_y)(x), (2.4.11)

if y(b) = y'(6) = = y(n-V(b) = 0(n = [fK(a)j + 1).

In particular, when 0 < *R(a) < 1, we have

(CTJ>"+2/)(a;) = (7?"+2/)(a:), when y(a) = 0, (2.4.12)

(cD%_y)(x) = (D?_y)(x), when y(6) = 0. (2.4.13)

If a — n E No and the usual derivative j / n) (x) of order n exists, then (cD"+y) (x)coincides with y^n\x), while (CD£_y)(x) coincides with j / n' ( : r) with exactness tothe constant multiplier (—l)n:

{cDl+y){x)=y^\x)and{cD^_y){x) = {-l)ny(n\x) (n G N). (2.4.14)

The Caputo fractional derivatives (cD"+y)(x) and (cD"_y)(x) are defined forfunctions y(x) for which the Riemann-Liouville fractional derivatives of the right-hand sides of (2.4.1) and (2.4.2) exist. In particular, they are defined for y(x)belonging to the space ACn[a,b] of absolutely continuous functions defined in(1.1.7). Thus the following statement holds.

Theorem 2.1 Let Sft(a) ^ 0 and let n be given by (2.4.3). If y(x) € ACn[a,b],then the Caputo fractional derivatives (GD%+y)(x) and (cD%_y){x) exist almosteverywhere on [a,b].

(a) If a $ No, (cD"+y)(x) and (cD%_y)(x) are represented by

and

- =: (-l)n{IâDny)(x), (2.4.16)

respectively, where D = d/dx and n = [D\(cej\ + 1.

In particular, when 0 < SH(a) < 1 and y(x) E AC[a,b],

^ ) r ( ^ = ; <* °and

(b) 7/ a = n G No, then (cD"+y)(x) and (cD%_y)(x) are represented by(2.4.14). In particular,

(cD°a+y)(x) = (cD°b_y)(x) = y(x). (2.4.19)

Chapter 2 (§ 2.4) 93

Proof. Let a g No. Using (2.4.1) and (2.1.5), integrating by parts the inner integraland differentiating (which is possible by the conditions of the theorem), we have

icDaa+y){x)

- tyT(n — a) \dx n - a

-E^-«>'k=0

\t=a

fJ a

n — a dtfc=0

dt

T(nl (-) "" ' f (* - tr-°-> \y>(t) -- a) [dx) Ja

{X t} [y [t> - 1)!- a)»-i] dt

Using the above argument again, we derive that

(cD°+y)(x) =I [Tl

* Hx-tr-^yÔi) Ja

which yields the result in (2.4.15). Relation (2.4.16) is proved similarly.

If a = n £ No, then (2.4.1), in accordance with the first formula in (2.1.7),takes the form:

-)"dx I

n - l

k=o

(2.4.20)

and from Lemma 1.1 we derive (cD™+y)(x) = y^(x), which yields the first resultin (2.4.14). The second one is proved similarly.

The following statement is analogous to that of Theorem 2.1 for functions y(x)in the space C" [a, b] of continuously differentiable functions on [a, 6] up to ordern.

Theorem 2.2 LeifR(a) ^ 0 and let n be given by (2.4.3). Also let y(x) £ Cn[a,b].Then the Caputo fractional derivatives (cD°+y)(x) and (cD£_y)(x) are continuouson [a,b]: (cD%+y)(x) £ C[a,b] and (cD%_y)(x) £ C[a,b].

(a) If a <£ No, then {cD%+y){x) and (cD%_y)(x) are represented by (2.4.15)and (2.4.16), respectively. Moreover,

{cD"+y){a) = (cD"_y)(b) - 0. (2.4.21)

In particular, they have, respectively, the forms (2.4.17) and (2.4.18) for0 < m(a) < 1.

(b) If a = n £ No, then the fractional derivatives (cD2+y)(x) and (cD^_y){x)have representations given in (2.4.14). In particular, the relations in (2.4.19) hold.


Proof. Let a £ No- Formulas (2.4.15) and (2.4.16) are proved as in Theorem2.1. The continuity of the functions (cD"+y)(x) and (cD"_y)(x) on [a,b] followsfrom the representations (2.4.15) and (2.4.16), according to Lemma 2.8(a) withf(t) = y(n\t) G C[a,b] and 7 = 0. The relations in (2.4.21) follow from thefollowing inequalities:

and

which are valid for any x G [a,b] and proved directly by using (2.1.1) and (2.1.2)(with n replaced by n - a), and (2.1.6) and (2.1.16) with /? = 1, and (1.1.19).

When a G No, the first relation in (2.4.14) follows from (2.4.21) and Lemma1.3 with 7 = 0. The second one is proved similarly.

Corollar y 2.3 Let 9t(a) _ 0 and let n be given by (2.4.3).

(a) If a ^ No, then the Caputo fractional differentiation operators CD"+ andCD%_ are bounded from the space Cn[a,b] to the spaces Ca[a,b] and Cb[a,b]respectively, defined by (1.1.25) and (1.1.26). Moreover,

II CDaa+y\\c. M l l/llc - and \\ cD^_y\\Cb = ka\\ y\\c~, (2.4.24)

where(

ka = \T(n-a)\[n-m(a) + lY (2" 4" 25)

(b) If a = n G No, then the operators GD™+ and CD%_ are bounded fromCn[a,b] to C[a,b]. Moreover,

|| cDna+y\\c = || i/||c- and || ^ " . y l l c = II y | | c- (2-4.26)

Proof. Assertion (a) follows from Theorem 2.2(a) if we take into account the esti-mates (4.2.22) and (4.2.23), definitions (1.1.25) and (1.1.26) of the spaces Ca[a,b]and Cb[a,b], definitions (1.1.19) and (1.1.20), and the estimate ||y(n)||c ^ ||j/||c»-Assertion (b) follows from Theorem 2.2(b) and from the relations in (2.4.14), inaccordance with (1.1.19) and (1.1.20).

Remark 2.12 For n - 1 < a < n [n G N), the derivative (cD"+y)(x) in the form(2.4.15) was defined by Caputo [121] and presented in his book [122], and therefore,the derivatives (cD"+y)(x) and (cD%_y)(x) are called Caputo derivatives. In thisregard, see also Mainardi [520] and Podlubny ([682], Section 2.4.1).

Remark 2.13 For a ^ 0 and y(x) G ACn[a,b], the relation of the form (2.4.15)was proved by Diethelm ([173], Theorem 3.1) for the Caputo derivative {D"ay)(x)defined by (2.4.1) with n = [a]:

(D?ay){x) =n - 1

(2.4.27)

Chapter 2 (§ 2.4) 95

The Caputo derivatives (c D%+y)(x) and (cD£_y)(x) have properties similarto those of the Riemann-Liouville fractional derivatives (D"+y)(x) and (D%_y)(x)given in (2.1.17) and (2.1.19), but different from those in (2.1.20).

Propert y 2.16 Let £R(a) > 0 and let n be given by (2.4.3). Also let 9t(/?) > 0.Then the following relations hold:

(cDaa+{t - af-1) (x) = r ( ^ }

Q ) ( z - af-1 (*(/?) > n), (2.4.28)

(cDZ_(b - tf-1) (x) = rff?a)(* > - xf-1 (W(/3) > n) (2.4.29)

and

(cD%+(t-a)k){x) = 0 and [c D%_{t-a)k){x) = 0 (k = 0,1,- , n -1 ). (2.4.30)

In particular,{cD%+l){x) = 0 and (c'D%_l){x) = 0. (2.4.31)

When £H(a) £ No and when a £ N, the Caputo fractional differentiation oper-ators CD"+ and c D%_ provide operations inverse to the Riemann-Liouville frac-tional integration operators J"+ and /"„ , given in (2.1.1) and (2.1.2), from theleft. But they do not have such a property when 9$a) £ No and 3(a) ^ 0. Thefollowing analog of Lemma 2.4 and Lemma 2.9(b) is valid.

Lemma 2.21 Let £R(a) > 0 and let y(x) £ L ^ a , b) or y(x) £ C[a, b].

(a) // m(a) g N or a £ N, then

(C'Daa+I%+y){x) = y(x) and (cD^_I^_y)(x) = y(x). (2.4.32)

(b) Ifm{a) £ N and 3{a) / 0, then

(cD2+I" +y){x) = y{x) - V ' ^ + \ x - a)n~a (2.4.33)

and(Ta+1-n*,\(h — \

b - x)n~a. (2.4.34)

Proof. Let y{x) £ Lâ^b) (y(x) £ C[a,b$, and let <R(a) ^ N, n = [^(a)] + 1 orn £ N and fc = 0,1, , n — 1. By Property 2.2 (Lemma 2.9(c)), we have

Since i/(a;) £ Loc(a,b) (y(x) £ C[a, 6]), then for a.e. (for any) x £ [a, &], analogousto (2.4.22), we get

(2.4.36)


for any k = 0,1, , n — 1 = [9t(a)], and hence

(o+) = 0 (k = 0,1, , n - 1). (2.4.37)

Thus, using (2.4.10) for SH(a) £ N and (2.4.20) for n e N, with y(x) replaced by(I" +y)(x), and applying Lemma 2.4 (Lemma 2.9(b)), according to which the firstrelation in (2.1.31) holds, we derive

(cDaa+I%+y)(x) = (DZ+IZ+y)(x) = y(x), (2.4.38)

which yields the first formula in (2.4.32). The second one is proved similarly.

Let now a = m + i9 (m G N; 6^0). Then n = m + 1 ^ 2 and, analogous to(2.4.35) and (2.4.36), we have

(IZ+y)w(x) = (i:+ ky)(x) (k = 0,1, , n - 2) (2.4.39)

and

where K = \\y\\Lx (= \\y\\c) (fc = 0,1, , m - 1) and hence

( / Q » W (o+) = 0 (* = 0,1, , n - 2). (2.4.41)

Using this formula, from (2.4.1) we obtain (2.4.33). (2.4.34) is proved similarly.

The next statement is an analog of Lemma 2.6(b) and Lemma 2.9(d).

Lemma 2.22 Let !R(a) > 0 and let n be given by (2.4.3). If y(x) G ACn[a,b] ory{x) G Cn[a,b], then

(/aa+ CD«a+y){x) = y(x) - J2 y^fL ~ o)k (2.4.42)

and

(7?_ ^gLy)^ ) = y(a;) - J2 {b _ 3.)*. (2.4.43)

In particular, if 0 < fH(a) ^ 1 and y(x) e j4C[a, 6] or y(x) £ C[a, 6], iften

(/aQ+ ^ + y ) ( a ;) = J/(i) - y{a), and (ij»_ ^gLi/)(a;) = y ^) - y(6). (2.4.44)

Froo/. Let a £ N. If j/(a;) G Cn[a,6] (y(x) £ Cn[a,6]), then by Theorem 2.1(a)(Theorem 2.2(a)), {cD%+y)(x) and (cI>°j/)(a;) are represented by (2.4.15) and(2.4.16), respectively. Then, using the semigroup property in (2.1.30), being heldby Lemma 2.3 (Lemma 2.9(a)), and (2.1.8), we have

( C GDaa+y){x) = (I°+I^+aDny){x) = {i: +Dn

a+y){x) (2.4.45)

and

(J?_ CD<Z_y){x) = {-l)n(I?_l£aDny)(x) = (if_£>?_i/)(a;). (2.4.46)

Chapter 2 (§ 2.4) 97

Then, by (2.1.41) and (2.1.48), from (2.4.45) and (2.4.46) we derive (2.4.42) and(2.4.43), respectively.

If a € N, then applying Theorem 2.1(b) (Theorem 2.2(b)) and using (2.1.41)and (2.1.48), we obtain (2.4.42) and (2.4.43).

Remark 2.14 For a > 0, the first formula in (2.4.32) and the relation (2.4.42)for the Caputo fractional derivative (D"ay)(x) of the form (2.4.23) were obtainedby Diethelm ([173], Theorems 3 and 4). We only note that these theorems areformulated for a ^ 0, but the case a = 0 needs an additional explanation.

We have defined the Caputo derivatives on a finite interval [a, b] by (2.4.1)and (2.4.2) and shown, in Theorems 2.1 and 2.2, that they can be representedin the forms (2.4.14) or (2.4.15) and (2.4.16), provided that f(x) € ACn[a,b] andf{x) G Cn[a, b}. Formulas (2.4.15) and (2.4.16) can be used for the definition of theCaputo fractional derivatives on the half axis E+ and on the whole axis E. Thusthe corresponding Caputo fractional derivative of order a eC (with *R(a) > 0 anda ^ N) on the half axis E+ and on the whole axis E can be defined as follows:

and

1 \n po' / /

- a ) Jx(2.4.50)

v ~-»,K-I Y{n-a)jx {t-x)a+ l~n v y'

respectively.When 0 < <R(a) < 1, the relations (2.4.47)-(2.4.48) and (2.4.49)-(2.4.50) take

the following forms:

{x e M+) (2.4.52)

and

^ ) f ( f ^ F (>:e E»- (24'54)


For a — n G N, we define the Caputo derivatives (cDQ+y)(x), (cD™y)(x) andc^y)(x) by

{CDn+y){x) := yW{x), CDn_y){x) := (-l)ny^(x) (x G R+) (2.4.55)

and

(c0Sl/)(aO:=y(n)(a:), (C^2/ )W := ( - l )V n )(^ ) (* € R). (2.4.56)

The Caputo derivatives (c D+y)(x) and (c'D°!_y){x) have properties similar tothose in (2.3.11) and (2.2.13).

Property 2.17 //fK(a) > 0 and A > 0, then

(GD%ext) (x) = XaeXx and (cDZe~xt) (x) = Xae~Xx. (2.4.57)

The Mittag-LefHer function Ea[X(x — a)a] is invariant with respect to the Ca-puto derivative CD%+, but it is not the case for the Caputo derivative CD°_. Infact, the following assertion holds.

Lemma 2.23 If a > 0, a £ R and A G C, then

(CD%+Ea[\(t - a)a}) (x) = XEa[X(x - a)a] (2.4.58)

and(CData-lEa ( A r a )) = 1 ^ ^ ^X~a) . (2.4.59)

In particular, when a = n G N,

DnEn[X{x - a)n] = En[X(x - a)"] (2.4.60)

Dn [tn-1En (Xrn)) (x) = -En>1.n (Xx~n) = ~Ên (Xx~n) . (2.4.61)

Proof. Relations (2.4.58) and (2.4.59) are proved directly by using the definition(1.8.1) of the Mittag-Leffler function Ea(z) and by the term-by-term differentiationof the series in the left-hand sides of (2.4.58) and (2.4.59).

The following assertion, which yields the Laplace transform of the Caputofractional derivative cD^+y, is derived from Lemma 2.13(a) and (1.4.9).

Lemma 2.24 Let a > 0, n - 1 < a ^ n (n G N) be such that y(x) G Cn(R+),y(n)(x) G Li(Q,b) for any b > 0, the estimate (2.2.35) holds for y(n){x), theLaplace transforms (Cy)(p) and C[Dny(t)] exist, and limx^+oo(D

ky)(x) = 0 fork = 0,1, , n — 1. Then the following relation holds:

(CcD%+y)(s) = 8a(Cy){s) - £ sa-k-l(Dky)(0). (2.4.62)

fc=0

In particular, if 0 < a ^ 1, then

(£cD%+y)(s) = sa(Cy)(s) - ^ " ^ ( O ) . (2.4.63)

Chapter 2 (§ 2.5) 99

Similarly, from Lemma 2.14, (1.4.37)-(1.4.38), and Remark 2.9, we derive theMellin transform of the Caputo fractional derivatives cD$+y and cD°_y.

Lemma 2.25 Let a > 0, n - 1 < a ^ n (n £ N) be such that y(x) £ Cn(R+),y{x) £ Xl+n_a(R+), and let there exist {My{n)){s + n-a) and (My){s - a).

(a) IfDK(s) < 1 - <R(n - a), then

-a)

(2.4.64)

- a)

n - l

fc=0

In particular,

(M c.

(b)

DS+,)

Ifftis

{M UD

r ( i -when 0 < a

/ \ r ( 1 +

[s) r( i

) > 0? then

a0+

<

a—

y)W =

n _s))

i ,

-s)s) [M

— 8)

"" ' ,,„_* rw

i n particular, when 0 < a < 1,

_ Q )

(2.4.67)

2.5 Fractional Integrals and Fractional Derivati-ves of a Function with Respect to AnotherFunction

In this section we present the definitions and some properties of the fractionalintegrals and fractional derivatives of a function / with respect to another functiong. Some of these definitions and results were given in Samko et al. ([729], Section18.2).

Let (a,b) (—oo ^ a 0. Also let g(x) be an increasing and positive monotone functionon (a, 6], having a continuous derivative g'(x) on (a,b). The left- and right-sidedfractional integrals of a function f with respect to another function g on [a, b] aredefined by


and

respectively. When a = 0 and 6 = oo, we shall use the following notations:

l C » / ) W := FfS) f b m - ^ ' - (. > o; *(«) > 0); (2.5.4)while, for a = — oo and 6 = oo, we have

Integrals (2.5.1) and (2.5.2) are called the Riemann-Liouville fractional integrals ona finite interval [a,b], (2.5.3) and (2.5.4) the Liouville fractional integrals on a half-axis R+, while (2.5.5) and (2.5.6) are called the Liouville fractional integrals on thewhole axis M.. If g'(x) 0 (-oo ^ a < x < 6 ^ o o ), then the operators in (2.5.1)-

(2.5.2), (2.5.3)-(2.5.4), and (2.5.5)-(2.5.6) are expressed via the Riemann-Liouvilleoperators (2.1.1)-(2.1.2), the Liouvill e operators (2.2.1)-(2.2.2) and (2.3.1)-(2.3.2)by

(I°+Wf){x) = (Qgl^+Q-'mx), (I^.J)(x) = (Qg^.Q^mx), (2.5.7)

Vo+;gf)(x) = (QôH^/X*) , (I°.,gf)(x) = (Q9Q+oo)-Q^f)(x), (2.5.8)

and

{IIJ){X) = (Qgl^ + Q^mx), {I1J){X) = (Qgfy+^Q^mx),(2.5.9)

where Qg is the substitution operator

(Qgf)(x) = M*)], (2-5.10)

and Q~x is its inverse operator. Therefore, the properties of the integrals (2.5.1)-(2.5.6) follow from the corresponding properties of the Riemann-Liouville andLiouvill e fractional integrals given in Sections 2.1-2.3. For example, the followingassertions hold, generalizing those in (2.1.16), (2.1.18), (2.3.10), and (2.2.14).

Propert y 2.18 Let <K(a) > 0 and 9t(/?) > 0.

lgW-gWF+P-K (2.5.11)

Chapter 2 (§ 2.5) 101

(b)lff-(x) = \g(b)-g(x)]0-1, then

gWF+e-K (2.5.12)

Propert y 2.19 Let <R(a) > 0 and A > 0. Then

{I«.geXg(t)){x) = \-aex^x) (2.5.13)

andg x ) = \-ae-x^x\ (2.5.14)

The semigroup property also holds.

Lemma 2.26 Let 9t(a) > 0 and EH(/3) > 0. Then the relations

(i:+wl!+wf)(x) = (i££/)(s), W-jLjK*) = (#+?/)(*) (2-5.15)

and

hold for "sufficiently good" functions f{x).

Let g'(x) 0 (-00 a < x < b oo) and «R(a) 0 (a ^ 0). Also letn = [9\(a)] + 1 and D = d/dx. The Riemann-Liouville and Liouvill e fractionalderivatives of a function y with respect to g of order a (SK(a) ^ 0; a ^ 0),corresponding to the Riemann-Liouville and Liouvill e integrals in (2.5.1)-(2.5.2),(2.5.3)-(2.5.4), and (2.5.5)-(2.5.6), are defined by

(D2+,gy)(x) := (-

9'(t)y(t)dt

i / i \ n r

and

" g'g'(t)y(t)dt

x g'(t)y(t)dt


T(n-a

and

P?;,»)(*) := ( ~,

9'(t)y(t)dt

™ ( ' }' ( 5 }

respectively.

When g(x) = x, (2.5.17) and (2.5.18) coincide with the Riemann-Liouvillefractional derivatives (2.1.5) and (2.1.6):

{Daa+.xy){x) = (D%+y)(x) and (D?_.xy)(x) = (D?_y)(x), (2.5.23)

(2.5.19) and (2.5.20) coincide with the Liouvill e fractional derivatives (2.2.3) and(2.2.4):

(D%+.xy)(x) = (D%+y)(x) and (Da_.xy)(x) = (D1y)(x), (2.5.24)

and (2.5.21) and (2.5.22) coincide with the Liouvill e fractional derivatives (2.3.3)and (2.3.4):

{D%.xy){x) = {Da+y){x) and {D°L,xy){x) = (Da_y){x). (2.5.25)

When a = n € N, the above general fractional derivatives (2.5.17)-(2.5.22) havethe following forms:

(Dna+,gy)(x) = (Dn

+.tgy){x) = ( ^ ) £ ) VW, (2-5-26)

(2.5.27)


(Dl+.igy)(x) = [D\. tgy){x) = ^ | , (2.5.28)9 \x)

(Dl_.gy)(x) = (Dl_.tgy)(x) = -V-^\. (2.5.29)9 \x)

The following results, generalizing those in (2.1.17), (2.1.19) and (2.3.11),(2.2.15), are analogous to those in Properties 2.18 and 2.19:

Chapter 2 (§ 2.5) 103

Property 2.20 Let 9t(a) 0 (a 0) and <ft(/?) > 0.

(a) //y+(z) = [3(2:) - 5(a)f-1, tfien

(^+;9y+)(z) = r^{P_]

a) [g(x) - gWf-*-1. (2.5.30)

(b) Ify-(x) = [g{b) - g(x)f-\ then

g{x)f-a-\ (2.5.31)

Property 2.21 Let fR(a) ^ 0 (a 0) and A > 0. Then

(D^.geXg{t))(x) = XaeXs (2.5.32)

and(D1.ge-X9^)(x) = Xae-X^x\ (2.5.33)

Analogous to the properties of the Riemann-Liouville and Liouvill e fractionalderivatives, presented in Sections 2.1-2.3, there exist the corresponding propertiesfor the general fractional derivatives in (2.5.17)-(2.5.22).

In conclusion, we consider the following special cases of the general integralsand derivatives (2.5.3) and (2.5.4) with g(x) = xa (u > 0):

j ^ y Jx {t_ily-a (* > 0; W(a) > 0) (2.5.35)and

^ T ^ (x>0)' (2-5-36)

a) Z 0 (a # 0); n = [SR(a)] + 1; 2? = £ )

We note that IQ+.X* f and I".x<,f are connected with the fractional integralsJ0V and II f by

i ( ) /)(x) (x > 0) (2.5.38)

and

(JV/)(z) = (l-f(t1/a)) {xa) = (NaI»N1/af)(x) (x > 0), (2.5.39)

where Na is an elementary operator given by (1.4.27).

As a particular case of (2.5.11), (2.5.30) and (2.5.14), (2.5.33), we have thefollowing results.


Property 2.22 Let a > 0 and 9t(a) ^ 0 ( a / 0 ).(a) 7/9t(a) > 0 and 9S(/?) > 0 , then

and

(b) 7/<R(a) ^ 0 (a / 0) and A > 0, then

(2.5.40)

and

(2.5.42)

(2.5.43)

Using the representations (2.5.38) and (2.5.39) and the relation (1.4.29), andtaking (2.2.36) and (2.2.37) into account, we obtain the Mellin transform of thegeneral fractional integrals (2.5.34) and (2.5.35).

Lemma 2.27 Let !R(a) > 0, a > 0 and f(x) £ XS1+

(a) //9t(s) < a[l-m(a)}, then

(b) 7/<R(s) >0, then

Remark 2.15 The formulas for the Mellin transform of the fractional deriva-tives (2.5.36) and (2.5.37) are more complicated than the relations in (2.5.44) and(2.5.45). For example, if <K(a) ^ 0, (a ^ 0), n = [9t(a)] + 1, and a > 0, then theMellin transform of {D^+.xay){x) for "sufficiently good" functions y{x) is given by

(MD%+.^y)(s) = - a a)

n - l

fc=i \ i=i

In particular, when

s-(k+Va(x1-'7D)n-1-k

(2-5.46)

(a;)] = 0 (jfe = 0,1, , n - 1) (2.5.47)

Chapter 2 (§ 2.6) 105

and

{x1-< r D)n~x-k [IZ+? x.y) (x)] = 0 (fc = 0 , l , - - -, n - l ) , (2.5.48)

the formula (2.5.46) is simplified to the form

(MDZ+.^y)(s) = T{1 + a~,S[ a)(My)(s - aa). (2.5.49)1 v1 ~~ S/(J)

When a = 1, then (2.5.49) coincides with (2.2.43), and (2.5.46) coincides with(2.2.47).

2.6 Erdelyi-Kober Type Fractional Integrals andFractional Derivatives

In this section we present the definitions and some properties of the Erdelyi-Kobertype fractional integrals and fractional derivatives and their special cases. Some ofthese definitions and results were presented in Samko et al. ([729], Section 18.1),Kiryakova [417] and McBride [566].

Let (a, 6) (—oo ^ a 0, a > 0, and rj e C. We consider the left- and right-sidedintegrals of order a G C (£H(a) > 0) defined by

a{a+ri) i-x

iaTjaand

Cb f(r(l-a-ri)-l f(f)f]f

( y - ^ i - a (0 a < « < 6 g oo), (2.6.2)

respectively. When a = — oo and b = oo, we shall use the following notations:-a{a+ri)

and

Integrals (2.6.1) and (2.6.2), as well as (2.6.3) and (2.6.4), are called the Erdelyi-Kober type fractional integrals. If a = 2, a = 0, and b — oo, then integrals (2.6.1)

and (2.6.4) are denoted by

/„«/)(* ) := (I^ 2J)(x) = r(Q ) ^ ( g a_ ^ - o (* > 0) (2-6.5)


andf l -

(x>0). (2.6.6)

These operators are called the Erdelyi-Kober operators. When a = 1, a = 0, andb = oo, the integrals in (2.6.1) and (2.6.4) take the forms

and

^\! ( 0 ) (2.6.8)

These operators are known as the Kober operators or the Kober-Erdelyi operators.

Using the operators Ma and Na defined in (1.4.26) and (1.4.27), we can rewrite(2.6.1) and (2.6.2) in terms of the Riemann-Liouville fractional integrals (2.1.1)and (2.1.2) in the form

( J ^ M / ) ( I ) = ( J V , M . O . ^ + M , % / ) ( I ) ( 0 ^ a < z < 6 ^ o o) (2.6.9)

and

(Ig_.attlf)(x) = {Nl,MvIg,_M-a-r,N1/af)(x) ( 0 ^ a < x < ^ c o ), (2.6.10)

respectively. Similarly, (2.6.3) and (2.6.4) have the following representations interms of the Liouvill e fractional integrals (2.3.1) and (2.3.2):

(/?;,,„/)(*) = (NaM.a.vI^M1lNll<r f){x) (x > 0), (2.6.11)

{I1,aJ){x) = {NrMÎIM^NyJ^x) (x > 0). (2.6.12)

In particular, the integrals in (2.6.5) and (2.6.6) are represented via the Liouvill efractional integrals (2.2.1) and (2.2.2) as follows:

(Iv,af)(x) = (AT2M_Q_r;/0a+M f,iV 1/2/)(iC) (x > 0), (2.6.13)

(Kr,,af)(x) = (i\r2M,JfM_a_I ,iV 1/2/)(a:) (* > 0), (2.6.14)

while (2.6.7) and (2.6.8) take the forms

tfU/X*) = (M-a-r,I&.M vf)(x) (x>0), (2.6.15)

(K~af)(x) = {M1lItM_a^f){x) {x>Q). (2.6.16)

In accordance with (2.6.9)-(2.6.10), the properties of the Erdelyi-Kober typefractional integrals (2.6.1)-(2.6.2) can be derived from the corresponding propertiesof the Riemann-Liouville fractional integrals. Similarly, from (2.6.11)-(2.6.12) andthe properties of the Liouvill e fractional integrals (2.3.1)-(2.3.2), we can obtainthe properties of the operators (2.6.3)-(2.6.4). In particular, the properties of theoperators (2.6.5)-(2.6.6) and (2.6.7)-(2.6.8) can also be derived from the propertiesof Liouvill e fractional integrals (2.2.1) and (2.2.2). We now present some of theseproperties. The boundedness of the operators I"+.ari and I"_.a v in the spaceLp(a.b) is given by the following result.

Chapter 2 (§ 2.6) 107

Lemma 2.28 Let 9=t(a) > 0 and 1 :g p < oo.

(a) If 0 < a —1 + l/(p<r), iften i/ie operator Io+.a vis bounded in LP(R+). In particular, the operators IVta and 7+a are bounded inLP(M.+ ) , provided that 9\(r)) > - 1 + l/(2p) and *R(r)) > - 1 + l/p, respectively.

(c) If a = 0, 6 = oo, and fH(r?) > —l/(pa), then the operator I--atV is boundedin LP(R+). In particular, the operators Kvâ and K~a are bounded in Lp(R

+),provided that 91(J?) > — l/(2p) and fH(»j) > —l/p, respectively.

The semigroup property also holds.

Lemma 2.29 Let D\(a) > 0, <H(/3) > 0, and 1 <; p < oo.

(a) 7/0 < a < 6 < oo and f{x) G Lp(a,b), then

(2-6.17)

(b) If a = 0, 6 = oo, / (a) 6 LP(IR+), and 9=1(77) > - 1 + l/(po-),

(IS+Mlg+M+JXx) = (I%tLf){x). (2.6.19)

7n particular, if9\{rj) > —1 + l/(2p), ifeen

(/,,a/,+a,fl/)(a:) = ( 7 ^ + 0 / ) ^ ), (2.6.20)

w/iê, /or 91(77) > - 1 + l/p,

(CWH* ) = ( a+ /)(«)- (2-6.21)

(c) If a = 0, b = 00, / ( i ) G 7^,(1+), and 9=1(77) > - l / ( p^ ) , then

{I%,nlt,^+J){x) = (7^/)(z). (2.6.22)

In particular, if$\{rj) > —l/(2p), iften

( T ^ T ^ + ^ / K * ) = (^,Q+/3/)(a;), (2.6.23)

while, for 9i(r}) > —l/p,

f){x) = (K-a+ftf)(x). (2.6.24)

The weighted analog of the relation of fractional integration by parts (2.1.49)also holds.


Lemma 2.30 Let 9^(a) > 0 and 0 ^ a < b oo. Then the following relationsare valid for "sufficiently good" functions f{x) and g(x):

[ x°-lf{x){i: +.a^g){x)dx = f x°-lg{x){IZ_.aJ){x)dx. (2.6.25)J a Ja

If a = 0 and b = oo, then

f" x°-lg{x){I°.aJ){x)dx. (2.6.26)Jo Jo

In particular,

and

/

OO /-OO

xf(x)(Iv,ag)(x)dx = / xg{x)(K af)(x)dx (2.6.27)o

O /.OO

/ f(x)(I+ag)(x)dx = g(x)(K+af)(x)dx. (2.6.28)

Let m(a) 0 , ( a / 0) and n = [!R(Q)] + 1. Also let a > 0 and ?? G C. T/ieErdelyi-Kober type fractional derivatives, corresponding to the Erdelyi-Kober typefractional integrals (2.6.1) and (2.6.2), are defined, for 0 a < x < b oo, by

x<r(n+n) (/«-« y) (a;) (2.6.29)

and

n \ Tcr(n-r)~a) (jn-aaxo— \ U \ X \1b—,<T

respectively, with D = d/dx. When a = —oo and b = oo, these definitions arewritten as follows:

râ ;- («-^«) (rr^+ a_y) (x). (2.6.32)

When a = 2, relations (2.6.29) (with a = 0) and (2.6.32) take the forms

(D+ay)(x) := (D-+Aay)(x) = x~ ^ ^ " ^ ( n + , ) ( 7 ; ) + a i n_Q 2 /) (cc); (2.6.33)

-n,n-ay) (x), (2.6.34)

Chapter 2 (§ 2.6) 109

while, for a = 1, they are given by

{D+ ay){x) := (Daa+.^y){x) = x^Dnxn+ (In+a,n-ay) (x), (2.6.35)

(D~ay)(x) := {Da_.^y){x) = x^+a(-D)nxn^-a [Kn+a_n,n_ay) (x). (2.6.36)

If we suppose a = n € N in (2.6.29) and (2.6.30), then we obtain

, (2.6.37)

) . (2.6.38)

In particular, when a = 2, a = 0, and b = oo, we have

(D+ny)(x) = Z"2" ^ ^ % 2 ( " + " ) y ( ^ ) , (2.6.39)

^ , (2.6.40)

while, for a = 1, a = 0, and b = oo, we have

(D+ny)(x) = x-WV+Mx), (2-6.41)

(D-ny)(x) = x«+n(-D)nx-*y(x). (2.6.42)

The Erdelyi-Kober type fractional differentiation operators in (2.6.29) and(2.6.30) provide operations inverse to the Erdelyi-Kober type operators (2.6.1)and (2.6.2) from the left.

Property 2.23 // 9t(a) > 0 and 0 ^ a < b oo, then, for "sufficiently good"functions f{x), the formulas

(D^,r,Ia+;aj) (*) = /(*) , (i??-;a,,^-;«r,,/) (*) = /W. (2-6-43)

{Do+;*, vIg+;*, vf) W = fix), (D1.,a,nIâJ) (x) = f{x) (2.6.44)hold. In particular, when a = 2,

(Z>+aW) (x) = /( i) , and (D-aKv,af) (x) = f(x), (2.6.45)

while, for a = 1,

(KJZJ) (*) = /(* W (f>-aK-af) (x) = f{x). (2.6.46)

The Mellin transform (1.4.23) of the Erdelyi-Kober type fractional integralsIo+;a-,rif a nd I-;a,r)f o n the half-axis E+ are given by the following lemma.


Lemma 2.31 Let 9t(a) > 0, a > 0 and rj £ C. Also let f(x) £ Lp(M+) with1 ^ j> 2, and suppose that s = (1/p) + IT (T £ R).

(a) //£R(T7 - s/a) > - 1 , then

' l ^ U - (2'6 47)

/n particular, when 9i(r] — s/2) > —1, tften

^ J - ^ ( s ) = r(i (|â-!/2)W^^ ' (2-6-48)

while, for 9\.(rj — s) > — 1,

(b) // «R(?7 + s/a) > 0, t/ien

/n particular, when 9"l(?7 + s/2) > 0, then

i/e, /or 9"l(r? + s) > 0,

2.7 Hadamard Type Fractional Integrals andFractional Derivatives

In this section we present the definitions and some properties of the Hadamardtype fractional integrals and fractional derivatives. Some of these definitions andresults were presented in Samko et al. ([729], Section 18.3), Butzer et al. ([113],[112], [114]), Kilbas [370], and Kilbas and Titura [405].

Let (a,b) (0 a 0 and \x e C. We consider the left-sided and right-sided integralsof fractional order a £ C (^(a) > 0) defined by

(Jaa+f)(x) := J* ( l og f ) ^1 (a < x < b) (2.7.1)

and

^ [ (log I}" X ( a < x < b), (2.7.2)

Chapter 2 (§ 2.7)

respectively. When a = 0 and b — oo, these relations are given by

and

fRrThe fractional integrals, more general than those in (2.7.3) and (2.7.4) with \i g C,are defined by

and

The integral in (2.7.3) was introduced by Hadamard [319] [J.Math.Pure etAppl.(1892)]. Therefore, the integrals (2.7.1), (2.7.2) and (2.7.3), (2.7.4) are oftenreferred to as the Hadamard fractional integrals of order a [see Samko et al. ([729],Section 18.3 and Section 23.1, notes to Section 18.3)]. The general integrals (2.7.5)and (2.7.6), introduced by Butzer et al. [113], are called the Hadamard typefractional integrals of order a.

Let 8 = xD (D = d/dx) be the -derivative (1.1.11). The left- and right-sidedHadamard fractional derivatives of order a £ C (9t(a) ^ 0) on (a, 6) are dennedby

and

where n = [fH(a)] + 1 [see Samko et al. ([729], Section 18.3)]. When o = 0 andb = oo, we have

(V%+y)(x) := 5n(J0Xay)(x) (x > 0; 5 = x-^j , (2.7.9)

(Va_y)(x) := (S)n(J?-ay)(x) (x > 0; 8 = x-^j . (2.7.10)

The Hadamard type fractional derivatives of order a, more general than those in(2.7.9) and (2.7.10) with fj.eC, are defined for 9t(a) ^ 0 by


a y)fx) *^ ir^f ^îc^^f Tn~a^)fx ) f2 7 12)

where x > 0, 5 = x-^, 9i(a) ^ 0, and J+~"y, J_~^"y are the Hadamard typefractional integrals (2.7.5) and (2.7.6) of order n — a (n = [£H(a)] + 1) [see Butzeret al. [113, 112, 114]].

When a = m £ N, then

(V™+y)(x) = (Smy)(x) and (V^_y)(x) = (-l)m(5my)(x) (2.7.13)

with 0 ^ a < a ; < 6 oo, and

CDol+,^y)(x) = x^Smx~^y(x) and (£>™ y){x) = x'1{-8)mx-^y{x) (2.7.14)

with x > 0.It can be directly verified that the Hadamard fractional integrals and frac-

tional derivatives (2.7.1), (2.7.7) and (2.7.2), (2.7.8) of the logarithmic functions[log(x/a)] l3~1 and [log(6/a;)]^~1 yield logarithmic functions of the same form.

Property 2.24 If9\(a) > 0, 9\{/3, and 0 < a > = n ^ * - ^ '(2.7.16)

and

g-j (2.7.18)

In particular, if (3 = 1 and $K(a) ^ 0, then the Hadamard fractional derivativesof a constant, in general, are not equal to zero:

(2.7.19)when 0 < 9t(a) < 1. On the other hand, for j = [9t(a)] + 1,

= 0 and f 27gL (log ^ " j (x) = 0. (2.7.20)

Chapter 2 (§ 2.7) 113

From (2.7.20) we derive the following result.

Corollar y 2.4 Let 9t(a) > 0, n = [SH(a)] + 1, and 0 < a "+y)(x) = 0 is valid if, and only if,

n

y(x) = YJC

where Cj £ R (j — 1, , n) are arbitrary constants.

In particular, when 0 < 9t(a) 1, i/ie relation (V"+y)(x) = 0 /jolds if, and

only if, y(x) = c (log ^) for any e e l .

(b) The equality (V%_y)(x) = 0 is valid if, and only if,

n ( b\

y(x) = Y^ di[ log ~)

where dj € E (j = 1, , n) are arbitrary constants.In particular, when 0 < 9\{a) ^ 1, the relation (V"__y)(x) = 0 holds if, and

only if, y(x) = d (log £) /or any d e l .

I t can also be directly verified that the Hadamard and Hadamard type frac-tional integrals and derivatives (2.7.3)-(2.7.6) and (2.7.9)-(2.7.12) of the powerfunction x@ yield the same function, apart from a constant multiplication factor.

Propert y 2.25 Let fR(a) > 0, /? G C, and \i £ C.

(a) If m(P + n) > 0, then

{X+JP) (x) = (A* + P)-"*? and (VS+^t?) (x) = (» + p)axe. (2.7.21)

In particular, if £R(/?) > 0, then

(Jo+t13) {x) = p-ax0 and {V%+t0) (x) = pax^. (2.7.22)

(b) IffR(fi-fj) <0, then

{J?J0) (x) = (M - P)~ax0 and {Va_Jp) (x) = (fi - /3)a^. (2.7.23)

In particular, if 9K(/3) < 0, then

(J"t0) (x) = (-/3)-ax0 and (Vit0) (x) = {-p)ax0. (2.7.24)

The Hadamard fractional integrals (2.7.1)-(2.7.2) with 0 < a 0, 1 p ^ oo, and 0 < a < b < oo.

T/ien i/ie operators J"+ and J£+ are bounded in Lp(a, b) as follows:

WX+fWp KxWfWj, and \\J?_f\\p K2\\f\\p, (2.7.25)

where

Kl = _ / PW-W'dt and K2 = - 1 - / ^(«)-1e-*/î .l r (a)l Jo lr(<*)l Jo

(2.7.26)

/n particular,

and ||J6a_/|U ^ /fH/IU, (2.7.27)

Lemma 2.33 Let fR(a) > 0, 1 ^ p ^ oo, c e 1, and / i6C.

(a) If9\(/j,) > c, then the operator So+tlt is bounded in X?(IR+) os follows:

& J ! ? c]-OT^). (2.7.28)

In particular, if c < 0, i/ie operator JQ+ is bounded in Xf!(W+) by

\\JS+f\\x> KtWfWx* (K+ = l-c}-*^), (2.7.29)

(b) If^(fi) > -c, then the operator J? is bounded in X^(E+) by

WJ^fWx? Z #2-11/11*? (#2" = [«(/*) + c]-m(")). (2.7.30)

7n particular, if c > 0, i/te operator J2 is bounded in XP(R+) by

c~m^). (2.7.31)

Remark 2.16 Lemma 2.33 was established in Butzer et al. ([113], Theorems 4and 6).

The Hadamard and Hadamard type fractional integrals (2.7.1)-(2.7.6) satisfythe following semigroup property.

Property 2.26 Let m{a) > 0, SR(/3) > 0, and 1 g p ^ oo.

(a) 7/0 < a < b < oo, tAen, /or / G Lp(a,b),

JZ.J& = J:fl (c g 0) and JSLJ*_f = Jba_+0f (c Z 0). (2.7.32)

( b ) Jr / / j £ C , c £ l ,a = 0 and b = oo, then, for f G XP(R+),

(2.7.33)7n particular, when /J, = 0,

JoWo+Z = ô++/3/ (c < 0), Jf Jf / = Ji+& f (c > 0). (2.7.34)

Chapter 2 (§ 2.7) 115

The conditions for the existence of the Hadamard fractional derivatives (2.7.7)and (2.7.8) are given by the following assertion, proved for a > 0 in Kilbas [370].Here the space of functions (1.1.13) is used.

Lemma 2.34 Let 9t(a) ^ 0 and n - [9t(a)] + 1. If y(x) G AC£[a,b] (0 < a "+y and T>^_y exist almosteverywhere on [a, b] and can be represented in the forms

and

n~a~1

respectively.

In particular, when 0 < Sft(a) < 1, then, for y(x) G AC[a, b],

and

When y(x) is an arbitrarily-often differentiate function and /i G K, theHadamard type fractional integration (2.7.5) and the Hadamard type fractionaldifferentiation (2.7.11) can be expressed in terms of an infinite series involving thegeneralized Stirling numbers [see Butzer et al. [115]].

Compositions between the operators of fractional differentiation (2.7.7)-(2.7.12)and fractional integration (2.7.1)-(2.7.6) are given by the following property.

Property 2.27 Let a G C and /? G C be such that 9t(a) > 9t(/?) > 0.(a) / / 0 < a < b < oo and 1 ^ p ^ oo, then, for f G Lp(a, b),

Vi+J?+f = J^f and Vljba_f = Jb

a_~0f. (2.7.39)

In particular, if (3 = m £N, then

V™+J?+f = X~mf and V^_Jba_f = Jb

a_~mf. (2.7.40)

(b) // fj, G C, c G R, a = 0 and b = oo, then, for f G XZ(R+),


-c).

0 >

(2-

-c)(2-

(2-

(2-

7.41)

7.42)

7.43)

7.44)

In particular, if f3 = m £ N,

V™+^Joa+J = X~™f ( 9 % ) > c ); V ™ t i J

while, when /J, = 0 and m G N,

2?£+ JoV = v/oT'3/ (c < o); ^ - / = ^ - " ^ / (c > o).

V$.Jo+f = Jo+mf (c < 0) a«r f v - J- f = J-~mf (c > 0).

The Hadamard and Hadamard type fractional derivatives (2.7.7), (2.7.8) and(2.7.11), (2.7.12) are operators inverse to the corresponding fractional integrals(2.7.1), (2.7.2) and (2.7.5), (2.7.6).

Property 2.28 Let 91(a) > 0.(a) // 0 < a c) and V I ^ J = f (9t( ) > -c ). (2.7.46)

/n particular, if \x = 0, then

V%+X+f = / (c < 0) and Pf Jf / = / (c> 0). (2.7.47)

The following result yields the formula for the composition of the fractionaldifferentiation operator J£+ with the fractional integration operator £>"+

Theorem 2.3 Let d\{a) > 0, n = - [-9t(a)] and 0 < a < b < oo. ,4/so Jet(Jo

n)T

Q?/)(a;) be the Hadamard type fractional integral of the form (2.7.1). Ify(x) eL(o,6) and (Jo

n+ay)(a:) e i4C?[a,6

fc 1

/n particular, if a = n £ N and y(a;) € AC™[a, b], then

n-l

k=0

M(i og£)\ (2.7.49)

Remark 2.17 For a real a > 0, the relation (2.7.48) was proved in Kilbas andTitiuora ([405], Theorem 9).

Chapter 2 (§ 2.7) 117

We now consider the following spaces of functions of the forms (2.1.36) and(2.1.37):

J?+ (Lp) := {g := J?+<p and tp G Lp(a, b)} , (2.7.50)

X- (LP) :=

and

In particular,

and

{9

if

:= J?_<p and

rrOt / VP\

i := {9 J-

J0\ (XP) :

<p G Lp(a,

= {g~J,

and „

= {5^.5

*)}

GA

(0<

<p and

"P(E+

and 1

a < b < 00),

)} (o = 0;6 = oo).

(2.7.51)

(2.7.52)

(2.7.53)

(2.7.54)

J* {XD := {9 := J^V and ^ G XCP(E+)} , (2.7.55)

then the relation in (2.7.48) can be simplified. From Property 2.28 we thus derivethe following assertion.

Lemma 2.35 Let SR(a) > 0.

(a) // 0 < a 

0

K

1+

= 0 and b =

c) = j/(a;) (y

= 2/(z) (2/ G

_y){x) =y{x)

00, i/ien

G 7a (Xp} f

(y G J^ (Xcp);

) = y(x)

R(/i) > c)

M) > -c)

); c < 0)

O O ).

(2/ € J6Q_

(2.7.56)

(2.7.57)

(2.7.58)

(2.7.59)

(2.7.60)

Now we consider the properties of the Hadamard fractional integrals (2.7.1),(2.7.2) and the Hadamard fractional derivatives (2.7.7), (2.7.8) in the spacesC7,iog[a,6] and C£7[a,b] defined in (1.1.27) and (1.1.28), respectively. The ex-istence of the fractional integrals J"+f, Jb°Lf in the space C7,iog[a, b] and of thefractional derivatives X>"+y, £>&_y in the space C™log[a, b] are given by the followingassertion.


Lemma 2.36 Let 0 < a 9t(a) > 0, then the fractional integration operators J°+ and J£_

are bounded from C7ii0g[a, b] into C7_Qiiog[a, 6]:

X-/lk_ Q, l o g ^ *i||/|c,,log (2.7.61)

Q) r[9t(a)]|r(i-91(7)1 ,9 7 fi9.

In particular, J£+ and J"_ are bounded in C7)i0g[a, b].

If 9^(7) ^ 9t(a), ifcen i/ie fractional integration operators J£+ and J"_ arebounded from C7iiog[a, b] into C[a, b]:

\\Jaa+f\\c S *2||/||c,,IO8 and\\Jba_f\\c k2\\f\Cl,ios, (2.7.63)

where

* lC'i) - 91(7)1 ,9 7 f i / n

/n particular, 3£+ an $b- are bounded in C7)i0g[a, b].

(b) If9\(a) 0, n = [9l(a)] + 1 and j/(a;) G C 7[a,6], i/ien the fractionalderivatives V"+y and T>£_y exist on (a,b] and can be represented by (2.7.35) and(2.7.36), respectively. In particular, when 0 ^ 9l(a) < 1 (a ^ 0) and y(x) 6C7ji0g|a, b], then V%+y and V£_y are given by (2.7.37) and (2.7.38), respectively.

Proof. Estimates (2.7.61) and (2.7.63) are proved by using the definition (1.1.27)of the space Clt\Og[a, b\. Statement (b) of Lemma 2.36 is proved by using Lemma1.5.

Assertions (a)-(c) of Lemma 2.37 below for fractional calculus operators (2.7.1),(2.7.2) and (2.7.7), (2.7.8), being analogous to those in Properties 2.26 and 2.28and Theorem 2.3, are proved directly.

Lemma 2.37 Let D\(a) > 0, 9*(/3) > 0, and 0 ^ 91(7) < 1. The followingassertions are then true:

(a) // f(x) G Cy,iOg[a,b], then the first and the second relations in (2.7.32)hold at any point x € (a, b] and x £ [a, b), respectively. When f(x) € C[a, b], theserelations are valid at any point x £ [a, b].

(b) If f(x) £ C7ii0g[a, b], then the first and the second equalities in (2.7.33)hold at any point x £ [a,b] and x £ [a,b), respectively. When f(x) £ C[a,b], thenthese equalities are valid at any point x £ [a, b].

(c) Let 9t(a) > 9l(/3) > 0. If f{x) £ C7,iog[a,6], then the first and the secondrelations in (2.7.39) hold at any point x £ (a, b] and x £ [a, b), respectively. Whenf(x) £ C[a, b], then these relations are valid at any point x £ [a,b\. In particular,when f3 = k £ N and9\(a) > k, the relations in (2.7.40) are valid in their respectivecases.

Chapter 2 (§ 2.7) 119

(d) Let n = [«R(a)] + 1. Also let fn-a(x) = {Jâf){x) and gn-a(x) ={J£~ag){x) be the fractional integrals (2.7.1) and (2.7.2) of order n — a.

If fix) 6 Cy,\Og[a,b] and fn-a(x) G C£ [a,b], then the relation (2.7.48) holdsat any point x G (a, 6]. In particular, when 0 < 9t(a) < 1 and f\-a(x) G C}7[a, 6],i/ien

(Jaa+2?-+/) (s) = (*) - ^ ^ (log I ) " " 1 . (2.7.65)

/ / f(x) G C[o,6] and fn~a{x) G C£[a,6], tften (2.7.48) /io/ds at any pointx G [a,b]. In particular, if f(x) G C"[a,b], the relation (2.7.49) is valid at anypoint x G [a,b\.

To conclude this section, we give relations for the Mellin transform (1.4.23) ofthe Hadamard type fractional integrals (Jot^f){x), {J-^f){x) defined in (2.7.5),(2.7.6) and for the corresponding Hadamard type fractional derivatives {VQ+ M / ) ( X ) ,

(D-tpf)(x) given in (2.7.11), (2.7.12). First we consider the Mellin transforms ofthe Hadamard type fractional integrals.

Lemma 2.38 Let fR(a) > 0 and / j £ C. Also let a function f(x) be such that itsMellin transform (A4f)(s) exists for s G C.

(a) // m(/j, -s)>0, then

(MJoa+J)(s) = (/, - s)-a(Mf)(s). (2.7.66)

In particular, when 9t(s) < 0, then

(MJoa+f)(s) = (-s)-a(Mf)(S). (2.7.67)

(b) If<R(n + s) >0, then

{MJ»J){s) = (M + s)-a(Mf)(s). (2.7.68)

In particular, when 9l(s) > 0, then

(MJ«f)(s)=s-a(Mf)(s). (2.7.69)

Proof. Let a > 0 and s G K be such that /j - s > 0. Using (1.4.23) and (2.7.5),changing the order of integration, and making the changes of variable log (j) = uand w(/i — s) — r, we have

r (a)

/»O

/

./o


a Ta-le-TdT{Mf){s), (2.7.70)j0

which, in accordance with (1.5.1), yields (2.7.66). This formula remains true forcomplex a G C and //, s G C by analytic continuation. Relation (2.7.68) is provedsimilarly.

Remark 2.18 For a function f(x) belonging to the space Xj!(M.-), defined in(1.1.3)-(1.1.4), with 1 g p ^ 2 and c € M, the relations (2.7.66), (2.7.67) and(2.7.68), (2.7.69) were proved in Butzer et al. ([114], Theorems 1 and 3).

The Mellin transforms of the Hadamard type fractional derivatives are givenby the following result.

Lemma 2.39 Let fR(a) > 0 and / j £C. Also let a function y(x) be such that itsMellin transform (M.y)(s) exists for s G C.

(a) J/SR(ju- s) > 0 and {MV%+tly)(s) exists, then

^y)(s) = (M - s)a(My)(s). (2.7.71)

In particular, when £K(s) < 0, then

(MV%+y)(s) = (-8)a(My)(s). (2.7.72)

(b) If<R(n + s) > 0 and {MV tliy){s) exists, then

(MVa_tfly)(s) = (M + sr(My)(s). (2.7.73)

In particular, when 9t(s) > 0, then

(MV1y)(s) = sa(My)(s). (2.7.74)

Proof. Let n = [*R{a)] + 1. By (2.7.11) and (1.4.26), we have

(s) = (MM-^MM-Vy) (s). (2.7.75)

Applying (1.4.28), (1.4.35) , (1.4.28), and (2.7.66), with a replaced by n - a, wefind that

g+^y) (s) = {M5nM»{Jon-«y) (s - n)

(s-vi) = (M - s )n (M(Jon^y) (s) = (/x-5)a(My)( S),

(2.7.76)which yields (2.7.71). The formula in (2.7.73) is proved similarly.

Chapter 2 (§ 2.8) 121

2.8 Griinwald-Letnikov Fractional DerivativesIn this section we give the definition of the Griinwald-Letnikov fractional deriva-tives and give some of their properties as presented in the book by Samko et al.([729], Section 20). The above operation of fractional differentiation is based on ageneralization of the usual differentiation of a function y(x) of order n G N of theform

^ ) M M (2.8.1)hn

Here (A%y)(x) is a finite difference of order n G No of a function y(x) with a stepft G K and centered at the point i £ l defined in (2.8.5).

Property (2.8.1) is used to define a fractional derivative by directly replacingn G N in (2.8.1) by a > 0. For this, ft™ is replaced by ha, while the finite difference(A%y)(x) is replaced by the difference (A%y)(x) of a fractional order a G M. definedby the following infinite series:

" )y{x-kh) ( i , fceI ;a>0), (2.8.2)k=o v J

where ( , 1 are the binomial coefficients (1.5.22). When ft > 0, the difference\ J

(2.8.2) is called left-sided difference while for ft < 0 it is called a right-sideddifference .

The series in (2.8.2) converges absolutely and uniformly for each a > 0 and forevery bounded function y(x). The fractional difference (A%y)(x) has the followingproperties.

Property 2.29 If a > 0 and (3 > 0, then the semigroup property

(A%A0hy)(x) = (Aa

h+0y)(x) (2.8.3)

is valid for any bounded function f(x).

Propert y 2.30 If a > 0 and y(x) G Li(E), then the Fourier transform (1.3.1) ofA^ is given by

){x) = (1 - e"h)a {Ty){x). (2.8.4)

In particular, when a = n G N, then, in accordance with (1.5.23) and (1.5.24),(2.8.2) coincides with (2.8.5):

n , v

(AJJy)(a;) = ^ ( - l ) * f nk \y{x-kh) (x,h G R; n G N). (2.8.5)

fc=0 ^ ^

By (1.5.22) and (1.5.23), we can define the difference (A%y)(x) in (2.8.2) for a = 0by

I = f{x). (2.8.6)


Following (2.8.1), the left- and right-sided Griinwald-Letnikov derivatives y+(x)

and y_ (x) are defined by

£>(*):= l i m ^ M (a>0) (2.8.7)

and

yaJ(X) := luaQ{-^0^ (a > 0), (2.8.8)

respectively. These constructions coincide with the Marchaud fractional deriva-tives for y(x)GLp(R) (1 g p < oo) [see Samko et al. ([729], Theorem 20.4)].

The definition (2.8.2) of the fractional difference (A^y)(x) assumes that thefunction y(x) is given at least on the half-axis. For the function y(x) given on afinite interval [a,b], such a difference can be defined as follows by a continuationof y(x) as a vanishing function beyond [a, b]:

(A%y)(x) = (Agj/'Xz) := ( - l ) f c £ y*(x-kh) (x,h£R;a> 0), (2.8.9)k=o ^ ^

where

0,

I t is acceptable to rewrite the fractional difference (2.8.9) in terms of the functiony(x) itself, avoiding its continuation as a vanishing function, in the forms

(Ala+y)(x) := J2 (-!)* ( fc ) v(x ~ kh) ( ^ K ; / » 0 ; a> 0) (2.8.11)

and

(Alb_y)(x) := J2 (-1)" (ak)y(x + kh) (x £ K; /i > 0; a > 0). (2.8.12)

Then, by analogy with (2.8.7) and (2.8.8), the left- and right-sided Grunwald-Letnikov fractional derivatives of order a > 0 on a finite interval [a, b] are definedby

»- i^ := hSo —^ (2A13)

and

P ( a ! ) : = S — ^ (2814)

respectively. Such Griinwald-Letnikov fractional derivatives coincide with the Mar-chaud fractional derivatives ([729], Theorem 20.6)] and can be represented in thefollowing form:

Chapter 2 (§ 2.9) 123

and

Here the equalities are understood in the sense of a special convergence in thenorm of Lp(a,b).

From (2.8.15) and (2.8.16) we obtain the following formulas of the form (2.1.20):

I )=^^(*-a)-a ( 0<a< l ), (2.8.17)k=0

2.9 Partial and Mixed Fractional Integrals andFractional Derivatives

In this section we give the definitions and some properties of multidimensional par-tial and mixed fractional integrals and fractional derivatives presented in the bookby Samko et al. ([729], Sections 24.1). Such operations of fractional integrationand fractional differentiation in the n-dimensional Euclidean space Rn (n € N) arenatural generalizations of the corresponding one-dimensional fractional integralsand fractional derivatives, being taken with respect to one or several variables.

Forx = (a;i,--- , xn) £ W1 (n G N\{1} ) and a = (on,--- , an) G C"(n € N\{1}) , we use the following notations:

*°:=*r-* - IXa^d^-rK)); J- := A -; (2.9.1)

[ a , b] = [01,61] x x [an,bn]; a = {au--- , an) £ K " ; b = (& i , - - - , bn) e l " ,(2 .9 .2)

a nd by x > a we m e an x \ > a\, , x n > an.

On the basis of (2.1.1) and (2.1.2), the partial Riemann-Liouville fractionalintegrals of order au € C (yK(ctk) > 0) with respect to the fcth variable Xk aredefined by

-- , xn)dtk {xk > ak)

(2.9.3)


and

C(2.9.4)

respectively. These definitions are valid for functions /(x) = , xn) definedfor Xk > ak and Xk < bk, respectively. By analogy with the one-dimensional case,the fractional integrals (2.9.3) and (2.9.4) are called left- and right-sided partialRiemann-Liouville fractional integrals.

Next we define the mixed Riemann-Liouville fractional integrals of order a €Cn (m(a) > 0) as

(4V)(x) = (i?l+ i2L+/)(x) = £ f ^ L . (x > a)(2.9.5)

and

£(2.9.6)

The mixed fractional integrals can be applied with respect to a part of the variables,i.e., ah = 0 for some k = 1, , n. In this case we set a.u = 0 for k = m +1, , nand Qfc G C (!SH(ajfe) > 0) for k = 1, , m. Using the following notations,

x1 -(xi,--- , xm); x" = ( zm + i , - - - , xn); a' = (a1,--- , am); (x')a' = x"

such mixed Riemann-Liouville fractional integrals of complex order a' are definedby

and

Analogous to (2.9.3) and (2.9.4), the fractional integrals (2.9.5), (2.9.7) and (2.9.6),(2.9.8) are called left- and right-sided mixed Riemann-Liouville fractional integrals.

The left- and right-sided partial Riemann-Liouville fractional derivatives oforder ctu € C (D\(ak) ^ 0) with respect to the fcth variable x are defined by

AJ]Lk fOXkJ Ja

, xn)dtk{XkT(Lk - ak) \dxkj Jak (xk - i fc)°*

(2.9.9)

Chapter 2 (§ 2.9) 125

and

dxj JXk

,--- , xn)dtk: - T(Lk - ak) V dxj JXk (tk - xk)°>

(2.9.10)where Lk = [<R(afc)] + 1.

In particular, when 0 < ak < 1, the relations (2.1.9) and (2.1.10) take thefollowing forms:

Jah

)(2.9.11)

,---, xk-i,tk,xk+i, , xn)dtkf-T(l-ak)dxkJah {tk-xk)°> {^<ok).

(2.9.12)If ak = Lk G No, (2.1.9) and (2.1.10) yield the usual partial derivatives as follows:

^ (JL^j(2.9.13)

The left- and right-sided mixed Riemann-Liouville fractional derivatives of or-der ak € C ($H(ajfc) ^ 0), corresponding to the mixed fractional integrals (2.9.5)and (2.9.6), are defined as

J \dxn) T(L-a)Jai Jan(2.9.14)

and

(2.9.15)where L = {Lu , Ln) and Lfc = [<H(afc)] + 1 (fe = 1, , n).

In particular, when 0 < a < 1, the relations (2.9.14) and (2.9.15) take thefollowing forms:


and

If a = L e Ng- := No x x No (n times), then (2.9.14) and (2.9.15) are reducedto

y(x) and (£>£y)(x) = (-l)l ml (J-^j j,(x). (2.9.18)

The left- and right-sided mixed Riemann-Liouville fractional derivatives of or-der cik € C (9t(afc) ^ 0), corresponding to the mixed fractional integrals (2.9.7)and (2.9.8), are given by

(2.9.19)with (x' > a') and

(2.9.20)where (x' < b'), L' = (Li , , Lm) and Lk = [<R(ak)] + 1 (k = 1, , m).

In particular, when 0 < a' < 1, the formulas (2.9.19) and (2.9.20) are reducedto the following forms:

= _d d 1 fXl r™ y(t',x")dt' ( x ' > a ' )

(2.9.21)and

Chapter 2 (§ 2 . 1 0 ) 1 2 7

^ f»~y(t>,x")dt'JXl JXm ( t ' - x ' ) a ' V ''

1 d_ 1dXl dxmT(l-ai)---r{l-am)JXl

(2.9.22)If a' = L '£No

m, then (2.9.19) and (2.9.20) are reduced to

and

^ — — ( A ) " (^)L%(x), (2.9.24)

respectively. We define above the partial and mixed Riemann-Liouville fractionalintegrals and fractional derivatives on the finite domain

[a,b] = [o!,6i] x x [an,bn].

The partial and mixed Liouville fractional derivatives of order a £ C on the octantK+...+ = { t = (<i, , tn) G Kn : h ^ 0, , tn ^ 0} are similarly defined. Forthis we must let ak = 0 and 6fc = oo in the formulas (2.9.3)-(2.9.8), (2.9.9)-(2.9.12),(2.9.14)-(2.9.17), and (2.9.19)-(2.9.22). By having ak = - co and bk = oo in theserelations, we define partial and mixed Liouville fractional derivatives of order a G Con the whole Euclidean space M.n. Most of the properties of the one-dimensionalRiemann-Liouville and Liouvill e fractional integrals and fractional derivatives, pre-sented in Sections 2.1-2.3, can be extended to these multidimensional cases. [See,in this regard, Samko et al. ([729], Sections 24.1-24.2)].

2.10 Riesz Fractional Integro-Differentiation

In this section we give the definitions and some properties of multidimensionalRiesz fractional integrals and fractional derivatives presented in the book by Samkoet al. ([729], Sections 25 and 26). Such operations of fractional integration andfractional differentiation in the n-dimensional Euclidean space E" (n G N) arefractional powers (—A)Q/2 of the Laplace operator (1.3.31). For a G C\{0} and"sufficiently good" functions / (x) = f(xi, , xn), such fractional operations aredefined in terms of the Fourier transform (1.3.22) by

9t(a) > 0, (2 10 11<R(a)<0. ^ - i U - i j

The operations Ia and DQ defined in (2.10.1) for 9t(a) > 0 are called the Rieszfractional integration and the Riesz fractional differentiation, respectively.

The Riesz fractional integration / " is realized in the form of the Riesz potentialdefined as the Fourier convolution (1.3.33) of the form

(/" / ) (x) = / fca(x " t ) / ( t )dt (3t(a) > 0). (2.10.2)


Here a function ka(x), called the Riesz kernel, is given by

i ( \-x\a~n rv — n =k 0 2 4

and the constant 7n(a) has the form

^ M ._ /7 n l ' " \

._ / 2«W2r[a/2][r((n - a)/2)]-\ a - n # 0,2,4, ,\ ( ) / / [a/2], a - n = 0,2,4, .

(2.10.4)

By the Fourier convolution theorem (see Theorem 1.4 in Section 1.3), we havethe following property.

Property 2.31 // !EH(a) > 0, then the Fourier transform of the Riesz potential(2.10.2) is given by

i . (2-10-5)

This formula is true for a function f belonging to Lizorkin's space $ defined inSection 1.2.

Corollary 2.5 Ifd\{a) > 2, then the following formula holds:

AIaf = -Ia~2f («H(a) > 2; / G $). (2.10.6)

For functions / from the Lizorkin space $, presented in (1.2.17), the semigroupproperty is also valid [see Samko et al. ([729], Theorem 25.1)].

Property 2.32 The Lizorkin space $ is invariant with respect to the Riesz po-tential Ia. Moreover, /"($) = $, and

Q. / G $) . (2.10.7)

When a — n 0,2,4, , the Riesz potential (2.10.2) takes the following form:

a) > 0; a "n ° ' 2 ' 4 ' ' ' ' ) - (2-10'8)When a > 0, then the following assertion holds.

Property 2.33 7/0 < a < n and 1 < p < n/a, then the Riesz potential (Iaf) (x)is defined for f G Lp{R

n).

The next statement, known as the Sobolev theorem, is a generalization of theHardy-Littlewood theorem given in Lemmas 2.10 and 2.15 [see Samko et al. ([729],Theorem 25.2)].

Chapter 2 (§ 2.10) 129

Theorem 2.4 Let a > 0, l ^ p ^ oo and 1 q ^ oo. The operator Ia is boundedfrom Lp(R

n) into Lq(Rn) if, and only if,

0 < a <n, 1 0, 1 < p < oo, 1 < r < oo, and -y,/! G ffi be such that

ap-n<1<n{p-l), - - - - - and ^ ^ = 1-^- - a. (2.10.11)p n r p r p

Then the operator Ia is bounded from Lp (Rn, |x|7) into Lq (R

n, |x|M) as follows:

1/r (r V/p

/(2.10.12)

/

where the constant c > 0 depends on a,n,p, r, 7, and /J,.

Let Rn be a compactification of M™ by an infinite point, that is, W™ = En U{°°} >and let p(x) be a nonnegative function o n i " . If 0 < a < 1, then the Riesz po-tential (2.10.8) is defined for /(x) belonging to a weighted space of Holderianfunctions Hx (M.n,p):

Hx (Rn,p) := {/(x ) : p(x)/(x) G Hx (]R») (0 < A < 1)} , (2.10.13)

wherex (]RH

: /(x) e C (R«) , |/(x + h)

(2.10.14)with c > 0. The following assertion is valid [see Samko et al. ([729], Theorem25.5)].

Theorem 2.6 Let 0 < a < 1, 0 < A < 1 and a + A < 1. Then the operator Ia isisomorphic from Hx (w1, (1 + |x |2)(n+a)/2) onto Hx+a (w1, (1 + ^


When 0 < a < n and 1 0) (2.10.15)

and

(Wtf)(x):= [ W(y,t)f{x-y)dy (t > 0), (2.10.16)JR"

respectively. Here P(x, t) is the Poisson kernel:

( | x |2 +%(n+1)/2 and a>n = « - W T ( ^ ) , (2.10.17)

while W(x,t) is the Gauss-Weierstrass kernel:

W(x,t) := (47rf)-"/2e-lxl2/4* . (2.10.18)

The following result holds [see Samko et al. ([729], Theorem 25.6)].

Theorem 2.7 If 0 < a < n and 1 < p < n/a, then the Riesz potential (7"/)(x)with / (x) € Lp(E.n) has the following representations:

(7Q/)(x) = - L r r-\Ptf){x)dt (2.10.19)r (a) Jo

and

/ ^ ^ ' d i . ( 2 .10 .20)

Moreover, the compositional relations of the Poisson and Gauss-Weierstrasstransforms with the Riesz potential are expressed in terms of the Liouville fractionalintegration operator (2.3.2) as follows:

(Pi /a/)(x) = {II (PT/)(x))) (t) (2.10.21)

and(WtI

af)W = (l-/2 (WV/)(x))) (t), (2.10.22)

where the operators I" and I_ are applied with respect to r > 0.

For a > 0, the Riesz fractional derivative D" in (2.10.1) is realized in the formof the hypersingular integral defined by

Here (A[y)(x ) is a finite difference of order I of a function y(x) with a vector stept e l " and centered at the point x £ l " :

(Aiy)(x) = {E- rt)7(x) := J^(~l)k ( [) I/(x - fct), (2.10.24)

Chapter 2 (§ 2.10) 131

coinciding with that in (2.8.5) for i , t £ E, while dn(l,a) is a constant defined by

o-a l+n/2 A,(n\

^ ) = r ( l + f ) r m 4 Jr (2-10.25)where

Note that the hypersingular integral (Day)(x) does not depend on the choiceof I (I > a). Such a construction is also called the Riesz fractional derivative oforder a > 0 in the sense that

{TT>ay){x) = \x\a{Ty){x) (a > 0) (2.10.27)

for "sufficiently good" functions y(x). In particular, this relation is valid for /belonging to the Lizorkin space $. Equality (2.10.27) is also valid for differentiablefunctions y(x). The following property holds [see Samko et al. ([729], Lemma25.3)].

Property 2.34 If y(x) belongs to the space C^°(Wn) of infinitely differentiablefunctions y(x) on M™ with a compact support , then the Fourier transform of(Daj/)(x) is given by

(FT>ay)(x) = \x\a(Fy)(x) (a > 0). (2.10.28)

The conditions for the existence of Da/ ) ( x ) are given by the following assertion[see Samko et al. ([729], Section 26.2)].

Lemma 2.40 Let a > 0 and [a] be the integer part of a. Also let a func-tion y(x) be bounded together with its derivatives (Dky)(x), (\k\ = [a] + 1).Then the hypersingular integral (Da/ ) (x) in (2.10.23) is absolutely convergent. IfI > 2[a/2], then this integral is only conditionally convergent.

The Riesz fractional derivative (2.10.23) yields an operator inverse to the Rieszpotential (2.10.2).

Property 2.35 The formula

D a / a / = / (a > 0) (2.10.29)

holds for "sufficiently good" functions f; in particular, for f belonging to the Li-zorkin space $.

Moreover, the inversion (2.10.29) is also valid for the Riesz potential (2.10.2) inthe frame of Lp-spaces: / (x) 6 Lp(M.n) for 1 ^ p < n/a. Here the Riesz fractionalderivative DQ is understood to be conditionally convergent in the sense that

Vay= lim D"y, (2.10.30)


where D"y is the truncated hypersingular integral defined by

(D°y)(x) := * / ^Jn+y dt (f > a; a > 0; £ > 0), (2.10.31)

and the limi t is taken in the norm of the space Lp(Rn).

The following result holds [see Samko et al. ([729], Theorem 26.3)].

Theorem 2.8 LetO<a<n and let y = Iaf with / (x) £ Lp(Rn) (1 ^ p < n/a).

Thenf(x) = (DQj/)(x) , (2.10.32)

where (Daj/)(x ) is understood in the sense of (2.10.30), with the limit being takenin the norm of the space Lp(R

n).

Corollar y 2.6 7/0 < a < n and /(x) G Lp{Rn) (1 p < n/a), then

/(x) = (Dar7)(x), (2.10.33)

where (DaIaf)(x) is understood in the sense of (2.10.30), and

(Da7a/)(x) = H m (D«/«/)(x), (2.10.34)

with the limit being taken in the norm of the space Lp(Rn).

2.11 Comments and ObservationsA . In the preceding sections of this chapter, we have not dealt with the fractional

calculus operators which are based essentially upon the Cauchy-Goursat In-tegral Formula. In recent years, these fractional calculus operators havebeen used rather extensively in obtaining particular solutions to numerousfamilies of homogeneous (as well as nonhomogeneous) linear ordinary andpartial differential equations which are associated, among others, with manyof the celebrated equations of mathematical physics such as (for example)the Gauss hypergeometric equation (1.6.13) and the Bessel equation (1.7.9).In the cases of (ordinary as well as partial) differential equations of higherorders, which have stemmed naturally from the Gauss hypergeometric equa-tion (1.6.13), the Bessel equation (1.7.9), and their many relatives and ex-tensions, there have been several seemingly independent attempts to presenta remarkably large number of widely-scattered results in a unified manner(see, for details, the work of Tu et al.[829] and the references cited by them;see also numerous other recent works on this subject, which are cited in theBibliography), item [B. ] From few years ago many interesting applicationsof the so called Fractional Fourier Transform (FFT) have been published.Under the name FFT are known several different definitions of generaliza-tion of the ordinary Fourier transform. This fractional Transform have beenwidely applied mainly in Optics and Signal Processing Theory. The excellentbook by Ozaktas et al. [658] is a good introduction in this matter (see, also,[657], [664], [119], [784], [904], [904], [110], and [111])

Chapter 2 (§ 2.11) 133

C. In many recent investigations (especially those using fractional calculus inspecific problems of applied mathematics and mathematical physics), use isfrequently made of such analytical tools as the Laplace and Fourier trans-forms (see Sections 1.3 and 1.4), and also of such generalized functions as(for example) the Dirac delta function 6(t), which occur naturally in the dif-ferential equations considered. In situations like these, it is important fromthe viewpoint of mathematical analysis that we choose the proper distribu-tional test-function spaces which can conveniently validate not only the useof the Laplace, Fourier, and other integral transforms as analytical tools,but also the presence of the generalized functions in the differential equationmodeling the physical problem (see also Section 1.2).

D. A number of such not-so-common functions as the Weierstrass function areknown to emerge as solutions of fractional differential equations. Here wehave chosen not to develop a detailed investigation of these functions.

E. The recent monograph by Martinez and Sanz [558] provides an excellentsource-book for the theory and applications of fractional (especially nega-tive and imaginary) powers of many different families of non-negative lin-ear operators as well as such differential operators as (for example) thoseof Riemann-Liouville and Weyl considered in this chapter, (see, also, [60],[437], [253], [719], [876]).

F. Control theory has been discipline where many mathematical ideas and meth-ods have melt to produce a rich crossing point of Engineering and Mathe-matics. Control theory is certainly one of the most interdisciplinary areas ofresearch and this field of Mathematics arises in most modern applications, forexample, in the control of Autonomous and Intelligent Robotic Vehicles (see,[258], [691], and [814]). In particular, during the last years the Fractionalcontrol models have produced very good solutions to so many interestingcontrol problems with applications in many branch of the Engineering.

Here we do not include explicit examples of such applications of the fractionalmodels, but we will included some references where the reader can find manyinteresting theoretical and applied results (see, for example, [653], [815],[204],[678], [565], [562], [612], [667], [683], [668], [816], [685], [685], [11], [357],and [598])


Chapter 3

ORDINARY FRACTIONALDIFFERENTIALEQUATIONS. EXISTENCEAND UNIQUENESSTHEOREMS

This chapter is devoted to proving the existence and uniqueness of solutions toCauchy type problems for ordinary differential equations of fractional order ona finite interval of the real axis in spaces of summable functions and continuousfunctions. Nonlinear and linear fractional differential equations in one-dimensionaland vectorial cases are considered. The corresponding results for the Cauchyproblem for ordinary differential equations are presented.

3.1 Introduction and a Brief Overview of Results

In this section we give a brief overview of the results of existence and uniquenesstheorems for differential equations of fractional order on a finite interval of thereal axis.

Most of the investigations in this field involve the existence and uniqueness ofsolutions to fractional differential equations with the Riemann-Liouville fractionalderivative (D%+y)(x) defined for (fR(a) > 0) by (2.1.5). The "model" nonlineardifferential equation of fractional order a (9l(a) > 0) on a finite interval [a, b] ofthe real axis R = (—oo, oo) has the form

(D5+y)(x) = f[x,y(x)} (<R(a) > 0; x > a), (3.1.1)

135


with initial conditions

( D ^ k y ) ( a + ) = b k , b k e C ( k = l , - - - , n ) , (3.1.2)

where n = fft(a) + 1 for a g N and a = n for a G N. The notation (D"+ ky)(a+)means that the limit is taken at almost all points of the right-sided neighborhood(a, a + e) (e > 0) of a as follows:

(D^ky)(a+) = xl\m+(D^ky)(x) (1 g * g n - 1), (3.1.3)

(D^ny)(a+) = mn+(Iây)(x) (a ? n); (D°a+y)(a+) = y(a) (a = n), (3.1.4)

where I"+ is the Riemann-Liouville fractional integration operator of order a G Cdefined by (2.1.1).

In particular, if a = n 6 N, then, in accordance with (2.1.7) and (3.1.4), theproblem in (3.1.1)-(3.1.2) is reduced to the usual Cauchy problem for the ordinarydifferential equation of order n 6 N:

y(n\x)=f[x,y(x)}, y(n-*)(o)=6fc) bk G C (k = 1, ,n). (3.1.5)

The problem (3.1.1)-(3.1.2) is therefore called, by analogy, a Cauchy type problem[see Samko et al.([729], Section 42)].

When 0 < 9\(a) < 1, the problem (3.1.1)-(3.1.2) takes the form

(DZ+y)(x) = f[x,y(*)], (£?*»)(«+) = & (b e Q (3.1.6)

and this problem can be rewritten as the weighted Cauchy type problem

(DZ+y)(x) = f[x,y(x)}, lim (x - af-ay{x) = c (c G C). (3.1.7)x—>a-\-

We discuss investigations of the above problems following their historical chronol-ogy. Essentially, they were based on reducing problem (3.1.1)-(3.1.2) to the fol-lowing nonlinear Volterra integral equation of the second kind:

Pitcher and Sewell [675] in 1938, first considered the nonlinear fractional dif-ferential equation (3.1.1) with 0 < a < 1, provided that f(x,y) is bounded in aspecial region G lying i n l x l and satisfies the Lipschitz condition with respecttoy:

|/0r,2,0 -f(x,y2)\Â\yi-y2\, (3.1.9)

where the constant A > 0 does not depend on x. They proved the existence ofthe continuous solution y(x) for the corresponding nonlinear integral equation ofthe form (3.1.8) with 0 < a < 1, n = 1, and b\ = 0. But the main result givenin Pitcher and Sewell ([675], Theorem 4.2) for the fractional differential equation

Chapter 3 (§ 3.1) 137

(3.1.1) with 0 < a < 1 was not correct because they used the compositional relationI" +D"+y — y instead of the correct one given in (2.1.40) to reduce the differentialequation to an integral equation. However, the work of Pitcher and Sewell [675]did contain the idea of reducing the solution of the fractional differential equation(3.1.1) to that of a Volterra integral equation (3.1.8).

Barrett [68] in 1954 first considered the Cauchy type problem for the lineardifferential equation

(D%+y)(x) - \y(x) = f(x) (n - 1 9t(a) < n, a / n - 1), (3.1.10)

where n € N is the smallest integer such that n > y\(a) > 0, with initial conditions(3.1.2). He proved in Barret ([68], Theorem 2.1) that, if f(x) belongs to L(a,b),then such a problem has the unique solution y(x) in some subspace of L(a, b) givenby

y{x) = £ bkxa-kEa,a_k+1 (X(x - a)a) + f\x - t)a-xEa,a [\(x - t)a] f(t)dt,

(3.1.11)where Ea,a-k+i{z) (k = I,--- , n) are the Mittag-LefHer functions defined by(1.8.17). Barrett's arguments were based on the formula (2.1.39) for the productI"+D" +f. By applying this formula, he implicitly used the method of reducingthe Cauchy type problem (3.1.10),(3.1.2) to a linear Volterra integral equation ofthe second kind of the form (3.1.8) with f[x,y(t)] = \y{t) + f(t) and then appliedthe method of successive approximations to derive the solution (3.1.11).

Al-Bassam [17] first considered the Cauchy type problem (3.1.6) for a real0< a ^ 1

{DZ+y)(x) = f[x,y{xj\ (0 < a g 1), (3.1.12)

(I la+

ay)(a+) = b, 6eE, (3.1.13)

in the space of continuous functions C[a, b], provided that f(x, y) is a real-valuedcontinuous function in a domain G c i x l such that sup(x y eG \f(x,y)\ ^ ooand that it satisfies the Lipschitz condition (3.1.9). Applying the operator I°+ toboth sides of (3.1.10), and using the relation (2.1.39) and the initial conditions(3.1.11), he reduced the problem (3.1.12)-(3.1.13) to the Volterra nonlinear integralequation (3.1.8) with n — 1:

Using the method of successive approximations, Al-Bassam [17] established theexistence of the continuous solution y(x) to the equation (3.1.14). Moreover, hewas probably the first to indicate that the method of contractive mapping couldbe applied to prove the uniqueness of the solution y(x) to (3.1.14), and formallypresented such a proof. Al-Bassam [17] also indicated - but did not prove - theequivalence of the Cauchy type problem (3.1.12)-(3.1.13) and the integral equation(3.1.14), and therefore his results on the existence and uniqueness of the continuous


solution y(x), formulated in Al-Bassam ([17], Theorem 1), could be true only forthe integral equation (3.1.14). We also note that the conditions suggested by Al-Bassam [17] are not suitable for solving the Cauchy type problem (3.1.12)-(3.1.13)in some simple cases; for example, for f[x,y(x)} = y(x).

The same remarks apply to the existence and uniqueness results formulatedwithout proof in Al-Bassam ([17], Theorems 2, 4, 5, 6) for more general Cauchytype problems of the form (3.1.1)-(3.1.2) for a real a > 0:

{D2+y)(x) = f[x, y(x)] {n-Ka^n; n G N), (3.1.15)

( D 2 + k y ) ( a + ) = b k , b k e R ( k = l , 2 , - - - , n ) , (3.1.16)

where the corresponding Volterra equation is given by (3.1.8) for the system ofequations (3.1.12), and for more general nonlinear and linear fractional differentialequations than (3.1.12). The above and some other results were presented inAl-Bassam ([18]-[21]) [see also Samko et al. ([729], Section 42.1)].

Dzhrbashyan and Nersesyan [217] studied the linear differential equation offractional order

n- l{Day){x) := (D^y)(x) + £ ak(x){D°—-*y)(x) + an(x)y(x) = f(x), (3.1.17)

with sequential fractional derivatives (Day)(x) and (Da'n-k-1y)(x) (k = 0,1, ,n — 1) defined in terms of the Riemann-Liouville fractional derivatives (2.1.5) by

D"" = D^D^-1 = 1, , n), £>CT° = D^T1, (3.1.18)

where

k

ak =Y,aJ - 1 (k = 0'1 ' - -- . n) ; 0 < a J = * (i = 0,1, , n) (3.1.19)j=o

(ak = ak- <Jk-i (fc = 1, , n); a0 = a0 + 1). (3.1.20)

Dzhrbashyan and Nersesyan [217] proved that, for a0 > 1 - an, the Cauchy typeproblem

(D°y)(x) = f{x), {Da*y){x)\x=0 = bk (k = 0,1, , n - 1) (3.1.21)

has a unique continuous solution y(x) on an interval [0,d], provided that thefunctions pk(x) (0 k n — 1) and f(x) satisfy some additional conditions. Inparticular, when pk(x) =0 (k = 0,1, , n), they obtained the explicit solution

VW = E fTTT^ + FT-T !\* ~ ty^fm (3.1.22)] ~ r ( l + ak) T(an) Ja

to the Cauchy type problem

(D^y){x) = f{x), ( C t / ) W |I = o = bk (fc = 0,1, , n - 1). (3.1.23)

Chapter 3 (§ 3.1) 139

The Cauchy type problems (3.1.1)-(3.1.2), (3.1.12)-(3.1.13), and (3.1.15)-(3.1.16) were studied by Al-Abedeen [12], Al-Abedeen and Arora [13], Arora andAlshamani [37], Tazali [811], Tazali and Karim [812], Hadid and Alshamani [321],Leskovskij [477], Semenchuk [757], Luszczki and Rzepecki [511], El-Sayed [222]-[224] and [226]), El-Sayed and Ibrahim [232], El-Sayed and Gafar [231]-[230], Ha-did [320], and others. But the above investigations were not complete, however.Most researchers have obtained results not for the initial value problems, but forthe corresponding Volterra integral equations. Some authors considered only par-ticular cases. Moreover, some of the results obtained contained errors in the proofof the equivalence of initial value problems and the Volterra integral equations andin the proof of the uniqueness theorem. In this regard, see the survey paper byKilbas and Trujill o ([407], Sections 4 and 5).

Delbosco and Rodino [164] considered the nonlinear Cauchy problem

(D$+y)(x) = f(x,y(x)); yik\0) = yk G R (k = 0,1, , [a]), (3.1.24)

with 0 ^ x ^ T, a > 0, and f(x,y) a continuous function on [0,1] x R. Theyproved the equivalence of this problem to the corresponding Volterra equation andused Schauder's fixed point theorem to prove that the equation considered has atleast a continuous solution y(x) denned on [0,5} for a suitable 0 < 8 1, providedthat xaf(x, y) is continuous on [0, l ] x l for some a (0 a < a < 1). Applying thecontractive mapping method, Delbosco and Rodino ([164], Theorem 3.4) showedthat if, additionally, f[x,y(x)} satisfies the following Lipschitz type condition,

\f[x,y(*)]-f[x,Y(x)]\ Z ^\y(x)-Y(x)\, (3.1.25)

then the equation in (3.1.24) has a unique solution y(x) G C[0,1]. They also provedthat if f[x,y(x)] = f[y(x)] is such that /(0) = 0 and the Lipschitz condition (3.1.9)holds, then the Cauchy problem

(D%+y)(x) = f[y(x)], y(a) = b (0 < a < 1; a > 0; b G R) (3.1.26)

and the weighted Cauchy type problem

{DS+y)(x) = f(y(x)), Yiinxl-ay(x) =c (0 < a < 1; c G M) (3.1.27)

have a unique solution y(x) such that x1~ay(x) G C[0, h] for any h > 0.

Hayek et al. [335] investigated the following Cauchy type problem for a systemof linear differential equations:

(D%+y)(x) = f(x,y(x)), y{a) = b (0 < a £ 1; a > 0; b G 1") (3.1.28)

with a real-valued vector function y(x), provided that f(x,y) is continuous andLipschitzian with respect to y. Applying the method of contractive mapping de-fined on a complete metric space, they proved the existence and uniqueness ofa continuous solution y(x) to this problem. In particular, they obtained such aresult for a system of linear differential equations:

(D8+y)(s:) = A{x)y{x) + B(x), y{a) =b (0 < a <, 1; a > 0; b G Rn) (3.1.29)


with continuous matrices A(x) and B(x).Using the approach of Al-Bassam [17] and Dzhrbashyan and Nersesyan [217],

Podlubny ([682], Section 3.2) considered the problem of the existence and unique-ness for the nonlinear Cauchy type problem of the form

(D°y)(x) = f(x,y(x)), (D°*y)(x)\x=0 = bk e C (* = 0,1,- , n - 1) (3.1.30)

with the sequential fractional derivatives defined in (3.1.17)-(3.1.18) in the space ofcontinuous functions. Podlubny [682] tried to prove the equivalence of the problem(3.1.30) and the corresponding Volterra nonlinear integral equation, but he did notgive sufficient conditions for such an equivalence.

The Cauchy type problem (3.1.1)-(3.1.2) with complex a £ C (9%) > 0) ona finite interval [a, b] of the real axis E was studied by Kilbas et al. ([376]-[374])in the space of summable functions L(a,b). The equivalence of this problem andthe nonlinear Volterra integral equation (3.1.8) was established. The existenceand uniqueness of the solution y(x) to such a problem was proved by using themethod of successive expansions. In particular, the corresponding results wereproved for the Cauchy problem (3.1.5) and for the Cauchy type problems (3.1.6)and (3.1.7). The results obtained were extended to the system of such problemsby Bonilla et al. [95]. The conditions for a unique solution y{x) to the Cauchytype problem (3.1.1)-(3.1.2), in particular to the Cauchy problem (3.1.5) and toCauchy type problems (3.1.6) and (3.1.7), in the weighted space of continuousfunctions Cn-a[a,b] and C"_a[a,b], denned in (1.1.22) and, [see, (3.3.1)], wereestablished by Kilbas et al. [375] and by Kilbas et al. [389]. These results wereextended by Kilbas and Marzan [378] to the Cauchy type problem for nonlinearfractional differential equations of order a S C (5H(a) > 0), more general thanthose in (3.1.1) and (3.1.17):

(DZ+y)(x) = f[x,y(x),(DZiv)(x),(D%v)(*),--- , {D^y){x)] (3.1.31)where 0 < 9t(ai) < SR(a2) < < 9t(am_i) < £H(a) and m ^ 2.

Diethelm ([173], Theorem 4.1) proved the uniqueness and existence of a localcontinuous solution C(0, h] to the Cauchy type problem (3.1.15)-(3.1.16), providedthat / is a continuous and Lipschitzian function. He used the above approach,based on the equivalence of the problem (3.1.15)-(3.1.16) to the Volterra integralequation (3.1.8), and applied a variant of Banach's fixed point theorem presentedin Section 1.13, Theorem 1.10.

The above investigations were devoted to fractional differential equations withthe Riemann-Liouville fractional derivative D°+y on a finite interval [a, b] of thereal axis R. Such equations with the Caputo fractional derivative cD"+y, definedin Section 2.4, are not studied extensively. Gorenflo and Mainardi [303] appliedthe Laplace transform to solve the fractional differential equation

(cD%+y)(x) - Xy(x) = f(x) (x > 0; a > 0; A > 0), (3.1.32)

with the Caputo fractional derivative (2.4.1) of order a > 0 and with the initialconditions

=bk{k = 0,l,---n-l; n-Ka^n; n e N). (3.1.33)

Chapter 3 (§ 3.1) 141

They discussed the key role of the Mittag-Leffler function (1.8.17) for the cases1 < a < 2 and 2 < a < 3. In this regard, see also the papers by Gorenflo andMainardi [304], Gorenflo and Rutman [311], and Gorenflo et al. [310]. Luchko andGorenflo [507] used the operational method to prove that the Cauchy problem(3.1.32)-(3.1.33) has the unique solution

n - l -x

y(x) = V bkxkEa k+i (AzQ) + / (x - t)a-lEa,a [\{x - t)a] f(t)dt, (3.1.34)

k=o

in terms of the Mittag-Lefler functions (1.8.17) in a special space of functions onthe half-axis R+. They also obtained the explicit solution to the Cauchy problemfor the more general fractional differential equation

m

(cD%+y)(x) -Y,Ck(CD^y)(x) = f(x) (a > a, > > am Z 0) (3.1.35)

via certain multivariate Mittag-Leffler functions.

Seredynska and Hanyga [758] considered the equation

y " ( x ) + k (CD%+) (x) + F(u) (Q^t^T; 0 < a 2) (3.1.36)

with constant k, and proved the existence and the uniqueness of a solution y(x, a) €C2[0,T] for the initial Cauchy conditions y(0) = yo, 2/'(0) = VQ- In this regard seealso Seredynska and Hanyga [759].

Diethelm and Ford [177] investigated the Cauchy problem for the nonlineardifferential equation of order a > 0

(cDg+y)(x)= f[x,y(x)} ( 0 ^ z ^ 6 < o o ), (3.1.37)

with initial conditions (3.1.33). They proved the uniqueness and existence of a localcontinuous solution y(x) € C[0, h] to this problem for continuous and Lipschitzian/ ; in this regard, see also Diethelm ([173], Theorems 5.4-5.5). The dependence ofthis solution y(x) on the order a, on the initial data (3.1.33), and on the function/ was investigated. Applications were given to present numerical schemes for thesolution y(x) to the simplest linear problem with 0 < a < 1:

(°D%+y){x) = Xy(x) + f(x) (0 ^ x b; A < 0), y(0) = b e R. (3.1.38)

Kilbas and Marzan ([380] and [382]) investigated the Cauchy problem of theform (3.1.37)-(3.1.33) with the Caputo derivative (2.4.1) of complex order a € C(9t(a) > 0):

(cD%+y)(x) = f[x,y(x)} ( a ^ x ^ b ) , (3.1.39)

y{k) (0) = bk G C (k = 0,1, , n - 1) (3.1.40)

where n = pH(a)] + 1 for a ^ N and n = a for a G N, on a finite interval [a, b]of M. They proved the equivalence of (3.1.38)-(3.1.37) and the Volterra integralequation of second kind of the form (3.1.8)


in the space C™"1^, b] and applied this result to establish conditions for a uniquesolution y(x) G C " " 1 ^ , ^ to the Cauchy problem (3.1.39)-(3.1.40).

Daftardar-Gejji and Babakhani [153] have studied the existence, uniqueness,and stability of solutions for the following system of fractional differential equa-tions:

{cD$+Y)(x)=AY(x), Y{0) = Yo, (3.1.42)

where £>Q+i s the Caputo derivative of order 0 < a < 1, Y(x) = , yn{%)}T

and A is n x n real matrix, and investigated the dependence of solutions on theinitial data for the corresponding nonlinear system of the form (3.1.42), in whichAY(x) is replaced by f[x,Y(x)]. In particular, they obtained the unique solu-tion of (3.1.42) in the form Y{x) = Ea {Axa), where Ea(A) is a a Mittag-Lefflerfunction (1.8.1) with matrix arguments:

k=0

(see, also [91], [147], [174], [513], [148], [179], [190], [184], and [187])Using the Adomian decomposition method, well-known for ordinary differential

equations [see, for example, the following papers [3],[4] and [318]] Daftardar-Gejjiand Jafari [154] derived analytical solutions of a more general system of differentialequations with Caputo derivatives and illustrated results obtained for equationsof the form (4.1.65) and (5.3.55).

Kilbas et al. [383] investigated the Cauchy problem for the nonlinear fractionaldifferential equation of the form (3.1.1) with the Hadamard fractional derivative(2.7.7) of complex order a £ C (9l(a) > 0):

{Vaa+y){x) = f [ x , y ( x )} ( a ^ x ^ b ) , (3.1.43)

{Vaa+

ky){a) = b k £ C (k = 1 , , n) (3.1.44)

where n = \p\{a)] +1 for a ^ N and n = a for a £ N, on a finite interval [a, b] of R.They proved the equivalence of (3.1.43)-(3.1.44) and a Volterra integral equationof the second kind of the form (3.1.8)

a)

(3.1.45)in the space XQ [a, b], denned by (1.1.3), and applied this result to establish condi-tions for a unique solution y(x) £ X^[a, b] to the Cauchy problem (3.1.43)-(3.1.44).

Fukunaga ([272] and [273]) considered the homogeneous linear differential equa-tion

m — 1

(D"™y) (x) + 2 ck (D"!ly) (x) =0 (x > 0; am > am_i > > ai > 0),

(3.1.46)

Chapter 3 (§ 3.1) 143

containing the right-sided Riemann-Liouville fractional derivatives D"™y (k =1, , m) given by (2.1.6). He gave the required conditions for special cases ofinitial value problems associated with equation (3.1.46), with c < 0, to have aunique solution y(x) in a closed interval [0,T] (T > 0) of the real axis M. and inthe half axis M.+.

A series of papers was devoted to the study of positive solutions of fractionaldifferential equations. Zhang [923] considered the Cauchy problem for a nonlinearequation (3.1.12) with a = 0 and 0 < a < 1:

(Dg+y) {x) = f[x,y(*)]\ 1/(0) = 0, (0 < x < 1), (3.1.47)

with a continuous function / . Using the equivalence of this problem and thecorresponding Volterra integral equation, proved by Delbosco and Rodino [164],he applied the so-called point theorem for the cone to prove the uniqueness of apositive solution y(x) > 0 of (3.1.47). Such a method was applied in Zhang [922]to derive conditions for one or several positive solutions to the Cauchy problem

(D}'sy) (x) = x"f\y{x)] (0 < x < 1; 0 < a < 1), t/(0) = 0, (3.1.48)

with the generalized fractional derivative D^'Sy, inverse to the Erdeelyi-Kobertype integral /Q+-# 7 defined by (2.6.1). Babakhani and Daftardar-Gejji [42], [43],studied the problem of the form (3.1.47), where D£+ is replaced by a linear com-bination of such Riemann-Liouville derivatives, and used a fixed point theoremto give conditions under which the problem has a unique positive solution and aunique, but not necessarily positive, solution. Daftardar-Gejji [152] studied theexistence of positive solutions for the system of fractional differential equations(3.1.47). We note that Atanackovic and Stankovic [38] have analyzed lateral mo-tion of an elastic column fixed at one end and loaded at the other in terms of asystem of fractional differential equations.

Note that Grin'ko [315] studied the nonlinear equation of the form (3.1.47),where the Riemann-Liouville derivative D$+y is replaced by a more general frac-tional differentiation operator involving the Gauss hypergeometric function (1.6.1).He proved the existence and uniqueness theorem for the solution of the consid-ered equation and constructed its approximate solution in the respective spaces ofHolder and continuous functions [see [729], Section 43.2]

In conclusion, we indicate the papers by Dzhrbashyan [206], Nakshushev ([613]and [616]), Aleroev ([29] and [30]), Veber [853], and Delbosco [163], which aredevoted to the investigation of Dirichet-type problems for ordinary fractional dif-ferential equations and papers by Veber ([847], [848], [849], [850], [852] and [851]),Didenko ([168], [167] and [169]), Kochura and Didenko [430], and Malakhovskyaand Shikhmanter [541], where ordinary fractional differential equations were stud-ied in spaces of generalized functions. In this regard, see Sections 2 and 3 of thesurvey paper by Kilbas and Trujill o [408]. Some authors constructed formal par-tial solutions to ordinary differential equations with other fractional derivatives aspresented in Chapter 2. Nishimoto [[629], Volume II , Chapter 6], Nishimoto et al[631], Srivatsava et al [792], [793] and Campos [118] constructed explicit solutions


of some particular fractional differential equations with the so-called fractionalderivatives of complex order (see, for example, Samko et al [729], Section 22.1). Aseries of papers by Wiener [879]-[888] were devoted to the investigation of ordinarylinear fractional differential equations and systems of such equations involving thefractional derivatives denned in the sense of a finite part of Hadamard (see, forexample, Kilbas and Trujill o [408], Section 4).

We also note that many authors have applied methods of fractional integro-differentiation to constructing solutions of ordinary and partial differential equa-tions, to investigating integro-differential equations, and to obtaining a unifiedtheory of special functions. The methods and results in these fields are presentedin Samko et al ([729], Chapter 8) and in Kiryakova [417]. We mention here the pa-pers by Al-Saqabi [25], by Al-Saqabi and Vu Kim Tuan [27] and by Kiryakova andAl-Saqabi [418], [419] and [26], where solutions in closed form were constructedfor certain integro-differential equations with the Riemann-Liouville and Erdelyi-Kober-type fractional integrals and derivatives given in (2.2.1), (2.2.3) and (2.6.1),(2.6.29), respectively.

We also indicate the paper by Kilbas et al [399], where the explicit solution ofthe Cauchy type problem for the equation

(D°+y) (x) = A f \ x - t ) a - x E l a [v{x - t)"] + f(x) ( a < x 0, and A, 7, p,w G C, and the initial conditions (3.1.2), wasestablished. That solution involves the generalized Mittag-LefHer function (1.9.1).The homogeneous equation corresponding to (3.1.49) (f(x) = 0) is a generalizationof the equation which describes the unsaturated behavior of the free electron laser(see [156], [157], [106], [104], [102], [103], [105] and [28]).

3.2 Equations with the Riemann-Liouville Frac-tional Derivative in the Space of SummableFunctions

In this section we give conditions for a unique global solution to the Cauchy typeproblem (3.1.1)-(3.1.2) in the space La(a,b) defined for a e C (9t(a) > 0) by

La(a,b):={yEL(a,b): Daa+y £ L(a,b)} . (3.2.1)

Here L(a,b) := Li(a,b) is the space of summable functions in a finite interval[a,b] of the real axis E defined by (1.1.1) with p = 1. We also generalize theobtained result to the Cauchy type problem for fractional differential equationsmore general than (3.1.1) and to the system Cauchy type problem (3.1.1)-(3.1.2).Our investigations are based on reducing the problems considered to Volterraintegral equations of the second kind and on using the Banach fixed point theorem.

Chapter 3 (§ 3.2) 145

3.2.1 Equivalence of the Cauchy Type Problem and the Vol-terra Integral Equation

In this subsection we prove that the Cauchy type problem (3.1.1)-(3.1.2) and thenonlinear Volterra integral equation (3.1.8) are equivalent in the sense that, ify(x) G L(a, b) satisfies one of these relations, then it also satisfies the other. Weprove such a result by assuming that a function f[x,y] belongs to L(a, b) for anyy G G C C For this we need the auxiliary assertion following from Lemma 2.1(a).

Lemma 3.1 The fractional integration operator I"+ with a G C (£K(a) > 0) isbounded in L(a, b):

In particular, for a > 0, the estimate (3.2.2) takes the form

Here, and elsewhere in this chapter, we shall understand all relations to holdalmost everywhere (a.e.) on [a,b\.

First we consider the Cauchy type problem (3.1.1)-(3.1.2) with real a > 0 givenin (3.1.15)-(3.1.16):

(D2+y)(x) = f[x,y(x)] (" > 0), (3.2.4)

(D2+ky)(a+) = bk, 6 f c € R ( fc = l , - - - , n = - [ - a ] ) . (3.2.5)

Theorem 3.1 Let a > 0, n = —[—a]. Let G be an open set in M. and let/ : ( o , l ) ] x G 4 l be a function such that f[x,y] G L(a, b) for any y G G.

If y(x) G L(a,b), then y(x) satisfies a.e. the relations (3.2.4) and (3.2.5) if,and only if, y(x) satisfies a.e. the integral equation (3.1.8).

Proof. First we prove the necessity. Let y(x) G L(a, b) satisfy a.e. the relations(3.2.4) and (3.2.5). Since f[x,y] G L(a,b), (3.2.4) means that there exists a.e.on [a,b] the fractional derivative (D"+y){x) G L(a,b). According to (2.1.10) and(2.1.7),

(D«+y){x) = (J-^j (i:+ ay)(x), n = -[-a)}, (I°a+y)(x) = y(x), (3.2.6)

and hence, by Lemma 1.1, (7™_j7ay)(:r) G ACn[a,b]. Thus we can apply Lemma2.5(b) (with f(x) replaced by y(x)) and, in accordance with (2.1.39), we have

(3.2.7)


By (2.1.5), we also have

' {I {XJ)-{a-i]y){x) = (Da= (Daa?y)(x). (3.2.8)

Using this relation and (3.2.5), we rewrite (3.2.7) in the form

(IS+D" a+y)(x) = y(x) - £ fF_V).{*th* - «)"" j

= ^ ) - £ r ( a - i + i) (a?" 0)°" J- (3'2-9)

By Lemma 3.1, the integral {I%+f[t,y(ij\){x) € L(a,b) exists a.e. on [a,b].Applying the operator I%+ to both sides of (3.2.4) and using (3.2.9) and (2.1.1),we obtain the equation (3.1.8), and hence the necessity is proved.

Now we prove the sufficiency. Let y(x) £ L(a, b) satisfy a.e. the equation(3.1.8). Applying the operator D"+ to both sides of (3.1.8), we have

T(a-j + l) ^ + ^ + + )

From here, in accordance with the formula (2.1.21) and Lemma 2.4 (with f(x)replaced by f[x,y(x)]), we arrive at the equation (3.2.4).

Now we show that the relations in (3.2.5) also hold. For this we apply theoperators D"+ k (k = 1, , n) to both sides of (3.1.8). I f l ^ A ^ n - 1 , then,in accordance with (2.1.17) and (2.1.21) and Property 2.2 (with f(x) replaced byf[x,y(x)]), we have

{Daa-

ky){x) = J2 T(a-jj + l) ( ^+ f c ( i ~ ar~j) {X) + (D:+ kla+f[

= EHence

(D^ky)(x) = jj^(x-a)k-> +y^J\x-t)k-1f[t,y(t)}dt. (3.2.10)

If k = n, then, in accordance with (3.1.4) and (2.1.16), similarly to (3.2.10), weobtain

j - ^ — f\x-t)n~'f[t,y(t)]dt. (3.2.11)

Taking in (3.2.10) and (3.2.11) a limi t as x —> a+ a.e., we obtain the relations in(3.2.5). Thus the sufficiency is proved, which completes the proof of Theorem 3.1.

Chapter 3 (§ 3.2) 147

Corollar y 3.1 Let 0 < a < I, let G be an open set in R, and let / : ( o , 6 ] x G - >lbe a function such that f[x,y] 6 L(a,b) for any y 6 G.

If y(x) G L(a,b), then y(x) satisfies a.e. the relations (3.1.12) and (3.1.13) if,and only if, y(x) satisfies a.e. the integral equation (3.1.14).

Corol lar y 3.2 Let n e N , let G be an open set in M, and let / : [ n , i i ] x G - > lbe a function such that f[x,y] G L(a,b) for any y G G.

If y(x) G L(a,b), then y(x) satisfies a.e. the relations in (3.1.5) with bk £ K(k = 1, , n) if, and only if, y(x) satisfies a.e. the integral equation

fjtr-'miwt (3.2.12)

Theorem 3.1 is clearly extended from real a > 0 to a complex a G C (9t(a) > 0)as follows:

Theorem 3.2 Let a G C and n - 1 < *R(a) < n (n G N). Let G be an open setin C and let f : (a,b] x G -> C 6e a function such that f[x,y] G L(a,b) for anyyeG.

If y(x) G L(a,b), then y(x) satisfies a.e. the relations (3.1.1) and (3.1.2) if,and only if, y(x) satisfies a.e. the equation (3.1.8).

In particular, if 0 < ![R(a) < 1, then y(x) satisfies a.e. the relations in (3.1.6)if, and only if, y(x) satisfies a.e. the equation (3.1.14).

Remark 3.1 The results of Theorems 3.1-3.2 and Corollaries 3.1-3.2 were ob-tained by Kilbas, Bonilla, and Trujill o ([376], Theorem 1 and Corollaries 1-2) and([374], Theorems 1-2 and Corollaries 1-2).

Remark 3.2 Theorem 3.2 yields conditions for the equivalence of the Cauchytype problem (3.1.1)-(3.1.2) and the Volterra integral equation (3.1.8) in the spaceL(a, b) for a G C and n —1 < 9\(a) < n (n G N). The problem of the equivalence ofthe Cauchy type problem (3.1.1)-(3.1.2) and the Volterra integral equation (3.1.8)for the case of complex order a = m + id (m G N; S e l; 6 j^ 0) is still open. Weonly indicate that in this case the Cauchy type problem (3.1.1)-(3.1.2) takes theform

(DZ+iey){x) = f [ x , y ( x ) \, (D™+ie-ky)(a+) = bk G C (k = 1 , 2 ,- ,m + 1 ),(3.2.13)

and it is clear that, for suitable functions y(x), this problem can be reduced, byusing (2.1.39), to the Volterra integral equation of the form (3.1.8) with n = m +1and a = m + id:

{x) -

(3.2.14)


3.2.2 Existence and Uniqueness of the Solution to the Cau-chy Type Problem

In this subsection we establish the existence of a unique solution to the Cauchy typeproblem (3.1.1)-(3.1.2) in the space LQ(a, b) defined in (3.2.1) under the conditionsof Theorems 3.1 and 3.2, and an additional Lipschitzian-type condition on f[x, y\with respect to the second variable: for all x G (a, b] and for all 1/1,1/2 £ G C C,

\f[x,yl\-f[x,y2]\Â\yl-y2\ (A>0), (3.2.15)

where A > 0 does not depend on x G [a,b]. First we derive a unique solution tothe Cauchy-problem (3.2.4)-(3.2.5) with a real a > 0.

Theorem 3.3 Let a > 0, n = —[—a]. Let G be an open set in E and let f :(a, b] x G —> M. be a function such that f[x,y] £ L(a,b) for any y G G and thecondition (3.2.15) is satisfied.

Then there exists a unique solution y(x) to the Cauchy type problem (3.2.4)-(3.2.5) in the space L° (a,b).

Proof. First we prove the existence of a unique solution y(x) G L(a,b). Accord-ing to Theorem 3.1, it is sufficient to prove the existence of a unique solutiony(x) G L(a,b) to the nonlinear Volterra integral equation (3.1.8). For this we ap-ply the known method, for nonlinear Volterra integral equations, of proving firstthe result on a part of the interval [a, b]; [for example, see Kolmogorov and Fomin[434]].

Equation (3.1.8) makes sense in any interval [a,xi] C [a,b] (a < x\ < b).Choose xi such that the inequality

holds, and then prove the existence of a unique solution y(x) G L(a,xi) to theequation (3.1.8) on the interval [a, xi]. For this we use the Banach fixed pointtheorem given as Theorem 1.9 in Section 1.13 for the space L(a,xi), which isclearly a complete metric space with the distance

d(yi,y2) = | | | / i - i /2| | i := f 1\yi(x)-y2(x)\dx. (3.2.17)J a

We rewrite the integral equation (3.1.8) in the form y(x) = (Ty)(x), where

with

»"(*) = E r ( a -i + l ) ( g " ^ (3-2'19)

Chapter 3 (§ 3.2) 149

To apply Theorem 1.9, we have to prove the following: (1) if y(x) G L(a, xi), then(Ty)(x) G L(a,xi); (2) for any 2/1,2/2 £ L(a,xi), the following estimate holds:

It follows from (3.2.19) that 2/0(2) € L(a, xi). Since f[x, y] G L(a, b), by Lemma3.1 (with b = x\ and g(t) = f[t, y{t)]), the integral in the right-hand side of (3.2.18)also belongs to L(a, x{), and hence (Ty)(x) G L(a, x\). Now we prove the estimate(3.2.20). By (3.2.18)-(3.2.19) and (3.2.1), using the Lipschitzian condition (3.2.15)and applying the relation (3.2.3) (with b = and g(x) = f[x, yi(x)} — f[x, 2/2(2)]),we have

\\Tyi -Ty2\\L{atXl) g \\lZ+[\f[t,yi(t)]-f[t,y 2)(t)}\]\

which yields (3.2.20). In accordance with (3.2.16), 0 < u < 1, and hence (byTheorem 1.9) there exists a unique solution y*{x) G L{a,x{) to the equation(3.1.8) on the interval [a, x\].

By Theorem 1.9, the solution y* is obtained as a limit of a convergent sequence(Tmy*0)(x):

lim ||Tm2/o-2/*IU(a,x1)=O, (3.2.21)m—>oo

where 2/0 {x) is any function in L(a, b). If at least one bk 0 in the initial condition(3.2.5), we can take 2/0(2) = J/o( ) with yo(x) defined by (3.2.19).

By (3.2.18), the sequence (Tm2/o)(s) is defined by the recursion formulas

If we denote ym(x) = {Tmyl)(x), then the last relation takes the form

ym(x) = yo(x) + f / [ * ' ^ : ! 1( ! ) i d t (m G N), (3.2.22)

and (3.2.21) can be rewritten as follows:

lim | | i /m-y' | |i (o,x1)=0. (3.2.23)m—f 00

This means that we actually applied the method of successive approximations tofind a unique solution y* (x) to the integral equation (3.1.8) on [a, xi}.

Next we consider the interval [xi, x2], where x2 = xi + hi and hi > 0 are suchthat x2 < b. Rewrite the equation (3.1.8) in the form

(x) -J- fX /[«'»(*)] * + V bj{X) ~ T(a) JX1 (x - t ) i - + jr[ T(a -j


— fV) Ja

f[t,y(t)]dt

Since the function y(i) is uniquely defined on the interval [a, Xi], the last integralcan be considered as the known function, and we rewrite the last equation as

yM=mM+J-f&im., ,3.2.25)

where ôi(^) as defined by

is the known function. Using the same arguments as above, we derive that thereexists a unique solution y*{x) £ L(xi,X2) to the equation (3.1.8) on the interval[a;i, X2]. Taking the next interval [x2, £3], where £3 = X2 + /12 and /12 > 0 are suchthat X3 < b, and repeating this process, we conclude that there exists a uniquesolution y*{x) G L(a, b) for the equation (3.1.8) on the interval [a, b].

Thus, there exists a unique solution y(x) = y*{x) G L(a,b) to the Volterraintegral equation (3.1.8) and hence to the Cauchy type problem (3.2.4)-(3.2.5).

To complete the proof of Theorem 3.3, we must show that such a uniquesolution y(x) G L(a,b) belongs to the space lia(a,b). In accordance with (3.2.1),it is sufficient to prove that (D"+y)(x) G L(a, b). By the above proof, the solutiony(x) G L(a, b) is a limit of the sequence ym(x) G L(a, b):

lim ||ym - y||i = 0, (3.2.27)m—»oo

with the choice of certain ym on each [a, , [XL-I, b\. By (3.2.4) and (3.2.15)we have

\\Daa+ym - Da

a+y\\x = \\f[x,ym] - f^y] A\\ym - y^. (3.2.28)

Thus, by (3.2.27), we get

lim \\D»+ym - D^ylU = 0, (3.2.29)

and hence (D%+y)(x) G L(a, b). This completes the proof of Theorem 3.3.

Corollar y 3.3 Let n G N, let G be an open set in C, and let / : [ t t , d ] x G - )Cbe a function such that f[x,y] G L(a,b) for any y G G and (3.2.15) holds.

Then there exists a unique solution y(x) to the Cauchy problem (3.1.5) in thespace L" (a, b):

Ln(a, b) = [y G L(a, b) : y™ G L(a, b)} . (3.2.30)

Theorem 3.3 is extended from a real a > 0 to a complex a G C ffi(a) > 0).

Chapter 3 (§ 3.2) 151

Theorem 3.4 Let a G C, n - 1 < fR(a) < n (n G N). Let G be an open set in Cand let f : (a,b] x G -> C be a function such that f[x,y] G L(a,b) for any y £ Gand the condition (3.2.15) holds.

Then there exists a unique solution y(x) to the Cauchy type problem (3.1.1)-(3.1.2) in the space La(a,b) defined in (3.2.1).

In particular, if 0 < £R(a) < 1, then there exists a unique solution y{x) to theCauchy type problem (3.1.6)

{Daa+y){x) = f[x,y(x)} (0 < 9t(a) < 1) (I1

a+ay)(a+) = 6 G C (3.2.31)

in the space LQ(a, b).

Proof. The proof of Theorem 3.4 is similar to the proof of Theorem 3.3, if we usethe inequality

A < 1 ( 3 2 3 2)A W(a)|r(a)| K ( 3 '2- 3 2)

instead of the one in (3.2.16).

3.2.3 The Weighted Cauchy Type Problem

When 0 < 9t(a) < 1, the result of Theorem 3.4 remains true for the weightedCauchy type problem (3.1.7), with c e C

{Daa+y){x) = f[x,y(x)} (0 < 3t(a) < 1), Km+ [(x - a)1""^*) ] = c. (3.2.33)

Its proof is based on the following preliminary assertion.

Lemma 3.2 Let a G C (0 < 9t(a) < 1) and let y(x) be a Lebesgue measurablefunction on [a, b].

(a) If there exists a.e. a limit

lim [{x - af-vyix)} = c (c G C), (3.2.34)

then there also exists a.e. a limit

(7i;aj/)(a+) := }™+{ll+ ay){x) = cT(a). (3.2.35)

(b) If there exists a.e. a limit

xlun+(I1a+

ay)(x)=b (6€C), (3.2.36)

and if there exists the limit limx_).o_|_ [(a; — a)l~ay(x)], then

lun+[(x-ay-"y(x)]= f ry (3.2.37)


Proof. Choose an arbitrary e > 0. By (3.2.34), there exists S = S(e) > 0 such that

\(t-a)âi

for a < t < a + 5. According to (2.1.16),

T(a) = (l%+(t - a)*-1) (x) (0<9t(a)<l). (3.2.39)

Using this equality and taking (2.1.1) into account, we have

\(l£ay) (x)-cT(a)\ = \(lây) (x) - c (lâ(t - a)""1) (x)\

= i r n 1 ^l r (! - a) Ja

If we choose a < x < a + 5, then a < t < x < a + S, and we can apply the estimate(3.2.26) and the formula (2.1.16) to obtain

CW(a)( i - a)^")" 1) (x) = e,

which proves the assertion (a) of Lemma 3.2.

Suppose that the limit in (3.2.37) is equal to c:

lim [(a: - a)1"Qj/(a;)] = c.

Then, by Lemma 3.2(a), we have

{I la+

ay){a+) := Jim+{l£ay){x) = c T(a),

and hence, in accordance with (3.2.36), c = b/T(a), which proves (3.2.37).

From Theorem 3.4 and Lemma 3.2, we obtain the existence and uniquenessresult for the weighted Cauchy type problem (3.2.33).

Theorem 3.5 Let a £ C and 0 < 9\(a) < 1. Let G be an open set in C and letf : (a, b] x G —\ C be a function such that f[x,y] € L(a,b) for any y £ G and(3.2.15) holds.

Then there exists a unique solution y(x) to the weighted Cauchy type problem(3.2.33) in the space La(a,b) defined in (3.2.1).

Proof. If y(x) satisfies the conditions (3.2.33), then, by Lemma 3.2(a), y(x) alsosatisfies the conditions (3.2.31) with b = cT(a):

(DZ+y)(x) = f[x,y(x)} (0 < 9t(a) < 1), (^+Qy)(a+) = cT(a) G C.

Chapter 3 (§ 3.2) 153

According to Theorem 3.4, there exists a unique solution y(x) G La(a,b) to thisproblem. By Lemma 3.2(b), y(x) is also a solution to the weighted Cauchy problem(3.2.33). This y{x) wil l be a unique solution to (3.2.33). Indeed, if we supposethat the weighted problem (3.2.33) has two different solutions in La(a,b), thenby Lemma 3.2(a), there will be also two different solutions to the Cauchy typeproblem (3.2.31) in LQ(a, b), which contradicts the uniqueness of the solution.

R e m a rk 3.3 The results in Theorems 3.3-3.5 and Corollaries 3.3-3.4 were estab-lished by Kilbas et al. in ([376], Theorems 2-6 and Corollaries 3-4) and ([374],Theorems 3-6 and Corollaries 3-6) under stricter conditions in special functionspaces of y{x).

Remark 3.4 In Theorem 3.4, we gave conditions for a unique solution to theCauchy type problem (3.1.1)-(3.1.2) in the space LQ(o,b) for a eC withn — 1 < *R(a) < n (n G N). The problem of a unique solution to the Cauchytype problem (3.1.1)-(3.1.2) in the case of complex order a = m + i9 (m G N;^ £ l ; 9 0) is still open [see Remark 3.2 in this regard].

3.2.4 Generalized Cauchy Type Problems

The results obtained in Sections 3.2.1-3.2.3 extend to Cauchy type problems moregeneral than (3.1.1)-(3.1.2) for the differential equation (3.1.31) with initial con-ditions (3.1.1):

(D°+y)(x) = f [x, y(x), {Daa\y){x), , {Da

a'+y){x)] , (3.2.40)

( D % - k y ) ( a + ) = b k , b k £ C (fc = l , - . . , n ) . (3.2.41)

The following assertion generalizing Theorems 3.1 and 3.2 holds.

Theorem 3.6 Let a G C (n - 1 < 9\(a) < n; n G N) , and let n = [SK(a)] + 1for a g N and n = a for a e N.

Let I G N and (Xj G C (j = 1, , I) be such that

0 = a0 < 9t(ai) < < <H(a;) < «R(a). (3.2.42)

Let G be an open set in C'+1 and let f : (a, 6] x G —> C be a function such thatf[x,y,yi,--- , yi\ G L(a,b) for any (y,yi,--- , Vi) £ G.

If y(x) £ L{a,b), then y{x) satisfies a.e. the relations (3.2.40) and (3.2.41) if,and only if, y(x) satisfies a.e. the integral equation

,3.2.43)


In particular, if 0 < <R(a) < 1, x > a and b0 G C, then y(x) satisfies a.e. therelations

(DZ+y)(z) = f [x,y{x),{Daa\y)(x\--- , {D?+y)(x)] ; (I1

a+ay){a+) = b0

(3.2.44)if, and only if, y{x) satisfies a.e. the integral equation

bo(x - a)"" 1 J _ rV[X) r(a + l) +T(a)Ja

dt

(3.2.45)

Proof. The proof of Theorem 3.6 is similar to that of Theorem 3.1, using theproperties of the Riemann-Liouville fractional integrals and derivatives.

The next statement generalizes Theorems 3.3-3.5.

Theorem 3.7 Let the conditions of Theorem 3.6 be valid, let f[x,y,yi, , y{\satisfy the Lipschitzian condition:

I

\f[x,y,yu---, yOix^-fix^Y,,---, Yt}\ g A,j= o

(3.2.46)

for all x G (a, 6] and y,yi, , yi; Y, Yi , , Yt e G, and where At > 0 does no£depend on x G [ a , 6 ], a nd /e£ ( J 9 "+ 3 y ) ( a +) = 6^. G C ( j = I , - - - , r i j ) 6e fixed

numbers, where nj = [^(a^)] + 1 for ctj £ N and rij = oij for ctj G N.

Then there exists a unique solution y(x) to the Cauchy type problem (3.2.40)-(3.2.41) in the space La(a,b).

In particular, if 0 < 9t(a) < 1 and (ll+ aj y)(a+) = bj G C (j = 1, , /) are/jared numbers, then there exists a unique solution y(x) G LQ(a, 6) io i/ie Cauchytype problem (3.2.44) and the weighted Cauchy type problem

{Daa+y){x) = f [x,y(x),(D^y)(x),--- , (D%y){x)] , (3.2.47)

lim [(x - a )1 - " ^ ) ] = c G C. (3.2.48)

Proof. Theorem 3.7 is proved in a way similar to the proofs of Theorems 3.3-3.5.By Theorem 3.6, it is sufficient to establish the existence of a unique solutiony(x) G L(a,b) to the integral equation (3.2.45). We choose x\ G (a,6) such thatthe condition

generalizing the one in (3.2.32), holds and apply the Banach fixed point theoremto prove the existence of a unique solution y{x) = y*(x) G L(a, x{). We use thespace L(a,b) and rewrite the equation (3.2.43) in the form y(x) = (Ty)(x), where

-L f1 \a) J a

Chapter 3 (§ 3.2) 155

By (3.2.46), (2.1.30) and (2.1.39) we have

(x)\

, Daa'+y2\\){x)

j=O

j=o

By the hypothesis of Theorem 3.7,hence, for any x G [a.b],

™^_ kjyi){a+)

3=0

Using this relation with x = Xi and (3.2.2) with b =

1 u\\yi -

= (D^ kiy2){a+), and

)(*)l

(3.2.51)

we derive the estimate

which, by (3.2.49) and Theorem 1.9, yields the existence of a unique solutionH*(x) to the equation (3.2.43) in L{a,X\). This solution is obtained as a limitof the convergent sequence Tmyo(x) = ym{x), for which the relations (3.2.21)and (3.2.23) hold. Next we show, using the same arguments as in the proof ofTheorem 3.3, that there exists a unique solution y(x) 6 L(a,b) to the integralequation (3.2.43) and hence to the Cauchy type problem (3.2.40)-(3.2.41) suchthat D"+y G L(a,b), which completes the proof of Theorem 3.7 in accordancewith (3.2.1).

In particular, if fR(a) < 1, then there exists a unique solution y(x) G LQ(a, 6) tothe Cauchy type problem (3.2.44). Then the same result for the weighted Cauchytype problem (3.2.47) is proved as in Theorem 3.5.

The results in Theorems 3.1-3.2 and Theorems 3.3-3.5 can be also extended tothe system Cauchy type problem (3.1.1)-(3.1.2):

(Da+Vr)(x) = f r [ x ; y i ( x ) , - - - ,ym(x)} (r = , m ) , (3.2.52)


(DXJkyk)(a+)=bjh ,bjk£C (k = l,---,m; j k = l,---,nk), (3 .2.53)

in a product of the spaces L a i ( a ,b ), , LQTr*(a,b) defined by (3.2.1). We shalluse the following notation:

Lm(a, b) := L(a, b) x x L{a, b) (m times), (3.2.54)

I> l [(a, b)m] = LQl (a, b) x x La™ (a, b). (3.2.55)

The following statements generalizing Theorems 3.1-3.2 and Theorems 3.3-3.5,respectively, hold.

Theorem 3.8 Let k = 1, , m, a € C, nk - 1 < 9t(afc) < nk (nk G N) orajt 6 N. Let G be open set in Cm and let fk {a, b] x G —> C 6e functions suchthat fk[x,yi,--- , J/m] £ L ( a , 6) / or a ny ( j / i , - - - , y m ) £ G.

If Vk{x) G L(a,b), then yk(x) satisfy a.e. the system of relations (3.2.52) and(3.2.53) if, and only if, yu{x) satisfy a.e. the system of integral equations

jk=l \ K JK '

(fc = l , - - - , m ) . (3.2.56)

/n particular, ifO< 9i{ak) < 1 and k = 1, , m , then yk(x) satisfy a.e. therelations

^ ^ — Jk[x, yi{X),- ,ym{X)\ (o.z.oi)

(I1a-

a"yk)(a+) = bk, bk £ C, (3.2.58)

if, and only if, yk(x) satisfy a.e. the system of integral equations

ar - i + -^— fx(3 2 5 9

for all k = 1, , m

Theorem 3.9 Let the conditions of Theorem 3.8 be valid and let fk(x,yi,- , ym)satisfy the Lipschitzian conditions

\fk{x,yi,---, ym)-fk{x,Yu---, y m ) | ^fc=i

(3.2.60)for all x e (a, b] and for all , ym; Yi , , Ym £ G, and where Ak > 0 donot depend on x E [a,b].

Then there exists a unique solution (yi,--- , ym) to the system Cauchy typeproblem (3.2.52)-(3.2.53) in the space LH[(a,6)m] .

Chapter 3 (§ 3.2) 157

In particular, if 0 < d\.(a.k) < 1, then there exists a unique solution(Vi,--- , Vm) e LlQl[(a,6)m] to the system Cauchy type problem (3.2.57)-(3.2.58)and to the weighted Cauchy type problem

(D^yk)(x) = fk[x\yi(x),---,ym(x)] (fc = 1, ,m), (3.2.61)

Km+[{x-a)1-a>yk{x)]=h,bk£C (k = 1, ,m). (3.2.62)

Proof. The proof of Theorem 3.8 is similar to the proofs of Theorems 3.1 and 3.2,using the properties of the Riemann-Liouville fractional integrals and derivatives.The proof of Theorem 3.9 is similar to the proofs of Theorems 3.3-3.5; we setXji G [a, b] (j = 1, , m) and use the inequality

^ « m(ak)\T(ak)\ ^ " " "

instead of the one in (3.2.32).

Remark 3.5 The results in Theorem 3.8 were obtained by Bonilla et al. ([95],Theorems 2.1-2-2).

Remark 3.6 The results of Theorems 3.9 were established by Bonilla et al. ([95],Theorems 3.1-3.4).

Remark 3.7 The results presented in Theorems 3.6 and 3.7 can be extended to asystem Cauchy type problem (3.2.40)-(3.2.41) and to a system of Volterra integralequations (3.2.43).

3.2.5 Cauchy Type Problems for Linear Equations

From Theorems 3.4-3.5 we derive the corresponding results for the Cauchy typeproblems for linear differential equations of fractional order a G C (9^(a) > 0).

Corollar y 3.4 Let a = n G N or a G C be such that n - 1 < SR(a) < n (n G N)and let g(x) G L(a, b).

If a(x) G Loolcbjb) or if a(x) is bounded on [a,b], then the Cauchy type problemfor the following linear differential equation of order a and bk G C (k = 1, , n)

(DZ+y){x) = a(x)y(x) + g(x); (D2+ky)(a+) = bk (3.2.64)

has a unique solution y{x) in the space La(a,6).

In particular, there exists a unique solution y(x) G LQ(a, 6) to the problem

{Daa+y){x) = X(x - afy{x) + g(x); (D°+ky)(a+) = bk, (3.2.65)

with complex A, /3 G C (<R(/?) ^ 0).


Corollar y 3.5 Let a = 1 or a G C (0 < SR(a) < 1). Letc,b0 G C andletg(x) G L(a,b).

If a(x) G Loc(a,b) or if a(x) is bounded on [a,b], then the Cauchy type problem

(D°+y)(x) = a(x)y(x) + g(x), (£+ay)(a+) = 60 (3.2.66)


{Daa+y){x) = a(x)y(x) + g(x), fai+ [(x - a )1 " " ^ ) ] = c (3.2.67)

have a unique solution y(x) in the space La(a,b).

In particular, either of the following Cauchy type problems,

{Daa+y){x) = X(x - afy(x) + g(x), (Il

a+ay)(a+) = b0, (3.2.68)

{Daa+y){x) = X(x - a)py(x) + g(x), hm+ [(x - a )1 " ^ * ) ] = c (3.2.69)

with A,/3 G C (9t(/3) ^ 0) has a unique solution y(x) G La(a,b).

The results given in Corollaries 3.4 and 3.5 can be extended to more generalfractional differential equations. Thus, from Theorem 3.7, we derive the followingassertions where we use the notation Da°y = y.

Corollar y 3.6 Let a G C (n - 1 < m{a) < n; n&N), and let n = pR(a)] + 1 fora $ N and n = a for a G N. Let I G N and ctj G C (j = 1, , I) be such that theconditions in (3.2.42) are satisfied, and let g(x) G L(a,b).

If a,j(x) G Loc(a,b) or if aj(x) (j = 0,1, , /) are bounded on [a,b], then thefollowing Cauchy type problem for the linear differential equation of order a, withbk G C and k = 1, , n

y)(x) = g(x); (Daa;

ky)(a+) = bk (3.2.70)

j=o

has a unique solution y(x) in the space La(a,b).

In particular, there exists a unique solution y(x) G La(a,b) to the Cauchy typeproblem for the equation

i

{Daa+y){x) + £ Xj(x - a)* {Da

a'+y){x) = g(x), (P^f c y) (a+) = bk (3.2.71)i=o

with Xj,/3j G C (9t(/3j) ^ 0) (j = 0, , I).

Coro l lar y 3.7 Let a = 1 or a G C be such that 0 < 9l(a) < 1. Let I G N anda.j G C (j = 1, , I) be such that the conditions in (3.2.42) are satisfied, and letg{x) eL(a,b).

Chapter 3 (§ 3.2) 159

If a,j(x) G Lx,(a,b) or if a,j(x) (j = 0 , 1 , - -- , I) are bounded on [a,b], andc, bo G C, then the Cauchy type problem

i

{D%+y)(x) + Y/aj(^(Daiy)(x) = g{x), (I1a+

Qy){a+) = b0 (3.2.72)3=0


I

{ua+y){X) -t- 2_JO'3\xj\ua+y)\X) — y{X), ûm i(x — a) y[x)\ — c (j.z.id)

j=o xâ

have a unique solution y(x) in the space La(a,b).

In particular, either of the following Cauchy type problems,

i

(D%+y){x) + 2Xj(x - af' (D^y){x) = g(x), (/Q1-Q

2/)(a+) = b0, (3.2.74)j=o

i

(D%+y){x) + Yj\j(x - afi{pl%y){x) = g(x), lim [(a; - af^yix)} = c3=0

(3.2.75)with \j,/3j G C (!H(/3j) ^ 0) (j = 0 , 1, ) has a unique solution y(x) G L a ( o , b).

From Theorem 3.9 we derive the results for the systems of Cauchy type prob-lems for linear differential equations of fractional order.

Corollar y 3.8 Let j = 1, , m. Also let a3- = rij G N or a.j G C be such thatrij - 1 < 5H(ofj) < rij (rij G N), and let gj(x) G L(a, b).

If aj(x) G £oo(a, b) or if a,j(x) are bounded on [a,b], then the system Cauchytype problem

{Daa'+yj){x) = aj(x)yj(x) + 9j(x) (3.2.76)

{D"f k} y){a+) = bkj G C (kj = 1, ,ny, j = 1, , TO) (3.2.77)

has a unique solution (yi,--- , ym) in the space h^[(a,b)m\.

In particular, the system Cauchy type problem

{p*\y){x) = Xj{x - afiyj(x) + gj{x) (3.2.78)

where Aj,/3y G C; 9l(/3j) ^ 0; j = 1, , m, and with the initial conditions(3.2.77) has a unique solution ( j / i , -- - , ym) G Ll al [ (a,b)m} .

Corollar y 3.9 Let j = 1, , TO. Also let ctj = 1 or ctj G C be such that0 < fR{otj) < 1, and let gj(x) G L(a, b).

Ifaj(x) G L00(a,b) or ifaj(x) are bounded on [a,b], then the system of Cauchytype problems, with bj, Cj G C and j = 1, , m

(D<^3+yj)(x) = a.j(x)yj(x) + gj(x), [I a+

a' yj)(a+) = bj (3.2.79)


and the system of weighted Cauchy type problems

x) = aj(x)yj(x) + 9j(x), \un[(x - af-^y^x)) = c3 (3.2.80)

have a unique solution (yi,--- , ym) in the space Ll a'[(a, b)m\.

In particular, either system of the following Cauchy type problems,

(D^yj)(x) = \j(x - a) yj(x) + 9j(x), {lll aiyj){a+) = bj, (3.2.81)

(D^yj)(x) = Xj(x - af'y^x) + 9j(x), Um[(z - af^y^x)} = Cj (3.2.82)

with j = I , - - - , m , Xj,pj e C (<*(& ) ^ 0) has {yx,--- , ym) e L l Q l [ (a ,6 )m] asthe unique solution.

Remark 3.8 It follows from the proofs of the theorems in this section that themethod of successive expansion can be applied to construct exact unique solutionsfor all Cauchy type problems considered in Section 3.2.5.

3.2.6 Miscellaneous Examples

In Sections 3.2.3-3.2.5 we gave sufficient conditions for the Cauchy type problemsand systems of such problems to have a unique solution in some subspaces of thespace of integrable functions. Actually these conditions show that the differentialequations of fractional order and systems of such equations can have integrablesolutions, provided that their right-hand sides are integrable. But these conditionsare not sufficient. Below we present two examples and discuss these examples inconnection with the results obtained in Sections 3.2.2 and 3.2.3.

Example 3.1 Consider the following differential equation of fractional ordera > 0:

(D2+y)(x) = X{x-af[y{x)}2 (x > a; A , / 3 e R ; A ^ 0 ). (3.2.83)

It can be directly shown that ifa+0 < 1, then this equation has the exact solution

Xy[X)~ XT(l-2a-p)

and this solution belongs to the space L(a,b).

In this case the right-hand side of the equation (3.2.83) takes the form

and this function, generally speaking, does not belong to the space L(a,b).

If we suppose that 2a + /3 < 1, then the right-hand side of the equation (3.2.83)belongs to the space L(a, b) and, since a + ft < 2a + j3, then a + ft < 1, and theequation (3.2.83) has the exact solution (3.2.84) which belongs to L(a,b).

Chapter 3 (§ 3.2) 161


{D°+y){x) = \{x - af[y{x)]1'2 (x > a; A,/3 G R; A # 0). (3.2.86)

It can be directly shown that, if 2(a + /3) > — 1, then this equation has the exactsolution

and this solution y(x) belongs to the space L(a,b).The right-hand side of the equation (3.2.86) is given by

which, generally speaking, does not belong to the space L(o, 6).If we suppose that a+2/3 > — 1, then the right-hand side of the equation (3.2.86)

belongs to the space L(a, b) and, since 2(a + /3) > a + 2/3, then 2(a + /3) > 1, andthe equation (3.2.86) has the exact solution (3.2.87), which belongs to L(a,b).

The above examples show that even if the right-hand sides f[x,y(x)] of frac-tional differential equations are not integrable on L(a,b), the solutions to theseequations can still belong to L(a, b).

On the other hand, it is possible to give a more detailed characterization of thesolution y(x) to fractional differential equations in the case when their right-handsides f[x,y(x)] are integrable. Namely, there are three possible cases:

1. The right-hand side f[x, y(x)] is integrable on [a, b] together with the solutiony(x), and they both have integrable singularities at the point a.

2. The right-hand side f[x,y(x)] is integrable on [a, b] with an integrable sin-gularity at point a and the solution y(x) is bounded on [a,b].

3. The right-hand side f[x, y(x)] is bounded on [a, b] together with the solutiony{x).

The first one case is presented in Example 3.1 with 0 < 2a + j3 < 1 and0 < a + /3 < 1, and in Example 3.2 with -1 <a + 2/3 < 0 and -1/2 < a + /3 < 0.

The second case is presented in Example 3.1 with 0 < 2a+/3 < 1 and a+/3 ^ 0,and in Example 3.2 with -Ka + 2/3 <0 and a 4- /3 ^ 0.

The third case is presented in Example 3.1 with 2a + @ ^ 0 (so that a + /3 0),and in Example 3.2 with a + 2/3 ^ 0 (so that a + /3 0).

The spaces for f[x, y(x)] and y(x) in Examples 3.1 and 3.2 can be characterizedin more detail by using the space C[a, b] of continuous functions and the weightedspace C7[a, b] of continuous functions defined in (1.1.21). Using these spaces, Cases1-3 in Example 3.1 are characterized as follows:

1. If 0 < 2a + fl < 1 and 0 < a + /3 < 1, then f[x,y(x)] 6 C2a+i3[a, b] andy{x) G Ca+p[a,b];


2. If 0 < 2a + 0 < 1 and a + 0 ^ 0, then f[x, y{x)] G C[a, b]C2a+i3[a, b] andy(x) £C[a,b\;

3. If 2a + 0 g 0, then f[x,y(x)] G C[a, b] and y(x) G C[a, 6].

Similarly, Cases 1-3 in Example 3.2 are characterized as follows:

1. If - 1 < a + 20 < 0 and -1 /2 < a + 0 < 0, then f[x, y{x)} G C_( Q + 2 /3) [a, b]&ndy(x) G C_(Q+/3)[a,6];

2. If - 1 < a + 20 < 0 and a + /3 ^ 0, then /[x,y(a;)] G C_(a+2/3)[a, b] and2/(a;) G C[a,6];

3. If a + 2/9 ^ 0, then /[a;, y(ar)] G C[a, 6] and y(a;) G C[a, b\.

The above arguments show that the spaces of continuous functions are prefer-able when considering solutions to differential equations of fractional order.

Remark 3.9 In this section, we established conditions for the existence of uniquesummable solutions of Cauchy type problems for nonlinear differential equationson the basis of their equivalence to the corresponding Volterra integral equationsand a Banach fixed point theorem. We began our investigation in Sections 3.2.1and 3.2.2 from the case of equations of positive order a > 0 with a real-valuedfunction f[x, y] with y in a domain G of 1 and real parameters A and bk G M.(k = 1, , -[—a]), and then extended them to the case of equations of complexorder a G C (n - 1 < 9\(a) < n) with a complex-valued function f[x, y] with y ina domain G of C and complex parameters A and bk G C (k = 1, , n). As wehave seen, the proofs of the corresponding results are similar in both cases.

Therefore, further in this Chapter, we shall present results for the Cauchy typeproblems for fractional differential equations in the first case. The correspondingresults in the second case can be derived analogously.

3.3 Equations with the Riemann-Liouville Frac-tional Derivative in the Space of ContinuousFunctions. Global Solution

In this section we give conditions for a unique solution y(x) to the Cauchy typeproblem (3.1.1)-(3.1.2) in the space C%_a[a,b] defined for n - 1 < a ^ n (n G N)by

CZ_a[a,b] = {y(x)eCn-a[a,b]: {Daa+y){x) G Cn.a[a, b}}. (3.3.1)

Here Cn-a[o., b] is a weighted space of continuous functions of the form (1.1.21):

Cn-a[a, b] = {g(x) : (x - a)n-ag(x) G C[a, b], \\g\\cn_a = \\(x - a)n~ag{x)\\c}.(3.3.2)

In particular, when a = n G N, Cg[a, b] coincides with the space Cn[a,b] offunctions g continuously differentiate up to order n on [a, b]: CJJ[a, b] := Cn[a, b].We give conditions for the existence of a global continuous solution y(x) to thisCauchy problem on [a, b]. As in Section 3.2, our approach is based on reducing thisproblem to a Volterra integral equation of the second kind of the form (3.1.8) and

Chapter 3 (§ 3.3) 163

on using the Banach fixed point theorem. The arguments here are basically thesame as the ones used in the previous section, using the properties of the Riemann-Liouvill e fractional integrals and derivatives in weighted spaces Cn-a[a, b] insteadof that in L(a,b). Therefore, we shall give schematic proofs in the main, exceptfor the proof of the uniqueness in Section 3.3.2 where the technique of workingwith functions in the spaces Cn-a[a, b] is demonstrated.

3.3.1 Equivalence of the Cauchy Type Problem and the Vol-terra Integral Equation

In this subsection we prove the equivalence of the Cauchy type problem (3.2.4)-(3.2.5) and the nonlinear Volterra integral equation (3.1.8) in the sense that, ify{x) G Cn-a[a,b] satisfies one of these relations, then it also satisfies the otherone. To establish such a result, we assume that a function f[x,y] belongs toCn-a [a, b] for any y G G C E. For this we need the auxiliary assertion whichfollows from Lemma 2.8(a).

Lemma 3.3 If j G M (0 7 < 1), then the fractional integration operator I"+

with a G C (9t(a) > 0) is bounded in Cj[a,b]:

l l^flll c * (b «)a l | g | 1^ ( 3 3 3)

First we establish the equivalence of the Cauchy type problem (3.2.4)-(3.2.5)and the integral equation (3.1.8) in the space (3.3.2).

Theorem 3.10 Let a > 0, n = —[—a]. Let G be an open set in M. and letf : (a, b] x G —> M. be a function such that f[x, y] G Cn-a[a, b] for any y G G.

If y{x) G Cn-a[a, b], theny(x) satisfies the relations (3.2.4) and (3.2.5) if, andonly if, y(x) satisfies the Volterra integral equation (3.1.8).

Proof. First we prove the necessity. Let y(x) G Cn-a[a, b] satisfy (3.2.4)-(3.2.5).Since f[x,y] G Cn-a[a,b], (3.2.4) means that there exists on [a, b] the fractionalderivative (D%+y)(x) G Cn-a[a,b). According to (2.1.10) and (2.1.7), the deriva-tive (D%+y)(x) has the form (3.2.6), and so by Lemma3.3, {I"+ ay)(x) G C%_a[a, b}.Hence we can apply Lemma 2.9(d) (with f(x) = y(x) and 7 = n — a) and inaccordance with (2.1.39) relation (3.2.7) holds. Using (3.2.5) and taking the samearguments as in the proof of Theorem 3.1, we represent (3.2.7) in the form (3.2.9):

{I* +D2+y){x) = y(x) - £ r{a*j. + 1)(x - a)"-*. (3.3.4)

By Lemma 3.3, the integral (/"+/[£,y(t)])(x) G Cn-a[a,b]. Applying the op-erator I°+ to both sides of (3.2.4) and using (3.3.4) and (2.1.1), we obtain theequation (3.1.8), and hence the necessity is proved.


Now we prove the sufficiency. Let y(x) G Cn-a[a, b] satisfy the equation (3.1.8).Applying the operator D%+ to both sides of (3.1.8), we have

J (D" {t ~ ^ {X) + (DI2Kr ( a j + l)(3.3.5)

From here, in accordance with the formula (2.1.21) and Lemma 2.9(b) (with f(x)replaced by f[x,y(x)]), we arrive at the equation (3.2.4).

Applying the operators D"+ k (k = 1, , n) to both sides of (3.1.8) and usingthe same arguments as in the proof of Theorem 3.1, we obtain relations of theforms (3.2.10) and (3.2.11):

Ji^jy fchy fj (3.3.6)

and

(^(Ê^^^^^ft 1-')""'/^^)] ^ (3-3.7)

Applying the limit as x —> a+ in (3.3.6) and (3.3.7), we obtain the relations in(3.2.5). Thus sufficiency is also proved, which completes the proof of Theorem3.10.

Corollar y 3.10 Let 0 < a < 1, let G be an open set in E and let f : (a,b]xG -» Ebe a function such that f[x, y] G C\-a[a, b] for any y G G.

If y(x) £ Ci-a[a,b], then y(x) satisfies the relations (3.1.12) and (3.1.13) if,and only if, y(x) satisfies the integral equation (3.1.14).

Coro l la r y 3 .11 Let n G N, let G be an open set in R and let / : [ a , i i ] x G - > I

be a function such that f[x,y] G C[a,6] for any y G G.

Ify(x) G C[a,b], then y{x) satisfies the relations in (3.1.5) if, and only if, y(x)satisfies the integral equation (3.2.12).

Remark 3.10 Theorem 3.10 and Corollaries 3.10-3.11 were proved in Kilbas etal. ([375], Theorems 1 and 2).

3.3.2 Existence and Uniqueness of the Global Solution tothe Cauchy Type Problem

In this subsection we establish the existence of a unique solution to the Cauchytype problem (3.2.4)-(3.2.5) in the space C"_a[o, b] defined in (3.3.1) under theconditions of Theorem 3.10, and an additional Lipschitzian condition (3.2.15).

We need the following preliminary assertion.

Chapter 3 (§ 3 . 3 ) 1 6 5

Lemma 3.4 Let 7 G M., a < c < b, g G C7[a, c] andg G C7[c, b]. Then g G C7[a, b]and

There holds the following result.

Theorem 3.11 Let a > 0 and n = —[—a]. Let G be an open set in M. and letf : (a, 6] x G —> M. be a function such that f[x,y] G Cn-a[a,b] for any y G G andthe condition (3.2.15) holds.

Then there exists a unique solution y(x) to the Cauchy type problem (3.2.4)-(3.2.5) in the space C°_a[a,b],

Proof. First we prove the existence of a unique solution y(x) G Cn-a[a,b]. Ac-cording to Theorem 3.10, it is sufficient to prove the existence of a unique solutiony(x) G Cn-Q[a, b] to the nonlinear Volterra integral equation (3.1.8). For this, asin the proof of Theorem 3.3, we apply the known method for nonlinear Volterraintegral equations.

Equation (3.1.8) makes sense in any interval [a,a;i] G [a, b] (a < X\ < b).Choose x\ such that the inequality

holds, and then prove the existence of a unique solution y{x) G Cn-a[a,X\] toequation (3.1.8) on the interval [a, x\\. For this we use the Banach fixed point the-orem (Theorem 1.9 in Section 1.13) for the space Cn-a[a, b], which is the completemetric space with the distance given by

d(yi,V2) = II2/1 - y2\\cn-a[a,Xl] max |(a;-a)n-a[yi(a;) - y2(x)]\. (3.3.9)x€[a,xi]

We rewrite the integral equation (3.1.8) in the form

y(x) = (Ty)(x), (3.3.10)

where T is the operator defined in (3.2.18) with yo{x) given by (3.2.19). To applyTheorem 1.9, we have to prove the following: (1) if y(x) G Cn-a[a,xi], then(Ty)(x) G Cn-a[a,xi]; and (2) for any 2/1,2/2 £ Cn-a[a,xi], the following estimateholds:

\\Tyi-Ty2\\cn_a[ a,x1] ^ w||yi-y2||cn_Q[o,a;i]. w = -An-afci -a)a—7- —r.J. \ZiQt Tl 1" -LI

(3.3.11)

It follows from (3.2.19) that yo{x) G Cn_a[o, xi] . Since f[x,y(x)] G Cn-a[a,xi],then, by Lemma 3.3 (with 7 = n — a, b = x\ and g(t) = f[t,y(t)]), the in-tegral in the right-hand side of (3.2.18) also belongs to Cn-a[a,xi], and hence(Ty)(x) G Cn_a[a,a:i]. Now we prove the estimate in (3.3.11). By (3.2.18)-(3.2.19) and (2.1.1), using the Lipschitzian condition (3.2.15) and applying the


relation (3.3.3) (with 7 = n — a, b = xi and g(x) = f[x,yi(x)] — f[x,y2(x)}), wehave

\\Tyi-Ty2\\cn_a[a,Xl}^\\l^

which yields the estimate (3.3.11). In accordance with (3.3.8), 0 < 1, andhence by Theorem 1.9, there exists a unique solution y*(x) = y*°(x) £ Cn-a[a, xi]to the equation (3.1.8) on the interval [a, xi].

By Theorem 1.9, this solution y* (x) is a limit of a convergent sequence(Tmy*0)(x):

Jirn ||Tmy* - y*\\cn.a[a,Xl] = 0, (3.3.12)

where y£(x) is any function in Cn-a[a,b]. If at least one bk 0 in the initialcondition (3.2.5), we can take y$(x) = yo{x) with yo{x) defined by (3.2.19). Thelast relation can be rewritten in the form

Jim^ \\ym ~ yllc-do.zi] = 0, (3.3.13)

where

ym(x) := (T™y*0)(x) = yo(x) + - L f / M ^ f f ^ (m e N). (3.3.14)1 \a) J a \x l)

Next we consider the interval [z i ,^ ] , where X2 = x\ + h\ and h\ > 0 aresuch that x2 < b. Rewrite the equation (3.1.8) in the form (3.2.25), where yoi{x)defined by (3.2.26) is the known function. Using the same arguments as above,we derive that there exists a unique solution y1(x) G Cn-a[xi,X2] to the equation(3.1.8) on the interval [xi, x2]- Taking the next interval [x2, x3], where x3 = x2 + h2

and h2 > 0 are such that £3 < 6, and repeating this process, we find that thereexists a unique solution y(x) to the equation (3.1.8) such that y(x) = y*k(x) andy*k{x) e Cn-a[xk-.i,xk] (k = I , - - - , L), where a = x0 < x\ < < xL = b.Then, by Lemma 3.4, there exists a unique solution y(x) £ Cn-a[a,b\ on thewhole interval [a, b].

Thus there exists a unique solution y(x) = y*(x) G Cn-a[a,b] to the Volterraintegral equation (3.1.8), and hence to the Cauchy type problem (3.2.4)-(3.2.5).

To complete the proof of Theorem 3.11, we must show that such a uniquesolution y{x) £ Cn-a[a, b] belongs to the space C^_a[a, b]. In accordance withthe definition (3.3.1), it is sufficient to prove that (D"+y)(x) £ Cn-a[a, b]. By theabove proof, the solution y(x) £ Cn-a[o-, b] is a limit of the sequence ym(x), whereym(x) := (TmyZ)(x) £ Cn-a[a,b]:

lim \\ym - y||o [a,6] = 0, (3.3.15)n-+oo

w i t h t he choice of ce r ta in ym on each [a, x 1], , [xL-i,b]. By (3.2.4) a nd (3.2.15),we have

\\Daa+ym - Da

a+y\\Cn_a = \\f[x,ym] - f[x,y]\\cn.a An_a\\ym - y\\Cn_a.

Chapter 3 (§ 3.3) 167

Thus, by (3.3.15),lirn^ \\D°+ym - Da

a+y\\Cn_a[aM = 0,

and hence (D"+y)(x) G Cn-a[a, b]. This completes the proof of Theorem 3.11.

Corollar y 3.12 LetO < a <1, letG be an open set inR and let f : (a,fi]xG^Mbe a function such that f[x,y] G Ci_Q[a, b] for any y G G and (3.2.15) holds.

Then there exists a unique solution y(x) to the Cauchy type problem (3.1.12)-(3.1.13) in the space Cf_a[a,b].

Corollar y 3.13 Let n G N, let G be an open set in E and let f : [a,b] x G -> Rbe a function such that f[x,y] G C[a,6] for any y G G and (3.2.15) is valid.

Then there exists a unique solution y(x) to the Cauchy problem (3.1.5) in thespace Cn[a, b].

3.3.3 The Weighted Cauchy Type Problem

When 0 < a < 1, the result of Corollary 3.12 remains true for the followingweighted Cauchy type problem (3.1.7), with c G C:

(D2+y)(x) = f[x,y(x)}; hm+ [(x - o)1-Qy(i) ] = c (0 < SH(a) < 1). (3.3.16)

Its proof is based on the following preliminary assertion.

Lemma 3.5 Let 0 < a < 1) and let y(x) G Ci-a[a,b].(a)//

lim [(x - a)l-ay(x)] = c, c G E, (3.3.17)

£/ien(/o

1;Qy)(a+) := Jiun+(Iây)(x) = cT(a). (3.3.18)

(b) //lim (7 +

Q2/)(a;) = 6, 6 G E, (3.3.19)

and if there exists the limit limxâ+ [(a; — a)1~ay(x)\, then

nm [(x — a) y{x)\ — —, r. (o.o.zv)

Proof. The proof of this lemma is similar to that of Lemma 3.2.From Corollary 3.12 and Lemma 3.5, we obtain the existence and uniqueness

result for the weighted Cauchy type problem (3.3.16).

Theorem 3.12 Let 0 < a < 1, let G be an open set in E and let f : (a, b] x G -> 1be a function such that f[x, y] G Ci-a[a, b] for any y G G and the relation (3.2.15)holds.

Then there exists a unique solution y(x) to the weighted Cauchy type problem(3.3.16) in the space C?_a[a,b\.


Proof. By applying Theorem 3.11 and Lemma 3.5, Theorem 3.12 is proved just asTheorem 3.5.

Remark 3.11 The results in Theorems 3.11-3.12 and Corollaries 3.12-3.13 wereestablished by Kilbas et al. ([375], Theorems 3-6) under stricter conditions in somespecial function spaces of y(x).

Remark 3.12 It follows from the proof of Theorem 3.11 that the method of suc-cessive expansions can be applied to construct the unique solutiony(x) = y*{x) £ Cn-a[a,b] to the Cauchy type problems (3.2.4)-(3.2.5).

Remark 3.13 The particular case of the weighted Cauchy type problem (3.3.16)with o = 0, 0 < a < 1, f[x,y(x)] = f[y{x)] in the form (3.1.27) was considered byDelbosco and Rodino [164], who proved its unique solution y(x) £ Ci-_a[0, h] forh > 0 [see Section 3.1 in this regard].

3.3.4 Generalized Cauchy Type Problems

The results obtained in Sections 3.3.1-3.3.2 are extended to the more generalCauchy type problems (3.2.40)-(3.2.41) than (3.1.1)-(3.1.2).

The following assertion generalizing Theorem 3.10 holds.

Theorem 3.13 Let a > 0 be such that n - 1 < a ^ n (n € N). Let I £ N\{1 }and ctj £ C (j = 1, , /) be such that

0 = a0 < ax < < at < a, (3.3.21)

Let G be an open set in M.l+1 and let f : (a, b] x G —> M. be a function such thatf[x,y,yi,--- , yi] £ Cn-a[a,b] for any (J / ,J / I , - -- , yi) £ G.

Ify(x) £ Cn-a[a,b], then y{x) satisfies the relations (3.2.40) and (3.2.41) if,and only if, y(x) satisfies the integral equation (3.2.43).

In particular, if 0 < a < 1, then y(x) £ Ci-a[a, b] satisfies the relations(3.2.44)-(3.2.45) if, and only if, y(x) satisfies the integral equation (3.2.45).

Proof. The proof of Theorem 3.13 is similar to that of Theorem 3.10, using theproperties of the Riemann-Liouville fractional integrals and derivatives.

The next statement generalizes Theorems 3.11-3.12.

Theorem 3.14 Let the conditions of Theorem 3.13 be valid, let f[x,y,yi, , y{\satisfy the Lipschitzian condition (3.2.46), and let (D"+ 3y)(a+) = 6^ £ K(j = 1, , rij) be fixed numbers, where rij = [9\(aj)} + 1 for ctj N and rij = ctjfor ctj £ N.

Then there exists a unique solution y(x) to the Cauchy type problem (3.2.40)-(3.2.41) in the space C%_a[a,b].

In particular, ifO < a < 1 and (Ia+a3y){a+) = bj £ K. (j = 1, , I) are fixed

numbers, then there exists a unique solution y(x) £ C"_a[a, b] to the Cauchy typeproblem (3.2.44) with bo £ R and the weighted Cauchy type problem (3.2.47) witheel.

Chapter 3 (§ 3.3) 169

Proof. The proof of Theorem 3.14 is similar to the proofs of Theorem 3.11-3.12 inthe same manner as Theorem 3.7. By Theorem 3.13, it is sufficient to establish theexistence of a unique solution y(x) G Cn-a[a, b] to the integral equation (3.2.43).We choose x\ G (a, b) such that the relation

holds, and rewrite the equation (3.2.43) in the form y(x) = (Ty)(x), where Tis operator (3.2.50). Using the relation (3.2.46) and the estimate (3.3.3) (with7 = n — a, b = x\ and a being replaced by a — aj), we have

\\Tyi -Ty2\\Cn_a[a,Xl] g AtY.WîlVi -i/2|]||c,_«[a,x1]j=0

sa-a- r(

g ! l - a)- ^ rwhich yields

IITyi - Ty2||Cn_Q[a,Xl ] g w||i/i - y2||cn_«[a,x1] (0 < w < 1).

Applying the Banach fixed point theorem (Theorem 1.9), we prove the existence ofa unique solution y(x) = y*(x) £ Cn-a[a,xi] for the equation (3.2.43) on [o,xx] .Further arguments are similar to those in Theorem 3.11.

The results of Theorem 3.10 and Theorems 3.11-3.12 can be also extendedto the system Cauchy type problem (3.2.52)-(3.2.53) in a product of the spacesC%l_ai[a,b], , C£™_am[a,6] denned in (3.3.1). We shall use the followingnotation:

Cn-a [[a, b]m] = Cni _Q1 [a, b] x x Cnm -aJa,b] , Co [[a, b]m] = C [[a, b]m],

C{:L [[a, b]m] = Can\_ai [a, b] x x C £ _ a m [a, b]. (3.3.22)

The following statements, generalizing Theorem 3.10 and Theorems 3.11-3.12,respectively, hold.

Theorem 3.15 Let j = , m, Uj £ K, rij - 1 < ctj ^ rij (rij G N). LetG be an open set in M.m and let fj : (a, 6] x G —¥ E be functions such thatfj [x, 2/i, , 2/m] e Cni -aj [a, b] for any (j/i , , ym) &G.

If yj(x) G Cnj-aj[a,b], then yj(x) satisfy the system of relations (3.2.52) and(3.2.53) if, and only if, yj(x) satisfy the system of integral equations (3.2.56).

In particular, if 0 < aj < 1 and yj(x) £ C\-aj [a, b\, then yj(x) satisfy thesystem of relations (3.2.57) and (3.2.58) if, and only if, yj(x) satisfy the systemof integral equations (3.2.59).


Theorem 3.16 Let the conditions of Theorem 3.15 be valid, and let there holdthe Lipschitzian conditions (3.2.60).

Then there exists a unique solution (t/i,--- , ym) to the system Cauchy type

problem (3.2.52)-(3.2.53) in the space Cl*[ a [[a,b] m].

In particular, if 0 < ctj < 1 (j = 1, , m), then there exists a unique solution

(yi, » 2/m) e C^[a [[a,b] m\] to the system Cauchy type problem (3.2.57)-(3.2.58)and to the system of weighted Cauchy type problems (3.2.61)-(3.2.62).

Remark 3.14 The results of Theorems 3.13 and 3.14 were proved by Kilbas andMarzan ([378], Theorems 1 and 3) under stricter conditions on / in some specialfunction spaces of y(x).

Remark 3.15 The results presented in Theorems 3.15 and 3.16 can be extendedto the system Cauchy type problem (3.2.40)-(3.2.41) and to the system of theVolterra integral equations (3.2.43).

3.3.5 Cauchy Type Problems for Linear Equations

In this subsection we present the existence and uniqueness of solutions for linearfractional differential equations considered in Section 3.2.5. First from Theorem3.11, Corollary 3.12 and Theorem 3.12, we derive the corresponding results for theCauchy type problems for linear differential equations of order a > 0.

Corollar y 3.14 Let a > 0 (n - 1 < a ^ n, n G N), and let a(x) G C[a,b] andg{x) G Cn-a[a,b].

Then the Cauchy type problem (3.2.64) for the linear differential equation oforder a has a unique solution y{x) in the space C"_a[a, b].

In particular, there exists a unique solution y{x) G C^_a[a, b] to the Cauchytype problem (3.2.65).

Corollar y 3.15 Let 0 < a ^ 1, and let a(x) G C[a,b] and g(x) G C\-a[a,b\.

Then the Cauchy type problem (3.2.66) and the weighted Cauchy type problem(3.2.67) have a unique solution y(x) in the space C"_a[a,b].

In particular, either of the Cauchy type problems (3.2.68) and (3.2.69) has aunique solution y(x) G C"_a[a, b].

The results of Corollaries 3.14 and 3.15 can be extended to more general frac-tional differential equations. Thus, from Theorem 3.14, we derive the followingassertions.

Corollar y 3.16 Let a > 0 (n - 1 < a g n; n G N). Let I G N and ctj > 0 (j =1, , I) be such that the conditions in (3.3.21) are satisfied. Let a,j(x) G C[a, b](j = 0,1, , I) and g(x) G Cn-a[a, b].

Then the Cauchy type problem (3.2.70) for the linear differential equation oforder a has a unique solution y(x) in the space C"_a[a, b].

In particular, there exists a unique solution y(x) G C"_a[a, b] to the Cauchytype problem (3.2.71).

Chapter 3 (§ 3.3) 171

Corollar y 3.17 Let 0 < a ^ 1. Let I G N and ctj G K (j = 1, , Z) 6e swc/i £/ia£(fte conditions in (3.3.21) are satisfied. Let aj(x) G C[a,b] (j = 0,1, , I) andg{x) G Ci_a[a,6].

Then the Cauchy type problem (3.2.72) and the weighted Cauchy type problem(3.2.73) have a unique solution y(x) in the space C"_a[a, 6].

In particular, either of the Cauchy type problems (3.2.74) and (3.2.75) has aunique solution y{x) G C"_a[a, &].

Theorem 3.16 yields the results for systems of Cauchy type problems of lineardifferential equations of fractional order.

Corollar y 3.18 Let j = 1,- , m, and let ctj > 0 be such n0 — 1 < ctj 5: nj(rij G N), and let a,j(x) G C[a,b] and gj(x) G Cnj-aj[a,b].

Then the system Cauchy type problem (3.2.76)-(3.2.77) has a unique solution

(yi, , Vm) in the space c £ [ a [[a, b]m].

In particular, the system Cauchy type problem (3.2.78)-(3.2.77) has a unique

solution (yi, , ym) G c £ lQ [[a, b]m].

Corollar y 3.19 Let j = 1,- , m, ctj > 0 be such that 0 < a31 1, and letaj(x) G C[a,b] and gj(x) G Ci-aj[a,b].

Then the system of Cauchy type problems (3.2.79) and the system of weightedCauchy type problems (3.2.80) have a unique solution (yi,--- , ym) in the spaceC \a\ !!„ Ulm]

In particular, either of the systems of Cauchy type problems (3.2.81) andweighted Cauchy type problems (3.2.82) has a unique solution

(y i , - - - , ym) & C^la[[a,b] m]

Remark 3.16 It follows from the proofs of the theorems in this section that themethod of successive expansions can be applied to construct exact unique solutionsto all Cauchy type problems considered in Section 3.3.

3.3.6 More Exact Spaces

In Sections 3.3.2 and 3.3.3, we gave sufficient conditions for the Cauchy typeproblems (3.2.4)-(3.2.5) and for the weighted Cauchy type problem (3.3.16) to havea unique solution y(x) G C"_a[a, b], provided that the right-hand side f[x,y(x)]of the fractional differential equation (3.2.4) belongs to the same space Cn-a[a,b]of weighted continuous functions y(x) on [a,b] defined in (3.3.2). When n - 1 <a < n (n G N), then f[x,y(x] and y(x) have a power singularity at the pointx = a and belong to the space L(a,b). This case reflects one of three possiblecases characterizing f[x,y(x] and y(x), indicated in Section 3.2.6 while consideringExamples 3.1 and 3.2:

Case 1. The right-hand side f[x, y(x)] is integrable on [a, b] together with thesolution y{x), and they both have integrable singularities at the point a.


Two other possible cases reflected in Section 3.2.6 are the following:Case 2. The right-hand side f[x,y(x)] is integrable on [a, b] with an integrable

singularity at the point a and the solution y(x) is bounded on [a, b].Case 3. The right-hand side f[x,y(x)} is bounded on [a,b] together with the

solution y (x).

From the above, we can derive that the spaces Cn-a[a, 6] for f[x,y(x)] andy(x), considered in Section 3.3.2, can be characterized more precisely. This alsofollows from the fact that the Riemann-Liouville fractional derivative D%+, as ausual derivative, can transform "good" functions at point a into "bad" functionsat a. For example, by (2.1.17),

^ (x - a)1-" ( a > 0 ), (3.3.23)K~a+*J\", T ( l -a)

so that, if 0 < a < 1, (x — a)1~a has an integrable singularity at point a.

In fact, Theorems 3.10, Corollary 3.12 and Theorems 3.11-3.12 remain trueif the condition f[-,y) : Cn-a[a, b] —> Cn-a[a, b] is replaced by a more generalcondition. Indeed, there holds the following assertion.

Property 3.1 Let a > 0 be such that n — 1 < a ^ n (n G N). Let G be anopen set in E and let f : (a, 6] x G —» R fee a function such that, for any y £ G,f[x, y] G C-yla, 6] with 7 6 l ( n - a ^ 7 < l ).

(a) If y(x) G Cn-a[a,b], theny(x) satisfies the relations (3.2.4) and (3.2.5) if,and only if, y(x) satisfies the integral equation (3.1.8).

(b) Let f[x,y] satisfy the Lipschitz condition (3.2.15).

Then there exists a unique solution y(x) to the Cauchy type problem (3.2.4)-(3.2.5) in the space C«_ai[a,b}:

CS_a,7[a, b] = {y{x) G Cn-a[a, b] : (D +y)(x) G C7[a, b]}. (3.3.24)

In particular, if 0 < a < 1, then there exists a unique solution of the Cauchytype problems (3.1.12)-(3.1.13) and (3.3.16) in Cf_Q>7[a,6].

(c) In particular, (a) and (b) hold for 7 = 0 when f[x,y] G C[a, 6] /or anyyeG.

Proof. The proof of (a) is similar to the proofs of Theorems 3.10 and 3.11 by usingLemma 2.9(a) instead of Lemma 3.3 for the proof of necessity.

To prove (b), we note that , y] : [a, b]xR -> Cy[a, b] with 7 G R (0 7 < 1),then the second term in (3.1.8) given by

f[t,y(t)]dt1 —(x-t)

in accordance with Lemma 2.9(a) and 2.9(b), belongs to C7_a[a, 6] for 7 > aand to C[a,b] for 7 ^ a. Thus it is clear that the first term yo(x) in the right-hand side of (3.1.8), defined by (3.2.19), belongs to Cn-a[a,b]. Since n G N, then

Chapter 3 (§ 3.3) 173

7 - a < n - a and n — a ^ 0. Then, by Property 1.1 in Section 1.1, the right-hand side of (3.1.8) belongs to Cn-a[a,b]. Therefore, the operator T denned by(3.2.18) maps Cn-a[a, b] onto Cn-a[a, b\. Now we choose Xi G (a, b) such that thecondition (3.3.8) is satisfied and repeat the proof of Theorem 3.11 to prove thatthere exists a unique solution y(x) to the Cauchy type problem (3.2.4)-(3.2.5) inthe space Cn-a[a, b] such that (3.3.15) holds.

Now we show that (D%+y)(x) G C7[o, b]. Since 7 ^ n — a, then applying(3.2.4), (3.2.15) and (1.1.37), we have

\\D°+ym - D°+y\\c^[a,b] = \\f[x,ym] - f[x,y]\\Cj[ a,b]

A\\ym - y\\Cy[a,b] S A(b - ay-n+a\\ym - y\\cn.a[a,b]

Thus, by the above two estimates and (3.3.15), we find that

\\Daa+ym - Da

a+y\\Cl[aM = 0-

Hence a unique solution y(x) G Cn-a[a,b] has the property (D%+y)(x) G C7[a,b].By (3.3.24), this means that y(x) G C"_a 7[a, b]. This completes the proof of (b).

From Property 3.1 we obtain the fourth possible case characterizing f[x,y(x)]and y(x) for the Cauchy type problems (3.2.4)-(3.2.5) and (3.3.16).

Case 4. The right-hand side f[x,y(x)] is bounded on [a, b] and the solutiony(x) is integrable on [a, b] with an integrable singularity at point a.

A property, similar to Property 3.1, exists for the generalized Cauchy type prob-lem (3.2.40)-(3.2.41), (3.2.47) and for systems of Cauchy type problems(3.2.52)-(3.2.53), (3.2.61)-(3.2.62).

Propert y 3.2 Let a > 0 be such that n - 1 < a ^ n (n G N). Let I G N\{1}and ctj > 0 (j — 1, , /) be such that the condition (3.3.21) is satisfied. Let Gbe an open set in Rl+1 and let f : (a, b] x G —> M. be a function such that for any{y,yi,--- , Vi) £ G, f[x,y,yi,--- , y{\ G C7[a,b] with 7 G E (n - a ^ 7 < 1).

(a) Ify(x) G Cn-a[a,b], then y{x) satisfies the relations (3.2.40) and (3.2.41)if, and only if, y(x) satisfies the integral equation (3.2.43).

(b) Let the Lipschitz condition (3.2.46) be satisfied, and for j = I , - - - ,nj, let(-0"+ J2/)(a+) = bkj G M. be fixed numbers, where nj — —[—ay].

Then there exists a unique solution y(x) to the Cauchy type problem (3.2.40)-(3.2.41) in the space C£_Q)7[a,6].

(c) In particular, (a) and (b) hold for 7 = 0 when f[x, y, y\, , yi] G C[a, b]for any (y , j / i , - - - , yi) G G.

Propert y 3.3 Let j = 1, , m, ctj > 0 be such that rij — 1 < ctj ^ rij (nj G N).Let G be an open set in KTO and let fj:(a,b]xG-^M. be functions such that forany (t/i, ym) G G, fj[x, j / i , , ym] £ Cli [a, b] with 7 G K (rij - a_,- ^ 7 < 1).

(a) If yj(x) G Cn_Qj[a, 6], i/ierz 2/ (2;) satisfy the system of relations (3.2.52)and (3.2.53) if, and only if, yj(x) satisfy the system of integral equations (3.2.56).


(b) Let fj(x,yi,--- , ym) satisfy the Lipschitz conditions (3.2.60).

Then there exists a unique solution (j/i,-- - , ym) to the system Cauchy type

problem (3.2.52)-(3.2.53) in the space C^[an {[a,b] m]:

c L Q i « , 7 [ M ] m ] = C £ _a i i 7 l [ a , 6] x x C ° : _ a m i 7 m[ a , 6 ] , (3.3.25)

where C " ' _a. 7\a,b] (j = I,--- , m) are defined in (3.3.24).

In particular, if 0 < a.j < 1, then there exists a unique solution (j/i , , ym) Gc i - a [[a>fo]™] i o i / l e m Cauchy type problem (3.2.57)-(3.2.58) and to the sys-tem of weighted Cauchy type problems (3.2.61)-(3.2.62).

(c) In particular, (a) and (b) hold for 7 = 0 when fj[x, j/ i , , ym] G C[a, b]for any (j/i,-- - , ym) £ G.

Cases 2 and 3 above appear when we consider the Cauchy type problem (3.2.4)-(3.2.5) with bn = 0. The results will be different for 0 < a ^ 1 and for a > 1.First we consider the Cauchy type problem (3.2.31) with 0 < a ^ 1:

{Daa+y){x) = f[x,y(x)] (0 < a g 1), (/o

1;°j/)(a+) = 0 (3.3.26)

and the weighted Cauchy type problem (3.2.33):

{D°+y){x) = f[x, y(x)} (0 < a 1), lim (x - af-ay{x) = 0. (3.3.27)

The integral equation (3.1.14), corresponding to the problem (3.3.26), takes theform

By Lemma 3.9(a), if f[x,y{x)} £ Cy[a,b] with any 7 G R (0 ^ 7 < 1), then theright-hand side of (3.3.28) and hence the solution y(x) belong to C7_a[a,6] for7 > a and to C[a, b] for 7 ^ a. From here we derive the result.

P rope r ty 3.4 Let 0 < a ^ 1 and 0 5! 7 < 1. Let G 6e an open set m K and Zei/ : (a, 6] x G —> M. be a function such that f[x, y] G Câ, b] for any y G G and theLipschitz condition (3.2.15) holds.

(a) // 7 > a, then the Cauchy type problems (3.3.26) and (3.3.27) have aunique solution y(x) in the space C"_Q 7[o, b]:

C"_a,7[a.b] = {9(x) e C7_a[a, 6] : (Z?°+ff)(a;) G C7[a, b}}. (3.3.29)

(b) 7/ 7 ^ a, i/ien t/ie Cauchy type problems (3.3.26) and (3.3.27) have aunique solution in the space Co)7[a, 6]:

C?,7[o,6] = {g(x) G C[a,b] : (Z}°+ff)(a;) G C7[o,6]} . (3.3.30)

(c) In particular, (b) holds for 7 = 0, provided that f[x,y] G C[a,b] for any

Chapter 3 (§ 3.3) 175

Proof. The derivation of Property 3.4 for the Cauchy type problem (3.3.26) issimilar to that of Property 3.1 due to the equivalence of this problem and theintegral equation (3.3.28) in the spaces C^-Q[a, b] and C[a, b] in Cases (a) and (b),respectively, in accordance with Property 1.1. The result for the weighted Cauchytype problem (3.3.27) follows from that for (3.3.26) on the basis of Lemma 3.5.

Next we consider the Cauchy type problem (3.3.1)-(3.3.2) with a > 1.

{Daa+y){x) = f[x,y{x)] (a > 1), (3.3.31)

with initial conditions

(D^-ky)(a+) = bk, & f c eR(* = l , - - . , n - l ) , bn = 0, (3.3.32)

where n = —[—a]. The Volterra integral equation (3.1.8) corresponding to thisproblem takes the form

rfe>fIf f[x,y(x)j € C-y[a,b] (0 ^ 7 < 1), then, by Lemma 2.9(a), the integral in theright-hand side of (3.3.33) is a continuous function on [a, b] (because 7 < 1 < a).The first term in the right-hand side of (3.3.33) is also a continuous function on[a, b]. Then we obtain the result.

Propert y 3.5 Let a > 1 and n = —[—a]. Let G be an open set in M. and letf : (a,b] x G —> M. be a function such that for any y E G, f[x,y] € C7[o, 6] with7 E M. (0 ^ 7 < 1) and the Lipschitz condition (3.2.15) is satisfied.

Then the Cauchy type problems (3.3.31) and (3.3.32) have a unique solutiony(x) E C%fi[a,b}.

In particular, this assertion holds for 7 = 0 when f[x,y] G C[a, b] for any

Proof. The proof of Property 3.5 is similar to that of Property 3.1 on the basisof the equivalence of the Cauchy type problem (3.3.31) and the integral equation(3.3.33) in the space C[a, b], by using Property 1.1 and Lemma 3.5.

Remark 3.17 Properties 3.4 and 3.5 contain Cases 2 and 3 above.

Properties, similar to Properties 3.4 and 3.5, are valid for the generalizedCauchy type problems (3.2.44), (3.2.47) and (3.2.40)-(3.2.41), respectively.

Property 3.6 Let 0 < a ^ 1 and 0 ^ 7 < 1. Let I E N\{1} and a, > 0(j = 1, , I) be such that the condition (3.3.21) is satisfied. Let G be an openset in E '+ 1 and let f : (a,b] x G —> M. be a function such that f[x,y,yi, , yi] EC~f[a,b] for any (y,j/i,--- , yi) E G, and the Lipschitz condition (3.2.46) is satis-fied. Also let (Ia+

a3 y)(a+) = bj € M. (j = 1, , I) be fixed numbers.


(a) If > a, then there exists a unique solution y(x) to the Cauchy typeproblems (3.2.44) with b0 = 0 and (3.2.47) with c = 0 in the space C7_a 7[a,b] .

(b) // 7 _ a, then there exists a unique solution y(x) to the Cauchy typeproblems (3.2.44) with bo = 0 and (3.2.47) with c = 0 in the space Cg7[o, b].

(c) In particular, (b) holds for 7 = 0, provided that f[x, y, j/ i , , yi] G C[a, b]for any (J / .J / I , - -- , Vi) G G.

Propert y 3.7 Let a > 1 and n = - [ - a ) ] . Let I G N\{1 } and otj > 0(j = 1, , /) 6e such that the condition (3.3.21) is satisfied. Let the fractionalderivatives D^'+y (j = 1, , /) exist on [a,b]. Let G be an open set in M.l+1 andlet f : (a, 6] x G —> E be a function such that f[x,y, j/ i , , yi] : C[a, b] —> C7[a, 6]/or any (y,2/i,-- ' > yi) £ G and the Lipschitz condition (3.2.46) is satisfied. Alsolet (7a +

a jy)(a+) = bj G M. (j = 1, , /) be fixed numbers.

Then the Cauchy type problem (3.2.40)-(3.2.41) with bn = 0 has a uniquesolution in the space CQ 0[a,b].

In particular, this assertion holds for 7 = 0, for any (y,yi,--- , yi) G G,provided that f[x,y,yir , J/J] G C[a,b].

Properties, similar to Properties 3.4 and 3.5, are also true for the systems ofCauchy type problems (3.2.61), (3.2.62) and (3.2.52)-(3.2.53), respectively.

Propert y 3.8 Let j = 1, , m, 0 < a,- _ 1, and 0 _ 7,- < 1. Lei G bean open set in Km. Also let the functions fj : (a, b] x G —\ K 6e suc/i i/iai/j[:r , 2/1, , ym] G C7 j [a, b] for any (j/i , , 2/TO) G G and i/ie Lipschitz conditions(3.2.60) are satisfied.

(a) 7/7j > Oj, iften the systems of Cauchy type problems (3.2.57)-(3.2.58) and(3.2.61)-(3.2.62) with bj = 0 (j = 1, , m) have a unique solution (j/i , , ym)in ifte space C]^la [[a,b] m]:

C l ra i « , 7 [ [ a , 6 ] r o ] = C ° l _ a i i 7 l [ a , 6] x x C ° " _ Q m > 7 m[ a , 6 ] , (3 .3.34)

where the spaces C"'_a. 7[ a, 6] are defined in (3.3.29).

(b) Ifjj ^ oij, then the systems of Cauchy type problems (3.2.57)-(3.2.58) and(3.2.61)-(3.2.62) with bj = 0 (j = 1, , m) have a unique solution (j/i , , ym)in the space c £7 [[a, b]m}:

cL a7 [[a, 6]ro] = C £7 1 [a, 6] x . .. x C«™m [a, 6], (3.3.35)

where the spaces CQ 7 [a, 6] are defined in (3.3.30).

(c) In particular, (b) holds for 7 = 0, provided that fj[x, j/ i , , ym] G C[a, b]for any ( i / i , - - - , y m ) G G.

Propert y 3.9 Let j = 1, , m, a,- > 1 and n = —[—oij], and let 0 ^ j3 < 1.Lei G be an open set in M.m and let the functions fj : (a, b] x G —> M 6e

Chapter 3 (§ 3.3) 177

that for any (j/i,-- - , ym) G G, fj[x,yi,--- , ym) G C7.[a,6], and i/ie Lipschitzconditions (3.2.60) are satisfied.

Then the system Cauchy type problem (3.2.52)-(3.2.53) with bnj = 0

(j = 1, , m) has a unique solution (j/i , , ym) G C ^ [[a, b]m].In particular, this assertion holds for jj = 0 when the functions

are such that fj[x,yi,-,ym] £ C[a,b] for any ( j / i , - - - , y m ) £ G.

Remark 3.18 The results in Properties 3.1-3.9 remain true for the Cauchy typeproblems of linear fractional differential equations and systems of such equationsconsidered in Section 3.3.5. In particular, Properties 3.1, 3.4, and 3.5 remain truefor the Cauchy type problems (3.2.64)-(3.2.69), Properties 3.2, 3.6, and 3.7 for theCauchy type problems (3.2.70)-(3.2.75), and Properties 3.3, 3.8, and 3.9 for thesystems of Cauchy type problems (3.2.76)-(3.2.82).

3.3.7 Further Examples

In Sections 3.2.3-3.2.5 we gave sufficient conditions for the Cauchy type problemsand systems of such problems to have a unique solution in some subspaces ofthe weighted spaces of continuous functions. Indeed, these conditions show thatthe differential equations of fractional order and systems of such equations canhave such solutions, provided that their right-hand sides are weighted continuousfunctions. But these conditions are not sufficient. Below we present two examplesand discuss them in connection with the results obtained in Sections 3.3.2 and3.3.3. The first example generalizes Examples 3.1 and 3.2 considered in Section3.2.6.


{D°+y){x) = X(x - af [y(x)]m (x > a; m > 0; m # 1) (3.3.36)

with real X,/3 G E (A / 0). Using (2.1.17), it is directly verified that, if

^ ^ - > - 1 , (3.3.37)1 — m

then the equation (3.3.36) has the explicit solution

y{x) = (a._a)(0+a)/(l-m)> (3.3.38)

Let

^ ^ (3-3.39)m -


The above solution y(x) G C~,-a[a, b] for 0 < 7 - a < 1, while y(x) G C[a,6] for7 — a fS 0:

y(x) G C7_a[a, 6], if 0 < 7 - a = ^ < 1, (3.3.40)

y(x)£C[a,b], if 7 - a = 4 0. (3.3.41)TO — 1

In this case the right-hand side of the equation (3.3.36) takes the form

f[x,y(x)} = A {x-a)(f)+am)l{1-m). (3.3.42)

The function /[a;,j/(a;)] G C7[a,6] when 0 < 7 < 1, and f[x,y(x)] G C[a, b] when7^0:

R J_ 7T)CY

f[x,y(x)]£C7[a,b], if 0 < 7 = , < 1, (3.3.43)TTh -L

/[^WleCffl,!], if 7 . (3.3.44)TO — 1

In accordance with (3.3.39), the following cases are possible for the spaces ofthe right-hand side (3.3.42) and of the solution (3.3.47):

(1) (3.3.43) and (3.3.40); (2) (3.3.43) and (3.3.41); (3) (3.3.4) and (3.3.41).

It is directly verified that the first case is possible when 0 < a < 1 and either ofthe following conditions holds:

TO>1, -a<f5<m-\- ma; (3.3.45)

0 < m < 1, m-l-ma</3< -a. (3.3.46)

The second case is possible when a > 0 and

TO > 1, — ma < (3 < TO — 1 — ma, /? —a;

or0 < T O < 1, m — 1 — ma < f3 < -ma, (3 - a.

When 0 < a < 1, these conditions take the following forms:

m > 1, -ma < f3 -a; (3.3.47)

0 < m < 1, -a^fi < -ma; (3.3.48)and if a 1, then

TO > 1, -ma < (3 < m - 1 - ma; (3.3.49)

0 < m < 1, TO- 1 -ma </3 < -ma. (3.3.50)

Chapter 3 (§ 3.3) 179

It follows from (3.3.41) and (3.3.44) that the third case in (3.3.7) is possible fora > 0 and either of the following conditions holds:

m > 1, p £ -ma; (3.3.51)

0 < m < 1, /3 -ma. (3.3.52)

It is directly verified that (3.3.38) is the explicit solution to the followingCauchy type problem:

(D:+y)(x)=X(x-af[y(x)]m, (ll+ ay)(a+) = 0 (0 < a < 1) (3.3.53)

and to the following weighted Cauchy type problem:

(D2+y)(x) = \(x-af[y(x)}m, lun+ [(x - a )1" ^ * ) ] = 0 (0 < a < 1), (3.3.54)

provided that any of the conditions (3.3.45),(3.3.46), (3.3.47), (3.3.48), (3.3.51)and (3.3.52) holds, and to the following Cauchy type problem

{Daa+y){x) = \{x-af[y{x)]m, (D^ky)(a+) = 0 (a > 0; k = 1, , n = -[-a]),

(3.3.55)provided that any of (3.3.49), (3.3.50), (3.3.51) and (3.3.52) is valid.

Now we establish the conditions for the uniqueness of the solution (3.3.38) tothe above problems by using Properties 3.4 and 3.5. For this we have to choosethe domain G and check when the Lipschitz condition (3.2.15) with

f(x,y) = X(x-afym (3.3.56)

is valid. We choose the following domain:

D = {{x,y)eC: a < x b, 0 < y < K(x - a)"; u e l ; K > 0} . (3.3.57)

This means that the domain G for y is unbounded for w < 0:

0<y < - u < 0} , (3.3.58){x — a)

and bounded for w 0:

G = {yeR: 0 < y < K{x - a)w; w > 0} . (3.3.59)

In particular, when w = 0,

G = {y€R: 0 < y < K}. (3.3.60)

To prove the Lipschitz condition (3.2.15) with f[x,y] given by (3.3.56), wehave, for any (x,yi), (x,y2) G D,

\f[x,yi) - f[x,y2}\ Z \X\(x - aY\y? - y?\. (3.3.61)


There hold the following relations:

| ! /1m- I /Jl |^m[max[| i /1|) | i /2| ] r-1| i /1- j /2| (m > 1), (3.3.62)

\yT-y?\ ^m[mm[\y1\,\y2\]}m-1\y1-y2\ (0 < m < 1). (3.3.63)

By (3.3.57), \y\ < K(x - a)". Therefore

max[|yi|,|y2|] <K(x-a)u, mm[\yi\,\y2\} <K(x-a)u,

and hence, for any m > 0 (m / 0),

\yT -y?\^m< mK(x - a)*"-1^? ~ vTl (3-3.64)

Substituting this estimate into (3.3.61), we obtain

\f[x,Vl] - f[x,y2]\ g \X\mK(x - af+^^ly, - y2\. (3.3.65)

If j3 + (m — l)w 0, then the last relation yields the following estimate:

\f[x,yi}-f[x,y2}\SA\yi-y2\, A = \X\mK(b - a)^™-1^. (3.3.66)

Thus the Lipschitz condition (3.2.15) is satisfied, provided that (3 + (m— l)cu 0.

Using the above results and Properties 3.4 and 3.5, we derive the uniquenessresult for the Cauchy type problems (3.3.53), (3.3.54) and (3.3.55).

Proposition 3.1 W 0 < a < 1, A,/3 e I (A / 0), m > 0 (m / 1) and7 = (/? + ma)/(m — 1). Let D be the domain (3.3.57), where w £ I is suchthat /? + (m - l)w 0.

(a) // either of the conditions (3.3.45) or (3.3.46) holds, then the Cauchy typeproblems (3.3.53) and (3.3.54) have a unique solution y(x) e C"_a 7[o, b] given by(3.3.38).

(b) // either of the conditions (3.3.47) or (3.3.48) is satisfied, then the Cauchytype problems (3.3.53) and (3.3.54) have a unique solution y(x) £ Cg7[a, b] givenby (3.3.38).

(c) // either of the conditions (3.3.51) or (3.3.52) is valid, then the Cauchytype problems (3.3.53) and (3.3.54) have a unique solution y(x) E CQ 0[a,b] givenby (3.3.38).

Proposition 3.2 Let a ^ 1, n = -[-a], A G 1 (A / 0), m > 0 (m / 1) andP G K. Let D be the domain (3.3.57), where w G E is such that j3 + (m- l)w 0.

If any of the conditions (3.3.49), (3.3.50), (3.3.51) and (3.3.52) holds, thenthe Cauchy type problem (3.3.55) has a unique solution y(x) G CQ0[a,!i] given by(3.3.38).

Remark 3.19 In particular, when m = 2 and m = 1/2, the conditions in (3.3.43)and (3.3.44) and the results in Propositions 3.1 and 3.2 yield more exact charac-terizations of weighted and usual spaces of continuous functions for the right-handside (3.3.42) and for the solution (3.3.38) of the Cauchy type problems (3.3.53)and (3.3.54) than was possible in Section 3.2 in the space of integrable functions.

Chapter 3 (§ 3.3) 181

Example 3.4 Consider the following nonlinear inhomogeneous fractional differ-ential equation of order a > 0:

(D%+y)(x) = \{x - af[y(x)]m + b{x - a ) " (x > a; m > 0 ; m / 1) (3 .3 .67)

with real a (A 7 0), 6, /3 and v. We suppose that

v = ^ , (3.3.68)1 — m

the condition (3.3.37) is satisfied and the transcendental equation

= 0 (3.3.69)+ 1 a ) [ a r + b ] r ( +— m J \1 — m

is solvable and £ = /J, is its solution. Then it is easily verified that the equation(3.3.67) has the solution

y(x) = n(x - o)0+*)/(i-"O . (3.3.70)

In this case the right-hand side of (3.3.67) takes the form

f[x,y{x)} = (A + b)(x - a)(0+°™)/(i-"O. (3.3.71)

The above solution y(x) and the function f[x,y(x)] belong to the spaces (3.3.40),(3.3.41) and (3.3.43), (3.3.44), respectively. Then, by using the same argumentsas in Example 3.3, from Properties 3.4 and 3.5 we obtain the uniqueness theoremfor the Cauchy type problem

(DZ+y)(x) = \(x-af[y(x)r + b(x-ay, (l£ay){a+) = 0 (0 < a < 1),(3.3.72)

for the weighted Cauchy type problem

{D°+y){x) = \(x-af[y(x)}m+b(x-ay, ^ + [(x - a)1""^*) ] = 0 (0 < a < 1),

(3.3.73)and for the Cauchy type problem

{Daa+y){x) = \{x-af[y{x)]m + b{x-aY, {Da

a-ky){a+) = 0, (3.3.74)

with a > 0 and k = 1, , n (n = —[—a]).

Proposition 3.3 Let 0 < a < 1, A,/3 e K (A 0), m > 0 (m 1) and7 = (/3 + ma)/(m — 1). Let D be the domain (3.3.57), where w £ I is suchthat p + (m — l)ui ^ 0. Let v be given by (3.3.68) and let the transcendentalequation (3.3.69) have a unique solution £ = /i.

(a) // either of the conditions (3.3.45) or (3.3.46) holds, then the Cauchy typeproblems (3.3.72) and (3.3.73) have a unique solution y(x) € C"_Q7[a, b] given by(3.3.70).


(b) // either of the conditions (3.3.47) or (3.3.48) is satisfied, then the Cauchytype problems (3.3.72) and (3.3.73) have a unique solution y(x) G CQ7[<2, b] givenby (3.3.70).

(c) // either of the conditions (3.3.51) or (3.3.52) is valid then the Cauchy typeproblems (3.3.72) and (3.3.73) have a unique solution y(x) G CQ 0[a, b] given by(3.3.70).

Proposit ion 3.4 let a > 0, A € 1 (A 7 0), m > 0 (m / 1) and ,0 6 1. Letv be given by (3.3.68). Let D be the domain (3.3.57), where u> G M. is such thatj3 (m — l)ijj, and let the transcendental equation (3.3.69) have a unique solution£ = **

/ / any of the conditions (3.3.49), (3.3.50), (3.3.51) and (3.3.52) holds, thenthe Cauchy type problem (3.3.74) has a unique solution y(x) G CgO[a, b] given by(3.3.70).

Remark 3.20 It is directly verified that (3.3.38) and (3.3.70) are solutions of theCauchy type problems (3.3.53)-(3.3.55) and (3.3.72)-(3.3.74) provided that thecondition (3.3.37) holds, or the equivalent one

/3 < m - 1 - a for m > l ; or m - 1 - a < /3 for 0 < m < l . (3.3.75)

In Propositions 3.1-3.4 we have proved the uniqueness of the above solutions in adomain D given in (3.3.57), provided that w g l b e such that j3 + (m — l)w ^ 0.The problem of the uniqueness of these solutions when /3 + (m — l)u < 0 is open.It is also an open problem to prove the uniqueness theorems for other domains.

We also note that the problem of uniqueness of the solutions (3.3.38) and(3.3.70) to the equations (3.3.36) and (3.3.67) was discussed by Kilbas and Saigo[396]. The solvability of the nonlinear equation (3.3.67) depends on the solvabilityof the transcendental equation (3.3.69). The positive solutions of such a transcen-dental equation were investigated by Karapetyants et al. [362].

3.4 Equations with the Riemann-Liouville Frac-tional Derivative in the Space of ContinuousFunctions. Semi-Global and Local Solutions

In Sections 3.2 and 3.3 we established sufficient conditions for the Cauchy typeproblems (3.1.1)-(3.1.2), (3.1.6), and (3.1.7), the generalized Cauchy type problems(3.2.40)-(3.2.41), (3.2.44), and (3.2.47), and the systems of Cauchy type problems(3.2.52)-(3.2.53), (3.2.57)-(3.2.58), and (3.2.61)-(3.2.62) to have a unique globalsolution in spaces of integrable and continuous functions. This section is devotedto the derivation of the conditions for the existence of a unique local continuoussolution to such problems when the initial conditions are applied at any pointXo G [a,b]. We shall use the same methods which were developed in Sections 3.2and 3.3 based on reducing the problems in question to Volterra integral equationsand using the Banach fixed point theorem. The results will be different in thecases XQ = a and xo > a. We begin with the simpler case.

Chapter 3 (§ 3.4) 183

3.4.1 The Cauchy Type Problem with Initial Conditions atthe Endpoint of the Interval. Semi-Global Solution

In this section we give conditions for the existence of a unique continuous solutiony(x) to problem (3.2.4)-(3.2.5) in the right neighborhood around point a. Suchsolutions given in part of the interval (a, b] we will call semi-global solutions. FromProperty 3.1 we derive the first result.

Theorem 3.17 Let a > 0 be such that n - 1 < a ^ n (n G N), and let f u e l(k = 1, , n). Let h G M. be a fixed number such that 0 < h < b — a, and let7 6 R (0 7 < 1). Let G be an open set in R and let f : {a,a + h] x G -» R bea function such that, for any y G G, f[x,y] G C^[a,b] and satisfies the Lipschitzcondition (3.2.15) for all x £ (a,a + h\.

If 7 ^ n — a, then there exists a unique solution y(x) to the Cauchy typeproblem (3.2.4)-(3.2.5) in the space C£_Q)7[a,a + h]:

C«_at7[a,a + h} = {y£Cn-a[a,a + h], D%+y G C7[a, a + h]} . (3.4.1)

The next result yields the semi-global solution to the Cauchy type problem(3.2.4)-(3.2.5) in the set G C R involving real numbers bu 6 1 in the initialcondition (3.2.5).

Theorem 3.18 Let a > 0 be such that n - 1 < a ^ n (n G N), and let K > 0,hi > 0 (0 < hi R be a functionsuch that (x — a)1f[x,y] G C(G) and the Lipschitz condition is satisfied: for any(x,yi), (x,y2) G G:

\f[x,yi]-f[x,y 2}\Â\yi-y2\ (A>0), (3.4.3)

where the constant A > 0 does not depend on x.

Then there exists an h, 0 < h hi, such that the Cauchy type problem (3.2.4)-(3.2.5) has a unique semi-global solution y(x) in the space C°_Q7[a, 0 + h] givenin (3.4.1).

Proof. As earlier in Section 3.3, first we prove the existence of a unique solutiony(x) G Cn-a[a, a + h]. By Property 3.1(b), it is sufficient to prove the existence ofsuch a unique solution to the Volterra equation (3.1.8). We again use the Banachfixed point theorem (Theorem 1.9 in Section 1.13).

LetM7= max \(x-a)y(x,y)\. (3.4.4)

{x,y)€G


We choose h such that

h = min

and

l'{ M 7r(i- 7)1/(71-7)'

<T(2a - n

Let U = Cn-a[a,a + h] be the space (1.1.21):

Cn-a[a, a + h] = I y(x) G C(a, a + h]

(3.4.5)

(3.4.6)

(x -

We introduce the subset C^_a[a, a + h] of Cn-a[a, a + h] as follows:

= < y(a;) G C(a, a + h] : maxI x£[a,a+h]

(x - a)- »y(x) -

(3.4.7)It directly follows that Cn-a[a, a+h] is the complete metric space with the distancegiven by

d{yi,y2) = \\yi-y2\\cn.a= max \(x - a)n-a[ yi(x) - y2(x)]\ (3.4.8)xe[a,a+h]

and that C^_a[a, a + h] is a. complete subset of Cn-a[a, a + h].We define in the space C^_a[a,a + h] the operator T by (3.2.18):

Using Theorem 1.9, we shall prove that there exists a unique functiony*(x) G C%_a[a,a + h] such that

(Ty*){x)=y*(x). (3.4.10)

For this it is sufficient to prove the following: (1) if y(x) G C°_Q[o, a + h], then(Ty)(x) G C°_a[a,a + h]; and (2) for any ylfy2 G Cn-a[a,a + h], the followingestimate holds:

- Ty2\\Cn_a £ w\\yi - J/2||cn_Q, 0 ^ (3.4.11)

Let y(x G C°_a[a,a + h]. Using (3.4.9) and (3.4.4), we have, for anyx e [a, a + h],

\f[t,y(t)]\dt

Chapter 3 (§ 3.4) 185

In accordance with (3.4.5), we also have

so that

maxxe[a,a+h]

and hence (Ty)(a;) G C°^_a[a,a + ft].Now we prove the estimate (3.4.11). Using (3.4.9), applying the Lipschitzian

condition (3.4.3), and taking (3.4.8) into account, we have

(x - a)n-a[(Tyi)(x) - (Ty2){x)] x-a)n-a r*l yi(t)]-y2(i\dt

<Ax - a)n~a fx (t - a)a-ndt

La)n~a fx (t - a)a-ndi(a) Ja \x ~ t) ~a \\yi -

<A [x - a)n-a(x - a)2a-"r(g - n + 1)T(2a-n + l)

T ( 2 a - n + l)

and hence

~ Ty2\\Cn_a ^

According to (3.4.6), w < 1, and hence, by Theorem 1.9, there exists a uniquesolution y*(x) e Cn_a[a, o + ft]. By Theorem 1.9, this solution is a limit of theconvergent sequence ym(x) = Tmyl{x):

lim \\ym - y*\\Cn-a =0, (3.4.13)

where y^(x) is any function in Cn-a[a, a + ft]. If at least one bk 0 in the initialcondition (3.2.5), we can take y$(x) = yo{x) with yo(x) denned by (3.2.19).

Further, it is proved similarly as in Property 3.1, that, if 7 ^ n — a, thenDay e C~,[a,a + h], and hence y(x) G C£_Q;7[a,a + ft].

Corollary 3.20 Let a G 1 (0 < a < 1), one? let K > 0, fti > 0 (0 < fti M 6e a function such that(x — o)7/[a;,y] 6 C((5) and the Lipschitz condition (3.4.3) is satisfied.

Then there exists an h, 0 < h ^ hi, such that the Cauchy type problem (3.1.6)and the weighted Cauchy type problem (3.1.7) have a unique semi-global solutiony(x)eC?_a^[a,a+h}.

Remark 3.21 When the conditions of Theorem 3.18 are satisfied with 7 = 0,Diethelm ([173], Theorem 4.1) proved the uniqueness of a solution y(x) £ C(0, h]to the Cauchy type problem (3.2.4)-(3.2.5) by applying Theorem 1.10 (see Section1.13) to the operator T defined by (3.4.9). The result in Theorem 3.18 provides amore precise characterization of the solution to the problem considered.

Remark 3.22 The results of Theorems 3.17-3.18 and Corollary 3.20 can be ex-tended to systems of Cauchy type problems (3.2.52)-(3.2.53), (3.2.57)-(3.2.58), and(3.2.61)-(3.2.62).

3.4.2 The Cauchy Problem with Initial Conditions at theInner Point of the Interval. Preliminaries

In Subsection 3.4.1 we considered the Cauchy type problem for the differentialequation (3.2.4) with the initial conditions (3.2.5) and with the weighted initialconditions given in (3.1.7). It does not seem to be possible to consider the usualCauchy conditions at the point a, because a solution y(x) to this problem, ingeneral, has a singularity at a and therefore it is not bounded and continuous atthe point a. Such usual conditions can be imposed at any inner point x0 G (a,b),because in this case the solution y(x) does not have a singularity at this point.

In this subsection we consider the nonlinear fractional differential equation(3.2.4) with the initial conditions given at the inner point XQ of a finite interval[a,b]:

(DZ+y)(x) = f[x,y(x)} (a > 0; x > a) (3.4.15)

y{k){x0) = bk £R(a<x0 <b; k = 0 , 1 ,- ,n - 1), (3.4.16)

In particular, for 0 < a < 1, the simplest such problem has the form

{D2+y){x) = f[x,y(x)] (0 < a < 1), y(x0) = boeR. (3.4.17)

We give the conditions for the existence of a local continuous solution y(x) to thisCauchy problem in a neighborhood around the point XQ.

When n G N, then, by (2.1.7), the problem (3.4.15)-(3.4.16) is reduced to theCauchy problem for the ordinary differential equation of order n G N on [a, b\:

y("\x) = f[x,y{x)], yW(xo) = bk£R{a<xo<b; k = 0 , 1 ,- , n - l ) . (3.4.18)

The existence of a local continuous solution y(x) to this problem in a neighborhoodof x0 is well known [see, for example, Erugin [251]]. The proof of this result is

Chapter 3 (§ 3.4) 187

based on the equivalence of (3.4.18) to the Volterra integral equation of the secondkind of the form (3.2.12):

y{x) =x0 j=0 J-

(a^x b). (3.4.19)

Therefore, we shall consider the Cauchy problem (3.4.15)-(3.4.16) for a $ Nwith n = [a] + 1. We show that this problem, for the case when the point XQdoes not coincide with the endpoint a, can also be reduced to a Volterra integralequation. Such a Volterra integral equation has the form (3.1.8):

g (3-4-20)

where I"+ is the Riemann-Liouville fractional integration operator (2.1.1), andconstants dj(j = 0,1, , n — 1) are solutions to the following system of algebraicequations:

n - l

^ T(a + j - n - k + 1)= (s0 - a)k [bk - (J°+*/[* , (3-4.21)

(fc = O, l , -" , n - l ) .

The determinant A n of this system is given by

A n =

r(a-2n+2) r(a-2n+3) ' " T(a-n+l)

Lemma 3.6 The following formula holds for the determinant An:

r(a-n+l)

T(a-n)

1

T(a-n+2)

r(a-n+l)

1

T(a-l)

1

(3.4.22)

A n =r(a)r(a-l)---r(a- n

n - l= n3=0

(n G N), (3.4.23)

w/iere 0! = 1 and j ! = 1 2 3 j (j G N).

/n particular,

1 1

v{ay r(a)r(a-l) ' T(a)r(a-l)r(a-2)'(3.4.24)

Proof. The formula (3.4.23) is proved fairly easily.

Using (3.4.21) and (3.4.23), the constants dj (j = 0,1, , n - 1) in (3.4.20)can be found by Cramer's rule.


Lemma 3.7 The constants dj (j = 0,1, , n — 1) in (3.4.20) are given by thefollowing relations:

[b0 - (l%+f[t,y(t)}) (xo)]- a) [h - (IZ+1 f[t,y(t)]) (x0)] r(a-n+l)

1

1

r(a-2n+3) r(a-n+l)

r(a-n+i)

r(a-n) r(a-n+l)

1 1 . . . (T _ n\n-l \U _ lja-n+1 erT(a-2n+2) T(a-2n+3) ^ ° " ' L""" 1 V*a+ J I

(3.4.25)

In particular, when 0 < a < 1, then

do = b0 - (Ia+f[t, y{t)}) (x0), (3.4.26)

and the equation (3.4.20) takes the following form:

y(x) = (l?+f[t, y(t)}) (x) + (j^j [bo - (/?+/[*, y{t)]) (x0)] . (3.4.27)

If 1 < a < 2, then

d0 = , _ . [b0 - {l"+f[t,y(t)]) (x0)] - °, > [h - {l%+ lf[t,y(t)]) (XQ)] ,

(3.4.28)

^ ) ^ )(3.4.29)

and (3.4.20) takes the following more complicated form:

y(x) = (lZ+f[t,y(t)\) (x)

Chapter 3 (§ 3.4) 189

where d0 and d\ are given by (3.4.28) and (3.4.29), respectively.

To prove the equivalence of the Cauchy problem (3.4.15)-(3.4.16) and theVolterra integral equation (3.4.20), we need a preliminary assertion which followsdirectly from Lemmas 2.9(b) and 2.9(d).

Lemma 3.8 Let a > 0 and n = —[—a]. Also let (I™+ag)(x) be the Riemann-Liouville fractional integral (2.1.1).

(a) If g(x) G C(a, b], then, at any point x G (a, b], the following formula holds:

(Daa+IZ+g){x)=g{x). (3.4.31)

(b) If g(x) G C(a,b] and (Iâg)(x) G C(a,b], then a relation of the form(2.1.39):

{I«+D«+g){x) = g(x) - r (^+

+ l^+l) (x ~ aT+i 'n (3.4.32)

holds at any point x G (a, b\.

In particular, ifO<a<l, then

{i: +Daa+g){x) = g(x) - {

x _ a)<*-i . (3.4.33)

3.4.3 Equivalence of the Cauchy Problem and the VolterraIntegral Equation

The following statement yields the equivalence of the Cauchy problem (3.4.15)-(3.4.16) and the Volterra integral equation (3.4.20) in a neighborhood of a pointx0 G (a, b).

Theorem 3.19 Let a > 0 be such that n - 1 < a < n (n G N), and let bk G K.(k = 0,1, , n — 1) be given constants. Let a < XQ 0 be suchthat [XQ — h, xo + h] C (0, b], and let U be an open and connected set in M. LetD = [xo — h, xo + h] x U and let f(x, y) be a function continuous in D.

If y(x) G C[x0 - h,x0 + h], then y(x) satisfies the relations in (3.4.15) and(3.4.16) if, and only if, y(x) satisfies the Volterra integral equation (3.4.20), inwhich the constants dj (j = 0,1, , n - 1) are given by (3.4.25).

Proof. Let y(x) satisfy the relations (3.4.15) and (3.4.16). Since a < xo — h andxo + h ^ b and y(x) G C[XQ — h,xo + h], we can extend a function y(x) to thecontinuous function y* on (a, 6]:

y*(x)£C{a,b], y*(x)=y{x) (x e[x0 - h,xo + h]). (3.4.34)

Therefore, we can apply Lemma 3.8(b). Using (3.4.32) with g(t) = y*(t), we have

(ia+Da+y ){x) = y ix) ~


at any point x E {a,b\. Thus, in accordance with (3.4.34), the relation

holds at any point x € [x0 - h, x0 + h].

Applying the operator I"+ to both sides of (3.4.14) and taking (3.4.35) intoaccount, we obtain, for x E [xo — h, xo + h],

ST \Aa+ y) \u^) ,_ â+i-n (rn_X < < ,

(3.4.36)Differentiation of this expression n - 1 times gives for k = 0,1, , n — 1

(3.4.37)Multiplyin g both sides of this relation by (ZQ — a)k and using (3.4.16), we have

xo)] (k = 0,1, , n - 1). (3.4.38)

Setting

U ) " (i = 0,1, , n - 1), (3.4.39)

from (3.4.36) and (3.4.38) we obtain the relations (3.4.20) and (3.4.21), respec-tively. According to Lemma 3.8, the constants dj (j = 0,1, , n - 1) in (3.4.20)are given by (3.4.25). Thus the necessity is proved.

Now let the relation (3.4.20) hold, in which the constants dj (j = 0,1, , n-1)are given by (3.4.25). Then, in accordance with the proof of the necessity, theseconstants satisfy the system (3.4.21). By the above proof, (3.4.21) is equivalent to(3.4.38), and substitution of (3.4.38) into (3.4.37) yields the relations in (3.4.16).Now we prove (3.4.15). Since y(x) G C[x0 — h,x0 + h] and f(x,y) is a continuousfunction in D = [x0 — h,xo + h] x U, the function f[x,y(x)] is also continuouson [x0 - h,x0 + h]. Then, similarly as (3.4.34), we can extend f[x,y(x)] to thecontinuous function f*[x,y(x] on (a,b]:

f*[x,y(x)}eC(a,b], f*[x,y*(x)} = f[x,y(x)} (x £ [x0 - h,x0 + h]). (3.4.40)

Chapter 3 (§ 3.4) 191

Thus the integral equation (3.4.20) can be extended from [x0 — h,xo + h] to (a, b]:

*)+£ r ( t t+ /i n+1) (T^ T(3.4.41)

We can apply Lemma 3.8(a) to the function g(t) = f*[t, y*(t)] and obtain, accord-ing to (3.4.31),

{Daa+I2+r[t,y*{t)]) (x) = f*[t,y*(x)}. (3.4.42)

By (2.1.21), we have

(Dfla+(t - a)a+j~ny) (x)=0 (j = 0,1, , n - 1). (3.4.43)

Applying the operator D"+ to both sides of (3.4.41) and using (3.4.42) and(3.4.43), we obtain

From here, in accordance with (3.4.34) and (3.4.40), we derive (3.4.15), and thesufficiency is proved. This completes the proof of Theorem 3.19.

Corollar y 3.21 Let 0 < a < 1 and b0 G R. Let a < x0 < b, let h > 0 be suchthat [XQ — h, xo + h] C (a, b], and let U be an open and connected set in M.. LetD = [XQ — h, XQ + h] x U and let f(x,y) be a continuous function on D.

If y(x) G C\XQ — h, Xo + h], then y{x) satisfies the relation in (3.4.17) at anypoint x £ [XQ — h, Xo + h] if, and only if, y(x) satisfies the Volterra integral equation(3.4.27).

Remark 3.23 The results of Theorem 3.19 and Corollary 3.21 can be extendedto systems of the Cauchy type problems (3.2.52)-(3.2.53), (3.2.57)-(3.2.58), and(3.2.61)-(3.2.62).

3.4.4 The Cauchy Problem with Initial Conditions at theInner Point of the Interval. The Uniqueness of Semi-Global and Local Solutions

Now we establish the existence of a unique semi-global solution y(x) to the Cauchyproblem (3.4.15)-(3.4.16) in the neighborhood of an inner point x0 of (a, b) and ofa unique local solution at the point x0. We begin from the conditions for a uniquecontinuous solution y (x) G C[xo — h,xo + h] under the conditions of Theorem 3.20,and an additional Lipschitzian-type condition of f(x, y) with respect to the secondvariable of the form (3.4.3).

First we derive a unique solution to the Cauchy type problem (3.4.17).

Theorem 3.20 Let 0 < a ^ 1, and let x0 € (a, b), K > 0, and hx > 0 be suchthat hi < min[a:o — a, b — xo], and let bo G K. Let D be a set

D = {{x,y) e l 2 : \x-xo\^h!, \{x - af-ay - {x0 - af-abo\ K) .(3.4.44)


Let f(x, y) : D —) E be a continuous function in D such that the Lipschitz condition(3.4.3) with constant A > 0 holds for any (x,yi), (x,j/2) G D.

Then there exists an h, 0 < h ^ such that the Cauchy type problem

{Daa+y){x) = f[x,y{x)} (x G [xo -h,xo + h]), y(x0) = b0 G K (3.4.45)

has a unique semi-global solution y(x) G C[XQ — h, xo + h].

Proof. Since f{x,y) : D —> IR is a continuous function in D, it is bounded in D,and we have

M = \\f(x,y)\\c{D)= max | / (a: ,y) |. (3.4.46)(i!l)6D

We choose /i > 0 such that

h = mm hi,-—, -——3 \4M(xo-a)

l/a 4Aha

< 1. (3.4.47)

By Corollary 3.21, it is sufficient to prove the existence of a unique solutiony(x) G C(J) to the corresponding Volterra integral equation of the form (3.4.27):

y(x) = {I%+g[t,y(t)]) (x) - ( -2——J [y(x0) - {I%+g[t,y(t)]) (x0)] . (3.4.48)

For this we use the Banach fixed point (Theorem 1.9 from Section 1.13). We

denote Ju = [xo — h,xo + h]. We introduce the subspace U = C a{J) of the spaceC(J):

Ch,a(J) = {y(x) G C(J) : ||(a: - o)1-Qi/(a;) - {x0 - a)1-ay(x0)\\C(j) K} .(3.4.49)

It directly follows that Ch,a(J) is a complete metric subspace of C(J) with thedistance given by

d(yi,V2) = 112/i -V2\\c(J) :=max|t/i(x) - y2{x)\. (3.4.50)

Let T be the operator defined, for y G Ch,a{J), by the left-hand side of (3.4.48):

(Ty)(x) = {l2+g[t,y(t)]) (*)

(3.4.51)Using Theorem 1.9, we shall prove that there exists a unique functiony*{x) G Ch a{J) such that

(Ty*){x)=y*{x). (3.4.52)

It is enough to prove that (Ty)(x) G ChtCX(J) and that T : Ch,a(J) -> Ch,a(J)satisfies the relation (1.13.2), which, in accordance with (3.4.50), takes the form

(0 ^ w < 1). (3.4.53)

Chapter 3 (§ 3.4) 193

First we show that {Ty)(x) G Ch,a(J). Since y(x) G C(J), thenf[x,y(x)] G C(J) since it is a composition of two continuous functions. Thus, byLemma 3.3 (with 7 = 0), (/£+/[*, y(t)]) (x) G C(J), and hence (Ty)(x) G C(J).Now we establish the following estimate:

IKi - a)l-a(Ty)(x) - (x0 - a)l-a(Ty)(x0)\\c{J) Z K. (3.4.54)

Using (3.4.51) and (3.4.46), we have, for any x G J = [XQ - S, x0 + 6],

\x - af-a(Ty)(x) - (x0 - af-a(Ty)(xo)\

M{x-a)l-a rxfX a-l

I xa—h

- ^ — [(x - ay-a(x -xo + h)a + (x0 - ay-aha]

Since x 5: x$ + h and, by the first relation in (3.4.47), h (xo — a)/3, then

\\(x - ay-a(Ty){x) - (x0 - a)l-a(Ty)(x0)\\c{J)

, 3Mha , , u _ a . 4Mha

and

- (x0 - aîTy^xo^j) g ^ ^ ^ (3-4-55)

By the first relation in (3.4.47), 4Mrfe°J^°~a) ^ K, and thus (3.4.55) yields (3.4.54),

and hence (Ty)(x) G Ch,a(J)-Next we establish the estimate (3.4.53) with

where A is the Lipschitz constant in (3.4.3). Let yi(x),y2(x) G Ch,a(J)- Using(3.4.51) and the Lipschitz condition (3.4.3), we have, for any x G J,

1 rr(a) yxo_


-V2\\c(J) {x-xo + h)a — a

x — a

l-a

Since x0 — h x x0 + h and h ^ (x0 - a)/3, then

\\(Tyi)(x) - (Ty2)(x)\\c{J)

T{a(2hY — a

- y2\\c(j)2 +

— a — h

x0 - a

l - a

ha

— a — h

l - a

Aha\\yi -y2\\c(j) 4Aha

2j = -

This proves the relation (3.4.53) with w given in (3.4.56).According to the second formula in (3.4.47), w < 1, and hence, by Theorem

1.9, there exists a unique function y*(x) £ Ch,a{J) such that the formula (3.4.52)holds. Thus y(x) = y*(x) is a unique solution to the Volterra integral equation(3.4.52), and hence, in accordance with (3.4.51), also to the integral equation(3.4.27).

From Theorem 3.20 we derive the existence of a unique continuous local solu-tion to the Cauchy problem (3.4.18).

Theorem 3.21 Let 0 < a ^ 1. Let U be an open and connected set in R and17 = (a,b) x U. Let f(x,y) : (a, b) x U —> M. be a continuous function satisfyingthe Lipschitz condition (3.4.3).

Then, for any {xo,yo) £ fi, there exists a positive number h > 0 such that[xo — h, Xo + h] C (o, b) and there exists a unique function y(x) : [XQ — h, Xo + h] —> Usuch that y(x) G C[XQ — h,xo + h] and

y(x0) = i/o, (3.4.57)

{D°+y){x) = f[x,y{x)] for any x £ [x0 - h,x0 + h]. (3.4.58)

Proof. Let P = {xo,yo) E 0 be any point on fi. We fix this point P and choosehi > 0 and K > 0, depending on P and such that hi < minf o — a,b — XQ],and denote by D the closed set (3.4.44). Since f(x,y) : (a, b) x U —> M. is acontinuous function satisfying the Lipschitz condition (3.4.3) on fi, this functionis also continuous and satisfies the Lipschitz condition on D. Then, on the basisof Theorem 3.20, there exists a positive number h > 0, depending on the point P,and there exists a unique continuous function y(x) £ C[xo — h, xo + h] such that therelations (3.4.57) and (3.4.58) hold. Since (xo,yo) is any point on Q,, at any suchpoint there exists a local solution y(x) to the Cauchy problem (3.4.57)-(3.4.58).

The next result, generalizing Theorem 3.20, gives conditions for a unique semi-global solution to the Cauchy problem (3.4.15)-(3.4.16).

Chapter 3 (§ 3.4) 195

Theorem 3.22 Let a > 0 be such that n - 1 < a ^ n (n G N). Let x0 £ (a,b),K > 0, and fti > 0 6e SMC/J i/iof /ii < min[xo — a, 6 — xo], and /ei ^ g M(A = 0,1, , n — 1). iei D be a set given by

D={{x,y)£R2 : \x - xo\ hu |(* - a)n~ay - (x0 - a)n~ab0\ g K] .(3.4.59)

Let f(x, y) : D —> M. be a continuous function in D such that the Lipschitz condition(3.4.3) with the constant A > 0 holds for any (x,yi), (x,2/2) G -D-

T/ien there exists an h, 0 < h h\, such that the Cauchy type problem

(D°+y){x) = f[x,y(x)} (x G [xo-h,xo+h]), y{k)(x0) = i i e K ( i = 0 , l , - ) n - l )(3.4.60)

has a unique solution y(x) 6 CT[XQ — h,xo + h}.

Proof. When a = n G N, this theorem is well known. If a £ N, Theorem 3.22is proved similarly as Theorem 3.20. Indeed, on the basis of Theorem 3.19, it issufficient to prove that there exists a unique solution y(x) G C"~l[x0 - h,x0 + h]to the integral equation (3.4.20). For this we use the fixed point theorem byintroducing the subspace U = C^~^{J) (J = [XQ — h, xo + h}) of the space Cn~1{J):

( n - l

CViJ) = V(x) G Cn~l(J) £ II(z " a)n+k-ayW(x)I k=0

-(x0 - a)n+k-ay(k\xo)\\c(j) g K) (3.4.61)

and taking for T the operator in the right-hand side of (3.4.20):

From Theorem 3.22 we derive the existence of a unique continuous local solu-tion to the Cauchy problems (3.4.15)-(3.4.16).

Theorem 3.23 Letn-1 < a ^n {n £ N), and let bk G E (fc = 0,1, , n - l ) begiven constants. Let U be an open and connected set in M and Cl = (a,b) x U. Letf{x,y) : (a, b) x U —> ffi. be a continuous function satisfying the Lipschitz condition(3.4.3).

Let r = n for a = n G N and r = n — 1 for a (£ N. Then, for any(ô,2/o) £ ^7 £/iere exists a positive number h > 0 sue/* t/ia£ [xo — h, Xo + h] C (a, b),and there exists a unique function y(x) : [xo — h, xo + h] —> U such thaty{x) G Cr[x0 — h,x0 + h] and

y{xo) = bo, y'{xo) = bu . . ., y(n-1){xo) = bn-u (3.4.63)

(D°+y){x) = f[x,y(x)] for any x e [x0 - h,x0 + h]. (3.4.64)

Proof. Theorem 3.23 is proved similarly as Theorem 3.21 by using Theorem 3.22.


Corollar y 3.22 Let n - 1 < a ^ n (n £ N). Let x0 £ (a,b) and bk £ K(k — 0,1, , n — 1). Lei a(a;),g(a;) £ C(a,b) and j3 0. Lei £/ fee an open andconnected set in M and /ei f{x,y) : (a,b) x U —> M. be a continuous function.

Let r = n for a = n £ N and r = n — 1 /or a $ N. Then, for any (#0,2/0) £ fi,i/iere exists a positive number h > 0 suc/i £/m£ [a:o — h,Xo + h] C (a, 6) and thereexists a unique function y(x) : [xo — h, xo + h] —¥ U such that

y(x) £ Cr[x0-h,x0 + h]; and y{k)(x0) = h £ E (fc = 0,1, , n - 1 ), (3.4.65)

(I£+l/)(a0 = (x - afa(x)y(x) + g(x) (x £ [x0 -h,xo + h]). (3.4.66)

In particular, if 0 < a < 1, then there exists a unique continuous functiony(x) £ C\XQ — h, xo + h] such that y{xo) — b £ M., and

{D«+y){x) = (x- afa{x)y{x) + g(x) (x £ [x0 -h,xo + h]). (3.4.67)

Remark 3.24 The space Ch,a(J) (J = [xo-h,xo + h]) with 0 < a < 1 in (3.4.49)characterizes the behavior of the unique solution y(x) to the Cauchy problem(3.4.17) near the point a in the case when h is chosen in such a way that xo isnear the point a. When 0 < a < 1 and a -> 1, then (3.4.49) tends to the spaceSh,i(J) = Sh(J):

Ch(J) = {y(x) £ C(J) : \\y(x) - (xo)\\c{j) K} . (3.4.68)

Such a space is usually applied in the proof of a unique solution to the Cauchyproblem (3.4.18) for the ordinary differential equation of order n £ N.

Similarly, the space C^~^{J) in (3.4.61) characterizes the behavior of theunique solution y(x) to the Cauchy problem (3.4.15)-(3.4.16) and of its deriva-tives j / f c ' (x) (k = 1, , n — 1) near the point a for the case when h is chosen insuch a way that zo is near the point a.

Remark 3.25 The results of Theorems 3.20-3.23 can be extended to the sys-tems of the Cauchy type problems (3.2.52)-(3.2.53), (3.2.57)-(3.2.58), and (3.2.61)-(3.2.62).

The results of Theorems 3.20-3.23 can also be extended to the vectorial casewhen a function y{x) is replaced by a vector-function y = (yi, j/2» > Dm)- Suchan extension of Theorem 3.23 in the case 0 < a < 1 was established by Hayek etal. ([335], Theorem 3.1).

3.4.5 A Set of Examples

In Sections 3.4.1 and 3.4.4 we gave sufficient conditions for a unique semi-globalcontinuous solution of Cauchy type problems with initial conditions at the endand inner points of a finite interval, respectively. Below we present two examples.The first one generalizes Examples 3.1 considered in Section 3.2.6.

Chapter 3 (§ 3.4) 197

Example 3.5 Consider the following Cauchy type problems for the differentialequation of fractional order 0 < a < 1:

{Daa+y){x) = \{x - af[y{x)]m + g(x) (x > a), (I1

a+ay)(a+) = b £ R, (3.4.69)

(D%+y)(x) = \{x - af[y{x)]m + g(x) (x > a), Jiim+ [(x - o)1-ay(a:)] = b £ R,

(3.4.70)with real m > 0 (m # 1), A £ R (A 0), /? £ K. such that ,0 ^ m(l - a). (3.4.69)and (3.4.70) are Cauchy type problem (3.1.6) and (3.1.7), respectively, with

f(x,y) = X(x-afym+g(x). (3.4.71)

Let K > 0, /ii > 0 (0 < /ii < 6 - a) and G be the set (3.4.14). Also letg(x) £ C-y[a, hi] with 1 — a 7 < 1. According to (3.4.14), for {x,y) £ G, we have

l y l ^ O c - o ) 0 - 1 , A=^-)+K ((x,y)eG). (3.4.72)

By (3.4.71) and (3.4.72), \f(x,y)\ ^ \X\Am(x - a)^+"»(«-i) + |ff(a;)| for any(cc,y) £ G, and hence, since f3 m(l — a), f(x,y) is bounded on G,

| / ( z , l O | ^ M, M=|A|^m( fo-a)/ 3+™(a-1) + | |5| |c {(x,y)eG). (3.4.73)

We note that, according to (3.4.72), \y\ ^ A(x - a ) " "1. Using this relation andthe formulas (3.3.62) and (3.3.63) in respective cases m > 1 and 0 < m < 1 it isdirectly verified that the Lipschitz condition (3.4.3) holds for the J(x,y) given by(3.4.71). Thus, from Corollary 3.20, we derive the result.

Proposition 3.5 Let 0 < a < 1, m > 0 (m 1) and j3 € R be such thatfi m ( l - a ), and Zei A £ E (A ^ 0) and b £ E. Let i f > 0, /ii > 0 (0 < hx < b-a),and let G be the set (3.4.14) and g(x) £ Cj[a,a + hi] with 1 — a ^ 7 < 1.

Then there exists an h, 0 < h hi, such that the Cauchy type problem (3.4.69)and the weighted Cauchy type problem (3.4.70) have a unique semi-global solutiony(x) in the space C"_a ^[a,a + h\.

In particular, the Cauchy type problems (3.4.69) and (3.4.70) with g(x) = 0:

(D-+y)(x) = X(x - af[y(x)r (x > a), (Iây)(a+) = b £ 1, (3.4.74)

{D2+y)(x) = \{x-af[y{x)]m (x > a), lim [(a; - a)l-ay(x)} = b £ R, (3.4.75)x—>a+

have a unique semi-global solution y(x) € C"_Q 7[a, a + h].

Remark 3.26 It was proved in Example 3.3 that the homogeneous differentialequation (3.3.36) has the explicit solution


Applying the same arguments as in the proof of Proposition 3.1, we obtain thefollowing result.

Proposition 3.6 Let 0 < a < 1, 0 < m < 1 and(3 G R be such that P ^ m ( l - a ),and let A G R (A 0) and b e t. Let K > 0, hi > 0 (0 < hi < b - a), let G bethe set (3.4.14), and let 1 - a ^ 7 < 1.

JTien there exists an h, 0 < h hi, such that the Cauchy type problem (3.4.74)and the weighted Cauchy type problem (3.4.75) have a unique semi-global solutiony(x) G C"_a ^[a,a + h] and this solution is given by (3.4.76).

Example 3.6 Consider the Cauchy problem for the following differential equationof fractional order (0 < a < 1):

(D2+y)(x) = \{x - af[y{x)]m + g(x) (x > a ), y(x0) = bo£R(a<xo< b),(3.4.77)

where m > 0 (m 1), A G R (A ^ 0) and /3 G R such that /? ^ m(l - a). (3.4.77)is Cauchy type problem (3.4.45) with f(x,y) = X(x — a)^[y(x)]m + g{x) given in(3.4.71). Let K > 0 and hi > 0 be such that hi < min[a;o -a,b-x0], let D be theset (3.4.44), and let g(x) G C[x0 -hi,xo + hi]. It is clear that f(x, y) G C{D). By(3.4.44), similarly as (3.4.73), it is shown that f{x,y) is bounded on D: for any(x,y) efl,

V + \\g\\c, A1 = \bo\(x - xo)^1 + K.(3.4.78)

Applying (3.4.78) and using (3.3.62) and (3.3.63) in respective cases m > 1 and0 < m < 1, it is directly verified that the right-hand side of (3.4.77) satisfies theLipschitz condition (3.4.3). Then Theorem 3.20 yields the following assertion.

Proposition 3.7 Let 0 < a < 1, m > 0 (m 1) and /3 G 1R be such thatP ^ m ( l - a ), and let X e M (A / 0) andb0 e l . LetK > 0 and hi > 0 be such thathi < mm[xo—a,b — Xo], and let D be the set (3.4.44) andg(x) G C[XQ — hi,xo + hi].

Then there exists an h, 0 < h ^ hi, such that the Cauchy problem (3.4.77) hasa unique semi-global solution y(x) G C[XQ — h,xo + h].

In particular, the Cauchy problem

(D%+y){x) = X{x - af[y{x)]m (x > a), y(x0) = b0 G K (a < x0 < b) (3.4.79)

has a unique semi-global solution y(x) G C[XQ — h, XQ + h].

3.5 Equations with the Caputo Derivative in theSpace of Continuously Differentiable Func-tions

In Sections 3.3 and 3.4 we gave the conditions for the existence of unique global andsemi-global and local continuous solutions for the Cauchy type problems (3.2.4)-(3.2.5), (3.1.6), (3.1.7) and (3.4.15)-(3.4.16), and (3.4.17) when the initial con-ditions are given at the Endpoint a of a finite interval [a, b] and at any point

Chapter 3 (§ 3.5) 199

XQ G (a, b), respectively. Here we consider similar problems for the nonlinear dif-ferential equations with the Caputo derivative (c D"+y)(x) denned in (2.4.1). Weshall use the same methods which were developed in Sections 3.4 based on reduc-ing the problems considered to Volterra integral equations and using the Banachfixed point theorem. The results will be different for the cases x0 = a and xo > a.We begin with the simpler case.

3.5.1 The Cauchy Problem with Initial Conditions at theEndpoint of the Interval. Global Solution

In this section we consider the nonlinear differential equation of order a > 0:

(cDZ+y)(x) = f[x,y(x)] (a > 0; a ^ x g b), (3.5.1)

involving the Caputo fractional derivative (cD%+y)(x), defined in (2.4.1), on afinite interval [a, b] of the real axis R, with the initial conditions

y ( k ) { a ) = b k , bk e l (fc = 0 , l , - - - , n - l ; n = - [ - a \ ) . (3.5.2)

We give the conditions for a unique solution y(x) to this problem in the spaceC^[a, b] denned for a > 0, r £ N and 7 £ R (0 7 < 1) by

C^r[a, b] = {y(x) G Cr[a, b] : cD°+y G C7[a, b}} , Cr/[a, b] = C;\a, b}. (3.5.3)

As in Sections 3.2-3.4, our investigations are based on reducing the problem con-sidered to the Volterra integral equation

f ,3.5.4)

When a € N, then, in accordance with (2.4.14), (cD%+y)(x) = y{n){x) for asuitable function y(x), and hence (3.5.1)-(3.5.2) is the Cauchy problem for theordinary differential equation of order n G N:

y^(x)=f[x,y{x)]{a^x^b), y<*>(a) = 6* e K (* = ,n - 1). (3.5.5)

The corresponding integral equation (3.5.3) takes the form

% ^ r (3.5.6)

First we establish an equivalence between problem (3.5.1)-(3.5.2) and the inte-gral equation (3.5.4) in the space Cr[a,b] of continuously differentiable functions.

Theorem 3.24 Let a > 0 and n = —[—a]. Let G be an open set in C and letf : (a,b] x G -) C be a function such that, for any y G G, f[x,y] G C7[a,b] with0 ^ 7 < 1 and 7 ^ a. Let r = n for a G N and r = n - 1 for a g" N.

If y{x) G Cr[a,b], then y(x) satisfies the relations (3.5.1) and (3.5.2) if, andonly if, y(x) satisfies the Volterra integral equation (3.5.4).


Proof. First we prove the necessity. Let a = n G N and y(x) G Cn[a,b] be thesolution to the Cauchy problem (3.5.5). Applying the operator J"+ to the firstrelation in (3.5.5) and taking into account (2.1.41) and the second relation in(3.5.5), we arrive at the integral equation (3.5.6). Inversely, if y(x) G Cn[a,b]satisfies (3.5.6), then, by term-by-term differentiation of (3.5.6), we have

n - l

Ej=k

(3-5.7)

for k = 1,- , n — 1. Taking the limit as x —> a, and taking into account thecontinuity of the integrands in (3.5.6) and (3.5.7), we obtain t/fc'(a) = bk fork = 0, l , - . ., n — 1. Differentiating the equation (3.5.6) n times, we obtainy(n\x) - f[x,y(x)]. Thus Theorem 3.24 is proved for a £ N.

Let now n — 1 < a < n and y(x) G Cn~1[a,b}. According to (2.4.1) and (2.1.5),

(X).

3=0

By the hypotheses of Theorem 3.24, f[x,y] G Cy[a,b], and it follows from (3.5.1)that (cD%+y)(x) G Cy[a,b], and hence, by Lemma 1.3,

«(*)-E^(*-a ) (x)€C»[a,b].

Applying Lemma 2.9(d) to g(t) = y(t) — X)?=o y i (t — a-Y, we obtain

a . Doc,y\ (x) — I -/ rt-4- Da\

n - l

- nV

3=0

(x)

n-l

3=0

where

- nM -

_ jn-a- I i a+

^ F(a - k + 1)

n - 1

(a:-a)a- f c , (3.5.8)

(3.5.9)i=o

Integrating (3.5.9) by parts, differentiating the obtained expression, and usingthe first formula in (2.1.33) with k — 1, we have

"" 1 uWfal-(< - o)>

j = 0

(x)

Chapter 3 (§ 3.5) 201

(z). (3.5.10)

Repeating this process n — k t imes (k = , n), we arrive at the followingrelation:

T1a+

n - 1a)

(t - ay~n+k

j=n — k

(x). (3.5.11)

Making the change of variable t = a + s(x — a), we obtain, for k = 1, , n,

j=n-k

(3.5.12)

Since a < n and y^n (x) G C[a, b] for k = 1, , n, then the last relationsyield y(-n~k^(a+) = 0 (k = 1, , n), and hence (3.5.8) takes the form

(x) = y(x) - - ay. (3.5.13)

Since f[x,y] G C7[a,b] and 7 ^ a, by Lemma 2.8(b),/ +[i,y(t)])(a;) G C[a,b}.Applying the operator I"+ to both sides of (3.5.1) and using (3.5.13) and theinitial conditions (3.5.2), we find that y(x) G Cn~1[a, b] is the solution to theintegral equation (3.5.4), and thus the necessity is proved.

Now let y(x) G Cn~x[a, b] be the solution to the integral equation (3.5.4). Letus first show that y(x) satisfies the initial conditions (3.5.2). Differentiating bothsides of (3.5.4) and taking (2.1.33) into account, for all k = 1, , n — 1, we have

,<*>(*) =j=k u

;(x-ay-" +-k)Ja (x-ty-"+k-

(3.5.14)

Making the change of variable t = a + s(x - a) in the integrals in (3.5.4) and(3.5.14), for all k = 1, , n — 1, we find that

n-l

j=k

(x — a)a k fl f[a +s(x — a),y(a + s(x — a))]dt' T(a-k) Jo (x - ty-t)1~a+k

Since a > n — 1 and f[x,y(x)] is continuous, the integrands in these relations arecontinuous, and by taking the limi t as x —> a+, we obtain relation (3.5.2).


Now we show that y(x) satisfies the equation (3.5.1). Applying the operatorD%+ to (3.5.4), taking (3.5.2) and (2.4.2) into account and using the first relationin (2.1.31), we arrive at the equation (3.5.1). Thus Theorem 3.24 is proved fora £N.

Corollar y 3.23 Let n G N, let G be an open set in M. and let f : (a, b] x G -> B.be a function such that, for any y G G, f[x, y] G Cy[a, b] with 0 ^ 7 < 1.

If y(x) £ Cn[a,b], then y(x) satisfies the relation (3.5.5) if, and only if, y(x)satisfies the Volterra integral equation (3.5.6).

Corollar y 3.24 Let 0 < a < 1 and 0 fs 7 ^ a. Let G be an open set in M. and letf : (a,b] x G —> M. be a function such that, for any y G G, f[x, y] G C7[a, b].

If y(x) G C[a, b], then y(x) satisfies the relations

(cDZ+y)(x)=f[x,y(x)], y(a)=bo£C, (3.5.15)

if, and only if, y(x) satisfies the Volterra integral equation

Now we establish the existence and uniqueness of the solution to the Cauchyproblem (3.5.1)-(3.5.2) in the space C"'n~1[a,b] under the conditions of Theo-rem 3.24 and an additional Lipschitzian condition (3.2.15). For this we need thepreliminary assertion similar to Lemma 3.4.

Lemma 3.9 Let n G N, a < c < b, g G Cn[a,c], and g G Cn[c,b}. Theng G Cn[a,b] and

\\g\\C'[a,b] max [||s||c»[a,c], ll$llc»[c,6]] (3-5.17)

Theorem 3.25 Let a > 0 and n = —[-a], and let 0 ^ 7 < 1 and 7 ^ a. Let Gbe an open set in C and let f : (a, 6] x G —> C be a function such that, for anyy EG, f[x,y] G C-y[a,b] and the Lipschitz condition (3.2.15) holds.

(i) If n — 1 < a < n (n G N), then there exists a unique solution y(x) to theCauchy problem (3.5.1)-(3.5.2) in the space C" ' " " 1^, &].

( i i ) / / a = n G N, then there exists a unique solution y(x) to the Cauchy problem(3.5.5) in the space C™[a,b].

(iii ) In particular, when 7 = 0 and f[x,y] G C[a, b], there exist unique solutionsto the Cauchy problem (3.5.1)-(3.5.2) in the space C " ' " - 1 ^ ] :

C^-^b] := C^-^b] = {y(x) G Cn-\a,b] : cDaa+y G C[a,b]} , (3.5.18)

and to the Cauchy problem (3.5.4) in the space Cn[a,b].

Proof. Let n — 1 < a < n (n £ N). First we prove the existence of a unique solutiony(x) G Cr[a,b]. According to Theorem 3.24, it is sufficient to prove the existenceof a unique solution y(x) G Cr[a, b] to the Volterra integral equation (3.5.4). For

Chapter 3 (§ 3.5) 203

this we use a method similar to that given in Theorems 3.3 and 3.11. Equation(3.5.4) makes sense in any interval [a, x\] G [a, b] (a < x\ < b). Choose x\ suchthat the inequality

n - l , _ Na_f c

Êna-fc + i ) < 1 (3-5-19)

k=o 'holds, A being given in (3.2.15), and prove the existence of a the unique solutiony(x) G C"" 1^, xi] to the equation (3.5.4) on the interval [a, xi]. For this weuse the Banach fixed point theorem (Theorem 1.9 in Section 1.13) for the spaceCn~1[a, b], which is the complete metric space with the distance given by

n - l

d{yi,y2) = ||S/1 — 2/2||c"—â.zi] : = E 1 1 ~ 2 ]|c[a,a;i]- (3.5.20)k=0

We rewrite the integral equation (3.5.4) in the form y(x) = (Ty)(x), where

(Ty)(x) = yo(x) + - ^ jT ^ | M yo(x) = g hj{x - ay. (3.5.21)hj

To apply Theorem 1.9, we have to prove the following: (1) if y(x) G C™"1^, a:i],then (Ty){x) G Cn-l[a,Xl}; and (2) for any yuy2 G C " - 1 ^ , ^ ] ,

\\Tyi - T^llc-Mo.x!] ^ wllyi " y2||c-i[al!Bl ] , (3-5.22)

with , , - A V " " 1 r(l-7)(xi-a)°-fc-1'with w - Ao } ^ k = 0 r(«-fe-7+i)

Let y(x) G C"" 1^, xi]. Differentiating (3.5.21) k times (k = 1,- , n — 1) andusing the first relation in (2.1.33), which is true by Lemma 2.9(c) with 7 = 0, wehave, for k = 0,1, , n — 1,

with J/Q '(#) = Y^j=k { -4)! ( ~ o.y~k. For any A; = 0,1, , n — 1, the first termin the right-hand side of (3.5.23) is clearly a continuous function on [a,Xi], andby Lemma 3.3(a) (with 7 = 0 and a replaced by a — k — 1), the second term isalso continuous on [a, xi] and the following estimates hold:

f (i[t 'yt?-°l = T(a T+iJntMtMcM, (3-5.24). " ~ K) Ja \ x ~ l! C\a x i l V ~ " I

for all k = 0,1, , n - 1. Hence (Ty)(a;) G C " " 1 ^ ^ ] . Now we prove (3.5.22).Using (3.5.20), (3.5.23), (3.5.24) and the Lipschitzian condition (3.2.15), we have


n - l

\\Tyi - Ty2\\c^[a,Xl] =fc=O

< ^ l l 1

n — 1 / \ Q, fc

~ -/[*,y2)(«]||c[a,x1]

n - l

^ o E r(a-Jfe-7+l) ll2/1

and hence, in accordance with (1.1.19),

1 / \ a —fc

2n—1

\\Tyi - Ty2\\Cn-i[a>Xl] Â0 a f c + l )fe=o * '

which yields the estimate (3.5.22). By (3.5.19), 0 < w < 1, and hence, by Theorem1.9, there exists a unique solution y*(x) — y*°(x) G Cn~l[a,x{\ to the equationsin (3.5.4) on the interval [a, x{\.

By Theorem 1.9, this solution y*{x) is a limi t of the convergent sequenceym(x) = (Tmy*)(x):

lim ||ym-j/*||c»-Ma,xi] =0,

where y£(x) is any function in Cn_Q[a, fc]. If at least one bk 0 in the initialcondition (3.5.2), we can take y^x) = j/o(a;) with yo(x) defined in (3.5.21).

Applying the same arguments as in the proofs of Theorems 3.3 and 3.11, andtaking Lemma 3.9 into account, we prove that there exists a unique solutiony(x) £ Cn~1[a, b] to the Cauchy problem (3.5.1)-(3.5.2), and this solution y(x)is a limit of the sequence ym(x) = (Tmy*)(x):

lim || j/m-y||c-Mo,6]=0. (3.5.25)

Using this formula and taking (3.5.1) and (3.2.15) into account, we have

\\cD2+ym - c Daa+y\\Ci[aM \\f[t,ym(t)} - f[t,y(t))\\cl{aM

^ A\\ym - y\\c^[a,b] ^ A{b - a)7||yTO - y\\C[ a,b} ^ A\\ym -

From this and (3.5.25) we obtain

li m || cDaa+ym -

Thus cD£+y e C7[o, b], and hence, in accordance with (3.5.3), y € C^'""1^, 6].This completes the proof of the assertion (i) of Theorem 3.25. The assertion (ii)

is proved similarly by using Corollary 3.23 and the Banach fixed point theorem.

Chapter 3 (§ 3.5) 205

Corollar y 3.25 Let n G N. Let G be an open set in R and let f : (a, b] x G -» Rbe a function such that, for any y G G, f[x, y] G C[a, b] and the condition (3.2.15)is valid. Then there exists a unique solution y(x) G Cn[a,b] to the Cauchy problem(3.5.4).

Corollar y 3.26 Let 0 < a < 1 and 0 ^ 7 ^ a. Let G be an open set in R and letf : (a, b] x G —> C be a function such that, for any y G G, f[x,y] G C^[a,b] andthe relation (3.2.15) holds.

Then there exists a unique solution y(x) to the Cauchy problem (3.5.15) in thespace C"[a,b]:

C°[a,b] := q-° [o , l] = {y(x) G C[a,b] : cDaa+y G C^[a,b]} . (3.5.26)

Corollar y 3.27 Let n-l<a^.n(n£N) and 0 ^ 7 < 1 be such that 7 ^ a,and let g(x) G Cj[a, b].

If a(x) G C[a,b], then the Cauchy problem

{cDaa+y){x) = o(x)y(a:) + g(x), yW(a) = bk (3.5.27)

with k = 0,1, , n — 1 and 6 G M, /ias a unique solution y(x) in the spaceC" ' " " 1^ , b] when a ^ N, and in the space C™[a, b] when a = n G N.


{cDaa+y){x) = X(x - afy{x) + g(x); y^(a) = bk, (3.5.28)

with k = 0,1, , n — 1, fefc G K, A G C and /3 0, has a unique solution y(x) inCa'n- I [a,&] when a £ N, rtile in C^[a,6] w/zen a = n G N.

Remark 3.27 The results presented in Section 3.5.1 were proved by Kilbas andMarzan ([380] and [382]).

Remark 3.28 The results of Theorems 3.24-3.25 and Corollaries 3.23-3.27 canbe extended to the systems of Cauchy problems (3.5.1)-(3.5.2), (3.5.5), (3.5.15),and (3.5.27)-(3.5.28).

3.5.2 The Cauchy Problems with Initial Conditions at theEnd and Inner Points of the Interval. Semi-Globaland Local Solutions

In this subsection we give the conditions for the existence of a unique semi-globaland local solution to Cauchy problems for differential equations with a Caputofractional derivative for the case where the initial conditions are given at anypoint Xo of the interval [a,b]. The first result gives a unique continuous solutiony(x) to the Cauchy problem (3.5.1)-(3.5.2) in the right neighborhood of a.


Theorem 3.26 Let a > 0, hx > 0, K > 0 and bk G R (k = 0,1, , n - 1). LetG be a set given by

G= \ (x,y) £ l 2 : a^xâ + hu \y - J2 %{? - a)'\ S K \ . (3.5.29)

I J' JLet 0 ^ 7 < 1 be such that 7 ^ a, and let f : G —> K be a function such that, forany y G G, f[x,y] G C^[a,b] and the Lipschitz condition (3.2.15) holds.

Then there exists h, 0 < h ^ such that the following assertions hold:(i ) If a ^ N and n = [a] + 1, then there exists a unique solution y(x) to the

Cauchy problem (3.5.1)-(3.5.2) in the space C^'n~1[a,a +h].

( i i ) / / a = n € N, then there exists a unique solution y(x) to the Cauchy problem(3.5.5) in the space C™[a,a + h].

(iii ) In particular, when 7 = 0 and f[x, y] G C[a, b], there exist unique solutionsto the Cauchy problem (3.5.1)-(3.5.2) in the space C"1" " 1^, a + h] and Cauchyproblem (3.5.4) in the space Cn[a,a + h\.

Proof. The proof is similar to that of Theorem 3.18. It is based on the equivalenceof the Cauchy problem (3.5.1)-(3.5.2) and the Volterra integral equation (3.5.4) inthe space C"'1^ + h] for a £ N, and of the Cauchy problem (3.5.5) and theintegral equation (3.5.6) for a € N. For this we take M = max ,y)eG \f{x,y)\,choose h as

L ( r ^ + 1 ) g)1 / a] (3.5.30)

and apply the Banach fixed point theorem to the operator T given by (3.5.21) toprove the uniqueness of the solution y(x) £ C[a,a + h] to the equation (3.5.4).

Corollar y 3.28 Let 0 < a < 1, and let hi > 0, K > 0, and b0 e K. Let G be theset given by

G = {(x,y) £ l 2 : a^xâ + h1, \y - b\ ^ K} . (3.5.31)

Let 0 ^ 7 < 1 be such that 7 ^ a, and let f : G —> M. be a function such that,for any y £ G, f[x,y] G C-y[a,b] and the Lipschitz condition (3.2.15) holds. Then

there exists an h, 0 < h ^ hi, such that the Cauchy problem (3.5.15) has a uniquesolution y(x) G C"[a, a + h].

Remark 3.29 When the conditions of Theorem 3.26 with 7 = 0 are satisfied,Diethelm and Ford ([177], Theorems 2.1 and 2.2) proved the existence a localcontinuous solution y(x) G C[a,a + h] [see also Diethelm ([173], Theorems 5.4 and5.5)]. The result in Theorem 3.26 characterizes more precisely the solution of theproblem considered.

The following result yields the conditions for a unique local continuous solu-tion to the Cauchy problem for the differential equation (3.5.1) with the initialconditions at any inner point XQ G (a,b):

(cD%+y)(x) = f[x,y(x)} (a > 0; a x b), (3.5.32)

Chapter 3 (§ 3.5) 207

y{k)(x0)=bk, 6fcGE(fc = 0 , l , - - - , n - l ). (3.5.33)

Theorem 3.27 Letn-1 < a ^ n (n G N), andletbk G E (fc = 0,1,- , n - 1) 6egiven constants. Let U be an open and connected set in E and Q, = (a, b) x U. Letf(x,y) : (a, b) x U —> E fee a continuous function satisfying the Lipschitz condition(3.2.15).

Then, for any (xo,yo) £ ^> iftere exists h > 0 sucft iftaf [:ro — ft, Xo + ^] C(a, 6) and iftere eM'sis a unique function y(x) : [xo — h,Xo + ft] —> C/ s«c/i thaty{x) G Cr[a;o — h, XQ + ft], where r = n for a = n G N and r = n — 1 for a $. N,and

y(a;o)=6o, 2 / ' M = 6 i , , ^ " " ^ ( ô ) = 6n- i , (3-5.34)

(c£>"+2/Xa:) = /[a:,2/(a:)] /or any a; G [a;0 - ft,a;0 + ft]. (3.5.35)

Proof. When a G N, then, by the first relation in (2.4.14), (cD%+y){x) = y(n^{x)and the result of Theorem 3.27 follows from that of Theorem 3.23.

Let now a $ N. First of all, we note that the constants y(k\a)(k = 0,1, , n — 1) can be expressed uniquely via the constants y^(xo) = bk

(k = 0,1, , n — 1). Indeed, by Theorem 3.24, the Cauchy problem

(cD%+y)(x)=f[x,y(x)] (a^x^b), y^(a) G C (k = 0 , 1 ,- ,n - 1) (3.5.36)

is equivalent to the following integral equation:

y{x) = J 2 y ^ P - ( x - ay + {l2+f[t,y(t)}) (x) ( a ^ x b). (3.5.37)i=o 2'

Differentiating this relation k times (k = 1, , n — 1) and using the first formulain (2.1.33), asserted by Lemma 2.9(c) (with 7 = 0), we have

S W (* = o,i,---, n-i).j=k [J lt)-

(3.5.38)Taking the limi t as x —> xo, we obtain

J2f i:-kf[t,y(t)])(xO) (fc = 0 , l , - - - , n - l ) ,j=k [J K>-

or, in accordance with (3.5.34),

n —1

T 0) (fc = 0 , l , - - - , n - l ). (3.5.39)j=k (J K}-

This formula yields the following recurrence relations for finding y^(a):


j=n-2{J n + i - ) -

( * = 0,1, . (3-5.40)j=O

Since XQ G (a, 6), then, for any x G (a, 6), the existence of the Caputo fractionalderivative (cD"+y)(x) is equivalent to the existence of the Riemann-Liouville frac-tional derivative (D%+y)(x). Then, by (2.4.6),

{cD«+y){x) = (D2+y)(x) - ] T " ^ " ^ ( ^ - a)fc"Q (a £ No; n = [a] + 1),k=o \ + '

(3.5.41)

with a ^ No and n = [a] + 1. Now we rewrite (3.5.35) in the form

(D°+y)(x) = g[x,y(x)}, (3.5.42)

for any x E [XQ — h, xo + h], and where

g[x, y(x)] = f[x, y(x)] + —^M—(x - a)k~a. (3.5.43)

k=o L[K a + i j

Therefore, the problem (3.5.34)-(3.5.35) is equivalent to the following Cauchyproblem:

y(xo) = bo, y'(xo) = h, , y("-1)(a;o) = 6n- i , (3.5.44)

(D"+y)(x) = g[x,y(x)] for any x G [x0 - h,x0 + h]. (3.5.45)

By the hypotheses of Theorem 3.27, f[x,y(x)] is a continuous function on (a, 6),the second term in (3.5.43) is also continuous on (a, b), and thus the functiong[x,y(x)] G C(a,b). Therefore, by Theorem 3.23, there exists a positive numberh > 0 such that [xo — h, XQ + h] C (a, b) and there exists a unique functiony(x) : [xo — h,x0 + h] -> U such that y(x) G Cr[x0 - h,x0 + h], and the relations(3.5.44) and (3.5.45) are valid. This completes the proof of Theorem 3.27.

Corollar y 3.29 Let 0 < a < 1, and let bo G M. be a given constant. Let U be anopen and connected set in M. and fi = (a, b) x U. Let f(x,y) : (a, b) x U —> E be acontinuous function satisfying the Lipschitz condition (3.2.15).

Then, for any (xo,yo) G 0, there exists h > 0 such that the real interval[XQ — h,xo + h] C (a, b) and there exists a unique function y(x) : [xo — h, xo + h] —> Usuch that y(x) G C[XQ — h, XQ + h], and for any x G [xo — h, Xo + h]

fDaa+y){x) = f[x, y(x)}; y(x0) = b0. (3.5.46)

Remark 3.30 The results of Theorems 3.26-3.27 and Corollaries 3.28-3.29 canbe extended to the systems of the corresponding Cauchy problems.

Chapter 3 (§ 3.5) 209

3.5.3 Illustrative Examples

In Sections 3.5.1 and 3.5.2 we gave the conditions when the Cauchy problemsfor differential equations with Caputo derivatives have a unique global and localsolution in the space of continuously differentiable functions and a semi-globalsolution in the space of continuous functions. Here we present several illustrativeexamples.

Example 3.7 Consider the following differential equation of fractional order a(a > 0):

fDZ+y){x) = \(x - af[y{x)]m (x > a; m > 0; m 1) (3.5.47)

with A , / 3 £ l ( A ^ 0). Using (2.4.28), it is directly verified that this equation hasthe following explicit solution:

By the definition (2.4.1) of the Caputo derivative cD"+y, this solution must becontinuous and hence bounded in a right neighborhood [a, a+e) (e > 0) of a, whichyields the condition a — 7 ^ 0, or {(3 + a)/(m — 1) 0, and hence y(x) € C[a, b].The right-hand side of the equation (3.5.47),

I[X 'V{X)] = A T ( 7 + 1) J {X a ) ' (3-5-49)

belongs to C7[a, b] and to C[a, b] in the respective cases 0 < 7 < 1 and 7 ^ 0. Thisleads to Cases (2) and (3), given in (3.3.7), which means that any of the conditions(3.3.47), (3.3.48), (3.3.49) and (3.3.50) is satisfied in the first case, while either of(3.3.51) and (3.3.52) in the second. The validity of the Lipschitz condition (3.2.15)was proved in Example 3.3 for the domain D denned by (3.3.57). Then Theorem3.25 yields the following results.

Proposition 3.8 Let 0 < a < 1, A,/? G R (A 0) and m > 0 ( m / l ) . Let D bethe domain (3.3.57), where OJ EM. is such that ft + (m — l)w 0.

(i) // either of the conditions (3.3.47) or (3.3.48) holds, then the Cauchyproblem

(cD»+y)(x) = X(x - af[y(x)]m (xâ), y(a) = 0 (3.5.50)

has a unique solution y(x) in the space C"[a, 6], 7 = (/? + ma)/(m — 1), and thissolution is given by (3.5.48).

(ii ) // either of the conditions (3.3.51) or (3.3.52) is valid, then the Cauchyproblem (3.5.50) has a unique solution y(x) £ CQ[O,b] given by (3.5.48).

Proposition 3.9 Let n - 1 < a < n (n e N\{1}) , A,/? G R (A 0) and m > 0(m 7 1). Let D be the domain (3.3.57), where us £ R is such that j3+{m— l)to ^ 0.


(i ) / / either of the conditions (3.3.49) or (3.3.50) holds, then the Cauchyproblem

{Daa+y){x) = X(x - af[y(x)]m (xâ), yO(o) = 0 (k = 0,1, , n - 1)

(3.5.51)has a unique solutiony(x) G C°'n~1[a,b], 7 = (f3 + ma)/(m — l), and this solutionis given by (3.5.48).

(ii ) // either of the conditions (3.3.51) or (3.3.52) is valid, then the Cauchyproblem (3.5.51) has a unique solution y(x) G CQ' " "1^ ,b] given by (3.5.48).


(cD2+y)(x) = X(x-afym(x)+b(x-a)v {xâ; m e l ; m / l) (3.5.52)

with A, b G E (A 0) and /?, v G E. Applying the same arguments as in the proofof Propositions 3.3-3.4, from Theorem 3.25 we obtain the uniqueness theorem forthe Cauchy problem for the equation (3.5.51).

Proposit ion 3.10 Let 0 < a < 1, A, b G E (A ^ 0), m > 0 (m / 1) and j3 £ E.Let D be the domain (3.3.57), where UJ G E is such that /3 + (m — l)w ^ 0. Letv = ((3 + am)l(\ — m) and let the transcendental equation (3.3.69) have a uniquesolution £ = /i.

(i ) // either of the conditions (3.3.47) or (3.3.48) holds, then the Cauchy prob-lem

[ cDaa+y){x) = X(x - af[y(x)]m + b(x - a)", y(a) = 0 (3.5.53)

has a unique solution y(x) in the space C"[a, 6], 7 = (/? + ma)/(m — 1), and thissolution is given by

y{x) = fi{x - a )a" 7 G C[a, b], (3.5.54)

(ii ) / / either of the conditions (3.3.51) or (3.3.52) is valid, then the Cauchyproblem (3.5.53) has a unique solution y(x) G Ca[a, b] given by (3.5.54).

Proposit ion 3.11 Let a ^ 1, A, b G E (A 0), m > 0 (m 1) and /3 G M.Let D be the domain (3.3.57), where to G E is such that (3 + (m — l)w ^ 0. Letv = (/3 + am)/(\ — m) and let the transcendental equation (3.3.69) have a uniquesolution £ = /U.

(i ) If either of the conditions (3.3.49) or (3.3.50) holds, then the Cauchyproblem

{D«+y){x) = \{x-aY[y(x)]m+b{x-aY; y{-k){a)=Q (xâ), (3.5.55)

for all k = 0,1, , n — 1, has a unique solution y(x) G C"'n~1[a,b], where7 = (/? + ma)l(m — 1), and this solution is given by (3.5.54).

(ii ) // either of the conditions (3.3.51) or (3.3.52) is valid, then the Cauchyproblem (3.5.55) has a unique solution y(x) G C Q ' "~ [a,6] given by (3.5.54).

Chapter 3 (§ 3.5) 211

Remark 3.31 By Propositions 3.1-3.4 and 3.8-3.11, the Cauchy type problems(3.3.53)-(3.3.55) and (3.3.72)-(3.3.74) for the differential equations (3.3.36) and(3.3.67) with the Riemann-Liouville derivatives D"+ and the Cauchy problems(3.5.50)-(3.5.51) and (3.5.53), (3.5.55) for the differential equations (3.5.47) and(3.5.52) with the Caputo derivatives °D"+ have the same solutions as in (3.3.38),(3.5.48) and as in (3.3.70), (3.5.54). But the solutions given in (3.3.38) and (3.3.70)can be continuous on [a, b] or continuous on (a, b] with an integrable singularityat the point a. The solutions in (3.5.48) and (3.5.54) can be only continuous on[a,b\.

Example 3.9 Consider the following Cauchy problems for the differential equa-tion of fractional order a > 0 (n — 1 < a ^ n, n G N):

(CD«+y)(x) = X(x - af[y(x))m + g(x), (a x b) (3.5.56)

y{k)(a) = bk, (3.5.57)

with bk G E, k = 0,1, , n - 1, m > 0 (m 1), A G R (A ^ 0) and (3 G E.Let K > 0, /ii > 0 (0 < hi < b - a) and G be the set (3.5.29). We suppose thatg(x) G C[a,a + h{\ and the constants a, /? and m are such that /? ^ m(l — a).(3.5.56)-(3.5.57) is the Cauchy problem (3.5.1)-(3.5.2). Analogous to Example3.5, it is proved that the right-hand side of (3.5.56): f(x, y) = X(x — a^y"1 + g(x)satisfies the Lipschitz condition (3.2.15) on G. Then, from Theorem 3.26, we derivethe uniqueness of a semi-global solution y(x) of the Cauchy problem (3.5.56)-(3.5.57).

Proposit ion 3.12 Let a > 0 (n - 1 < a < n; n G N), m > 0 (m ^ 1),and 13 e R be such that j3 ^ m(l - a), and let A £ E (A 0) and b,bk £ E(k = 0,1, , n - 1). Let K > 0, hx > 0 (0 < hi < b - a), let G be the set(3.5.29) and let g(x) G C7[a,o + hi], where 0 ^ 7 < 1 is such that 7 ^ a.

Then there exists an h, 0 < h ^ hi, such that the Cauchy problem (3.5.56)-(3.5.57) has a unique semi-global solution y(x) G C°'n~1[a,a + h].

In particular, if 0 < a < 1, the Cauchy problem

(cD%+y)(x) = \(x - af[y{x)]m + g(x) (a g x g b; 0 < a < 1), y(a) = bo£R,(3.5.58)

has a unique semi-global solution y(x) G C"_Q 7[a, a + h].

Example 3.10 Consider the Cauchy problems for the differential equation (3.5.56)of order a > 0 with the following initial conditions:

y{k){x0) = bk G E (k = 0,1, , n - 1). (3.5.59)

Here f(x, y) = A(a;-a)^j/m+5 is a continuous function on (a, b) xfi and satisfies theLipschitz condition (3.2.15). Then, from Theorem 3.27, we obtain the uniquenessof a local solution of the Cauchy problem (3.5.56), (3.5.59).


Proposition 3.13 Let a > 0 (n - 1 < a ^ n; n e N ) , m > 0 ( m / l) and (3 G M6e suc/i tAof /3 ^ m(l - a), te£ A € 1 (A ^ 0) and let bk G E (k = 0,1, , n - 1)fee given constants. Let U be an open and connected set in M. and g(x) G C(U).

Then, for any (xo,yo) G (a, b) x fi, there exists an h > 0 such that thereal interval [XQ — h, XQ + h] C (a, b) and there exists a unique real functiony(x) : [:ro — h,xo + h] —> U such that y(x) G CT[XQ — h,xo + h], with r = n fora = n G N and r = n — 1 for a £ N, and the relations (3.5.56) and (3.5.59) arevalid.

In particular, if 0 < a < 1, bo G M. and a ^x ^b, theny(x) G C[XQ — h, XQ + K\and it satisfies the following equation:

(CD2+y)(x) = \(x - af[y{x)]m + g(x); y(x0) = b0. (3.5.60)

3.6 Equations with the Hadamard FractionalDerivative in the Space of Continuous Func-tions

In this section we give the conditions for the existence of a unique global continuoussolution to the Cauchy type problem (3.1.43)-(3.1.44) in the space C".n_a^[a, b]denned, for n — 1 < a fS n (n G N) and 0 ^ 7 < 1, by

C?; » -a ,7M ] = {y(x) G Cn_a,log[a,6] : {T%+y)(x) G C7,log[a,6]} . (3.6.1)

Here T>"+y is the Hadamard fractional derivative (2.7.7), and C7ii0g[a, b] is theweighted space of continuous functions (1.1.27):

Gr,iogM] = {<?(*): (\og-Yg(x) e C[a,b], ||ff||c,ilog = (\og-Yg(x)\\ } .

(3.6.2)

We consider the following Cauchy type problem (3.1.43)-(3.1.44):

(V2+y){x) = f[x,y(x)} (x > a; a > 0), (3.6.3)

( V ^ - k y ) ( a + ) = b k , b k € R ( k = l , - - - , n ; n = - [ - a \ ) . (3.6.4)

The notation (V"^ky)(a+) means that the limi t is taken at all points of the right-sided neighborhood (a, a + e) (e > 0) of a:

{Vaa-

ky){a+) = x\im+(V2-ky)(x) (k = 1, ,n - 1), (3.6.5)

(2>°+ny)(a+) = Um+(Jây)(x) (a n); (V°a+y)(a+) = y(a) (a = n), (3.6.6)

where Ja+a is the Hadamard fractional operator of order n-a defined in (3.1.45).

Using the properties of the Hadamard integration and differentiation operatorsgiven in 2.35 and 2.36 and applying the same arguments as in the proofs of Theo-rems 3.1 and 3.10, we prove the equivalence of the problem (3.6.3)-(3.6.4) and theVolterra integral equation (3.1.45).

Chapter 3 (§ 3.6) 213

Theorem 3.28 Let a > 0, n = —[—a] and 0 ^ 7 < 1. Let G be an open set inIK and let f : (a, b] x G —> M. be a function such that f[x,y] £ C-y}\og[a,b] for anyycG.

If y(x) £ Cn-atiog[a,b], then y(x) satisfies the relations (3.6.3) and (3.6.4) if,and only if, y(x) satisfies the Volterra integral equation (3.1.58).

In particular, if 0 < a ^ 1 and y(x) € Ci-ai\og[a,b], then y(x) satisfies the

(V»+y)(x) = f[x,y(x)}, (Jây) (a+) = b e K (3.6.7)

if, and only if, y(x) satisfies the following integral equation:

To prove a uniqueness result for (3.6.3)-(3.6.4), we need a preliminary assertion.

Lemma 3.10 Let ) g C , a < c < b, g e C7jiog[a,c] and g £ C7,iog[c, b]. Theng G C7[a,b] and

llflrllc7,iog[o,6] ^ max [||ff||c7liog[o,c], IMIc7,log[c,&]]

Theorem 3.29 Let a > 0, n = —[—a] and 0 ^ 7 < 1 be such that 7 ^ n — a. LetG be an open set in C and let f : (a,b] x G —> C be a function such that, for anyy G G, f[x,y] G C7ôg[a,b] and the Lipschitz condition (3.2.15) holds.

Then there exists a unique solution y(x) to the Cauchy type problem (3.6.3)-(3.6.4) in the space Cfn_a^[a,b}.

In particular, if 0 < a < 1 and 7 ^ 1 — a, then there exists a unique solutiony{x) to the Cauchy type problem (3.6.7) in the space C^.1_Q,7[a,6].

Proof. The proof is similar to the proofs of Theorems 3.3 and 3.11 and is based onTheorem 3.28 and the Banach fixed point theorem (Theorem 1.9 in Section 1.13)for the space Cn_ajiog[a, x\], which is the complete metric space with the distancegiven by

. (3.6.9)<%i,y 2) = ||yi - 2/2||cn_a los[a,Xl]

Choosing xi (a < x\ < b) such that there hold the following inequalities:

wi := A (log £ i r ~ 7 + a ^ ~ 7 ) < 1 for 7 > a, (3.6.10)V a / 1(1-Ha — 7)

u2 := A (log ^ ) " " 7 r ^ " 7 ) , < 1 for 7 ^ a, (3.6.11)

\ a I 1 (1 + a — 7)rewriting the equation (3.1.45) in the form y(x) = (Ty)(x), where

(Ty)(x)=yo(x) + —J (log-) f[t,y(t)]j, (3.6.12)


7 — 1 '

and applying Theorem 1.9, we prove that, if y(x) £ Cn~a,\o%\P'-, #i]> then{Ty)(x) € Cn-aiiog[a,xi), and, for any t/i,t/2 e Cn_a,iOg[a,a;i], there hold thefollowing estimates:

liry i -Tl/2||cn_a,log[«,«1] ^^112/i - j / 2 | | C n _ Q l o g M l ] , 0 < uij < 1 (i = 1,2).(3.6.13)

This means that there exists a unique solution y*(x) £ Cn-at\og[a,xi] to theequation (3.1.45) such that

Jirn^ \\ym(x) - 2/(a:)||cn_Q,log[a,x1] = 0, (3.6.14)

ym{x):=y0(x) + ^j*(\og^)a~lf[t,{Tm-ly){t)]j (men). (3.6.15)

Considering the interval [0:1,0:2], where 0:2 = 0:1 + hi and hi > 0 are such thatX2 < b, and applying the same arguments as in proofs of Theorems 3.3 and 3.11, weprove, using Lemma 3.10, that there exists a unique solutiony{x) = y*{x) £ Cn-a,iOg[a,b] to the integral equation (3.1.45) and hence to theCauchy type problem (3.6.3)-(3.6.4). To complete the proof of Theorem 3.29, in ac-cordance with the definition (3.3.1), it is sufficient to prove that(Tâ+y)(x) £ ^ [0 ,6 ]. By the above proof, the solution y(x) £ Cn-atiOg[a, b] isa limi t of the sequence ym{x) £ Cn-aôg[a, b], and it is directly proved that

and hence (T>"+y)(x) £ C7iiog[a, &]. This completes the proof of Theorem 3.29.

Corollar y 3.30 Let n £ N and 0 ^ 7 < 1. Let G be an open set in M. and letf : (a,b]x G -> E be a function such that, for any y £ G, f[x,y] £ C7,iog[a,&] andthe relation (3.2.15) holds.

Then there exists a unique solution y(x) to the Cauchy problem

(Sny)(x) = f[x,y(x)}, (5n-ky)(a) = bk € E (* = I,-- , n) (3.6.16)

in the space C£[a, b] = {y £ C[a, b] : 6ny £ C[a, b]}.

The spaces of the solution y(x) of the Cauchy type problem (3.6.3)-(3.6.4) arecharacterized more precisely when bn = 0 and the results wil l be different for0 < a ^ 1 and for a > 1. First we consider the first case:

{Vaa+y){x) = f[x,y(x)] (0 < a 1), (Jây)(a+) = 0. (3.6.17)

The integral equation (3.6.8), corresponding to the problem (3.6.17), takes thefollowing form:

a-l rlf

Chapter 3 (§ 3.6) 215

By Lemma 2.36(a), if f[x,y(x)] G C~,tios[a,b] (0 ^ 7 < 1), then the right-handside of (3.3.28) and hence the solution y(x) belong to C-y-a[a, b] for 7 > a and toC[a, b] for 7 ^ a. From here we derive the result.

Theorem 3.30 Let 0 < a ^ 1 and 0 ^ 7 < 1. Let G be an open set in C and letf : (a, b] x G —» M. be a function such that, for any y 6 G, f[x, y] G C7iiog[a, b] andthe Lipschitz condition (3.2.15) is satisfied.

(a) // 7 > a, then the Cauchy type problem (3.6.17) has a unique solutiony(s)€C?.7_Qi7[a,6].

(b) If 7 ^ a, i/ien i/ie Cauchy type problem (3.6.17) has a unique solutiony{x) e Cf.0<^[a,b].

Similarly, we can prove the result for the problem (3.6.3)-(3.6.4) with a > 1:

(VZ+y)(x) = f[x,y(x)] (a > 1), (3.6.19)

(2^- f c 2/ ) (a+) = & f c eR( fc = l , - - - , n - l ; n = - [ - a ] ) , 6n = 0. (3.6.20)

Theorem 3.31 Let a > 1, n = —[—a]. Let G be an open set in M. and letf : (a, 6] x G -> R be a function such that, for any y £ G, f[x, y] G C7)i0g[a, 6] andi/ie Lipschitz condition (3.2.15) is satisfied.

Then the Cauchy type problem (3.6.19)-(3.6.20) has a unique solutiony(x) e Cf[a,b].

In particular, if f[x,y] G C[a, b] for any y £ G, then the Cauchy type problem(3.6.19)-(3.6.20) has a unique solution y(x) G C%[a,b].

When 0 < a < 1, the results of Theorem 3.29 remain true for the followingproblem:

(V:+y)(x)=f[x,y(x)](0<a<l), lim [(log *Y~a y(x)} = c G R. (3.6.21)

Theorem 3.32 Let 0 < a < 1 and 0 ^ 7 < 1 be such that 7 ^ 1 - a. Let G bean open set in R and Zei / : (a, b] x G —> M 6e a function such that, for any y G G,f[x,y] G C7iiOg[a,6] and the Lipschitz condition (3.2.15) is satisfied.

Then there exists a unique solution y(x) to the weighted Cauchy type problem(3.6.21) in the space C£;1_Qi7[a,6].

Theorem 3.30 yields the result for the weighted problem (3.6.21) with c = 0:

{Vaa+y){x) = f[x,y(x)] (0 < a < 1), lim [(log ^ )1" ° y(a;)J = 0. (3.6.22)

Theorem 3.33 Let 0 < a < 1 and 0 ^ 7 < 1. Let G be an o w -;i m R and let

f : (0,6] x G —> M be a function such that, for any y G G, f[x,y] G C7ii0g[a, 6] andthe Lipschitz condition (3.2.15) is satisfied.


(a) / / 7 > a, then the weighted Cauchy type problem (3.6.22) has a uniquesolution y(x) in the space C";7_Q 7[a, b].

(b) // 7 ^ a, then the weighted Cauchy type problem (3.6.22) has a uniquesolution y(x) in the space C".07[a, b].

(c) In particular, if for any y G G, f[x,y] G C[a, b], then the weighted Cauchytype problem (3.6.22) has a unique solution y{x) in the space C"[a,b].

The above results can be extended to the following equation, which is moregeneral than (3.6.3):

(VZ+y)(x) = f [x,y(x), (Kiv)^), W+V)(x)] (3-6.23)

Theorem 3.34 Let a > 0, n = —[-a] and 0 ^ 7 < 1 be such that 7 ^ n — a. LetI G N\{1 } and ctj > 0 (j = 1, , /) be such that the conditions in (3.3.21) aresatisfied. Let G be an open set in W.l+1 and let f : (a, b] x G —> M. be a function suchthat f[x, y, 2/1, , yi] £ C7,i0g[a, b] for any (y, j/i , , yi) G G, and the Lipschitzcondition (3.2.46) is satisfied.

Then there exists a unique solution y(x) to the Cauchy type problem (3.6.23)-(3.6.4) in the space C£n_Q)7[a,6].

In particular, if 0 < a < 1 and 7 ^ 1 — a, then there exists a unique solutiony(x) G C".1_a7[a, b] to the Cauchy type problem for the equation (3.6.23) with the

initial conditions {j'ây) (a+) = b G R and lima;_>.a (log ^) " y(x) = c £ l .

Theorem 3.35 Let a > 0 and 0 ^ 7 < 1. Let I 6 N\{1 } and ctj > 0 (j =I ,-- - , I) be such that the condition (3.3.21) is satisfied. Let G be an open setin E '+ 1 and let f : (a, b] x G —> M. be a function such that f[x,y,y\, , yi] GC7)iOg[a,b] for any (y,yi,--- , yi) G G, and the Lipschitz condition (3.2.46) issatisfied.

(a) // 0 < a 5= 1 and 7 > a, then there exists a unique solutiony(x) G C".7_Q7[a, b] to the Cauchy type problem for the equation (3.6.23) with

the initial conditions (Jây) (0+) = 0 and linix-â (log ^) a y(x) = 0.

(b) // 0 < a ^ 1 and 7 ^ a, then there exists a unique solutiony(x) G C^.07[a, b] to the Cauchy type problem for the equation (3.6.23) with the

initial conditions {jây) {a+) = 0 and limx^.a (log -) ° y(x) = 0.

(c) If a > 1 and n = —[—a], then the Cauchy type problem for the equation(3.6.23) with the initial conditions

(V^ky)(a+) = bk, bkeC (fc = l , 2 , - " , n - l ) , bn = 0. (3.6.24)

has a unique solution y(x) G C"[a, b].

(d) In particular, ifforany{y,yu--- , y{) G G, f[x,y,yi,--- , y{\ G Co,iog[a,6],then the Cauchy type problems in (a)-(c) have a unique solution y(x) G C"[a, b].

Chapter 3 (§ 3.6) 217

Corollar y 3.31 Let n, I G N and otj > 0 (ctj < n; j = 1, , I) be such that theconditions in (3.3.21) with a = n are satisfied. Let G be an open set in M.l+1 andlet f : (a,b] x G -4- W. be a function such that f[x,y,yi, , y{\ G Coj0g[a,b] forany {y,yi,--- , yi) G G, and the Lipschitz condition (3.2.46) is satisfied.

Then there exists a unique solution y(x) G Cg.Oy[a,b] for the Cauchy problem

(Sny)(x) = f [x,y{x),{Vaa\y){x),---, {Va

a'+y){x)} (x > a; 6 = x-^j ,

(3.6.25)(5n-ky)(a) = bk G K (* = 1, , n). (3.6.26)

Remark 3.32 The results in Theorems 3.28-3.35 can be extended to a system ofcorresponding Cauchy type problems. In this regard, we note that Annaby andButzer [36] studied a first-order system of equations of the form

(S + c)y(x) = f j [ x , y 1 ( x ) , - - - , y l ( x ) } ( c G E ; j = 1 , (3-6.27)

They developed the theory for (3.6.27): existence and uniqueness of solutions, lin-ear systems, Sturm-Liouville problems, asymptotic of eigenvalues, etc., and pro-posed an application to sampling theory. Indeed it will be an interesting problemto develop such a theory for the generalized fractional equation (3.6.23).

From Theorems 3.29-3.35 we can derive the corresponding results for theCauchy type problems for linear fractional differential equations. For example,from Theorem 3.34 we derive the following assertion.

Corollar y 3.32 Let a > 0, n = -[-a] and 0 ^ 7 < 1 be such that 7 ^ n — a.Let I G N\{1 } and ctj > 0 (j = 1, , /) be such that the conditions in (3.3.21)are satisfied, and let a,j(x) G C[a, b] (j = 1, , I) and g(x) G C7)iog[o, b].

Then the Cauchy type problem for the following linear differential equation oforder a,

xM*)=9(x) (*><* ) (3.6.28)

with the initial conditions (3.6.4), has a unique solution y(x) in the space

c?;n_a,>,&] .In particular, there exists a unique solution y(x) G C" .n_Q 7[a, b] to the Cauchy

type problem for the equation with Xj G M. and f3j ^ 0 (j = 0 , 1, , I):

1

x-af'y(x)=g(x) (x > a). (3.6.29)


(D%+y)(x) = A ( log -Y [y(x)]m (x>a>0; m > 0; m # 1) (3.6.30)


with A,,8 6 i (A / 0). Using (2.7.16), it is easily verified that, if the condition[(/? + a)/(l — m)] > —1 holds, then this equation has the explicit solution

(log^)( / 3+" ) / ( 1 " ° . (3.6.31)

It is clear that (3.6.31) is also the explicit solution to the following Cauchytype problems:

{V* a+y){x) = A (log Y [y(x)]m, (Jây)(a+) =0 (0 < a < 1), (3.6.32)

{Vaa+y){x) = A (log )0 [y(x)]m, ^n+ [(x - a )1" ^ * ) ] =0 (0 < a < 1),

(3.6.33)provided that any of the conditions (3.3.45), (3.3.46), (3.3.47), (3.3.48), (3.3.51)and (3.3.52) holds, and to the following Cauchy type problem of order a > 0:

{Vaa+y){x) = A (log f [y(x)}m, (P«-fey)(a+) = 0 (k = 1, , n = - [ - a ] ) ,

(3.6.34)provided that any of the conditions (3.3.49), (3.3.50), (3.3.51) and (3.3.52) is valid.

We choose the following domain:

D= {{x,y) G<C: a < x ^ b, 0 < y < K (log - ) " ; w £ l , K > o} .

(3.6.35)I t is directly verified that the Lipschitz condition (3.2.15) is satisfied, providedthat /3 + (m — l)w ^ 0. Using the same arguments as in the proofs of Propositions3.1-3.2, from Theorems 3.29-3.31 we derive the uniqueness result.

Proposition 3.14 Let 0 < a < 1, A G R (A / 0), m > 0 (m 1), /? G R and0 = 7 = [(/? + rna)/(m - 1)] < 1. Let D be the domain (3.6.35), where u G l i ssuch that j3 + (m — l)w 0.

(a) 7/ either of the conditions (3.3.45) or (3.3.46) holds, then the problems(3.6.32) and (3.6.33) have a unique solution y(x) G C".7_a [a, b] given by (3.6.31).

(b) If either of the conditions (3.3.47) or (3.3.48) is satisfied, then the problems(3.6.32) and (3.6.33) have a unique solution y(x) G C^.07[a, b] given by (3.6.31).

(c) If either of the conditions (3.3.51) or (3.3.52) is valid, then the problems(3.6.32) and (3.6.33) have a unique solution y(x) G C"[o, b] given by (3.6.31).

Proposition 3.15 Let a ^ 1, n = - [ -a], U I (A / 0), m > 0 (ra / 1),j 3 £ E and 0 ^ 7 < 1. Let D be the domain (3.6.35), where to G M. is such thatP + (m - l)w 0.

If any of the conditions (3.3.49), (3.3.50), (3.3.51) and (3.3.52) holds, then theproblem (3.6.34) has a unique solution y(x) G Coo[&)fr] given by (3.6.31).

Chapter 3 (§ 3.6) 219


(T%+y){x) = A ( log-)0ym(x) + b ( log- )" (x > a; m > 0; m 1) (3.6.36)\ a> \ a)

with a, A £ R (A 0), 6, /? and i/ G M . Applying the same arguments as in theproof of Propositions 3.3-3.4, from Theorems 3.29-3.31 we obtain the followingresults.

Proposition 3.16 Let 0 < a ^ 1, A,/?,6 £ R (A ^ 0), m > 0 (m # 1) and0 ^ 7 = [(/? + ma)/(m - 1)] < 1. Let D be the domain (3.6.35), where u e l i ssuch that ft + (m — l)w ^ 0. £e£ u = (/3 + ma)/(I — m) anrf /ei £/ie transcendentalequation (3.3.69) /ioz;e a unique solution £ = \x.

(a) // either of the conditions (3.3.45) or (3.3.46) holds, then the Cauchy typeproblems for the equation (3.6.36) with the initial conditions {Jây){a+) = 0and limj_ ).a+ [(a; — a)x~ay(x)] = 0 have a unique solution y(x) G C".y_a 7[a, b].

(b) // either of the conditions (3.3.47) or (3.3.48) is satisfied, then the Cauchytype problems in (a) have a unique solution y(x) £ C".07[a, b].

(c) If either of the conditions (3.3.51) or (3.3.52) is valid, then the Cauchytype problems in (a) have a unique solution y(x) G Cg[a,b].

In all cases (a)-(c), the unique solution y(x) is given by (3.6.31).

P r o p o s i t i on 3 .17 Let a > 1, X,/3,b G R (A ^ 0), m > 0 (m ^ 1) and0 ^ 7 = [(/? + ma)/(m - 1)] < 1. l ei Z) 6e ifte domain (3.3.57), where w G l i sSMC/I i/iai ft + (m — l)w ^ 0. Lei v = (ft + ma)/(I — m) and let the transcendentalequation (3.3.69) have a unique solution £ = /z.

If any of the conditions (3.3.49), (3.3.50), (3.3.51) and (3.3.52) holds, then theCauchy type problem for the equation (3.6.36) with the initial conditions (3.6.4)has a unique solution y(x) G C"[o, 6] given by (3.6.31).


Chapter 4

METHODS FOREXPLICITLY SOLVINGFRACTIONALDIFFERENTIALEQUATIONS

The present chapter is devoted to explicit and numerical solutions to fractional dif-ferential equations and boundary problems associated with them. The approachesbased on the reduction to Volterra integral equations, on compositional relations,and on operational calculus are presented to give explicit solutions to linear dif-ferential equations. For simplicity, we give results involving fractional differentialequations of real order a > 0 and given real numbers. They also remain true forfractional differential equations of complex order a (n — 1 < D\(a) < n; n € N)and complex numbers. Numerical treatment of fractional differential equations isalso discussed.

4.1 Method of Reduction to Volterra IntegralEquations

In this section we present the method for finding closed-form solutions to boundaryvalue problems for linear ordinary differential equations based on the reductionto Volterra integral equations. We derive explicit solutions to the Cauchy typeand Cauchy problems for linear ordinary fractional differential equations involvingthe Riemann-Liouville, Caputo, and Hadamard fractional derivatives on a finiteinterval [a, b] of the real axis E. The problems considered have a unique solutionin accordance with the results obtained in Sections 3.3, 3.5, and 3.6.

221


4.1.1 The Cauchy Type Problems for Differential Equationswith the Riemann-Liouville Fractional Derivatives

In this subsection we construct explicit solutions to linear fractional differentialequations with the Riemann-Liouville fractional derivative (D"+j/) (x) of ordera > 0 given by (2.1.10) in the space C%_a 7[o,6] (n = - [ - a ] , 0 7 < 1), dennedin (3.3.24).

First we consider the Cauchy type problem for the fractional differential equa-tion (3.1.10) of order a > 0 with the initial conditions (3.1.2):

(D%+y)(x) - Xy(x) = f(x) (a < x g b; a > 0; A £ E), (4.1.1)

{Daa+

ky){a+) = bk, (bk € E; k = 1, ,n = -[-a]). (4.1.2)

We suppose that f{x) G C-y[a,b] (0 7 < 1). Then, by Property 3.1(a), (4.1.1)-(4.1.2) is equivalent in the space Cn-a[a, b] to the following Volterra integral equa-tion:

[X a) +T(a)Ja (x-ty(4.1.3)

We apply the method of successive approximations to solve this integral equation.According to this method, we set

(4.1.4)x f(t)dt .

Jx—ty^ ( m G N )- ( 4 L 5 )

Using (2.1.1), (4.1.4) and taking (2.1.16) into account, we find for yi(x) that

yi(x) = yo(x) + X (lZ+yo) (x) + (/«+/) (x)

that is,

= t ^ E A ^ : ; r y + r g j jf (I - r" V(t)dt (4L6)J — 1 K— 1

Similarly, using (4.1.4)-(4.1.6), we find for 2/2(2) that

V2(x) = 2/0(2) + A (i?+y i ) (a) + (l2+f) (x)

that is,

Chapter 4 (§ 4.1) 223

2/2 (X) == E .(T — n\a-3

Taking into account the first relation in (2.1.30), we get

f Ltif(t)dt. (4.1.7)

Continuing this process, we derive the following relation for ym(x) (m G N):

A A*" 1 ,) a k - 1 f(t)dt.

(4.1.8)Taking the limit as m —) oo, we obtain the following explicit solution y(x) to theintegral equation (4.1.3):

A"- 1\ak-l

or, by replacing the index of summation k by k — 1,

k+a-jJ \ak+a—1 f(t)dt.

(4.1.9)Taking into account the relation (1.8.17), we rewrite this solution in terms of theMittag-Leffler function Ea^(z):

]f(t)dt.V(T\ — V ^ h.lT_n\a-i jp . \\IT _ n\a-]_L_ I (T^f\o--ljp \\lT_f\aLf\tJbJ / U-iXtlr Hi) J-JfV (~¥—7-1-1 * * I ^ * I I 1^ I \ ' "I JCt Ct I " \ "^ "I

3=1 Ja

(4.1.10)This yields an explicit solution to the Volterra integral equation (4.1.3) and henceto the Cauchy type problem (4.1.1)-(4.1.2).

It is clear that f[x,y] = Xy + f(x) satisfies the Lipschitz condition for any^lj £2 £ (a, b] and any y G G, where G is any open set of C. If 7 n — a, then,by Property 3.1(b) and Remark 3.18, there exists a unique solution to the Cauchytype problem (4.1.1)-(4.1.2) in the space C"_an[a,b]. Equation (4.1.10) yieldsthis solution. Thus we derive the following result.


Theorem 4.1 Let a > 0, n — —[—a] and 7 (0 7 < 1) be such that 7 ^ n — a.Also let A € E. If f £ C7[a,b], then the Cauchy type problem (4.1.1)-(4.1.2) has aunique solution y{x) £ C"_Q/y[o, b] and this solution is given by (4.1.10).

In particular, if f(x) = 0, then the Cauchy type problem involving the homo-geneous differential equation (4.1.1)

{D2+y)(x) - \y{x) =0 ( « 0 ; A e l ), (4.1.11)

with the initial conditions (4.1.2), has a unique solution y{x) £ C"_a[a,b](Cn

a.a[o, b] := C%_afi[a, b}) of the form

y(x) = £ bj(x - a)°~jEa,a-j+1 [X(x - a)a] . (4.1.12)J=I

Remark 4.1 Barrett [68] first constructed the explicit solutions (4.1.10) and(4.1.12) to the Cauchy type problems (4.1.1), (4.1.2) and (4.1.11), (4.1.2) in acertain subspace of L(a, b).

Example 4.1 The solution to the Cauchy type problem

(DZ+y)(x) - \y(x) = f(x), (£«-1j/)(a+) = b (b £ K) (4.1.13)

with 0 < a < 1 and A £ E has the following form:

y(x) = bix-aÊâ [X(x - a)a}+ [*(x-t)"- 1 Ea,a [X(x - t)a] f(t)dt, (4.1.14)J

while the solution to the problem

{Daa+y){x) - Xy(x) = 0, ( ^ 1

2 / ) ( a +) =b (b £ E) (4.1.15)

is given byy(x) = b(x - a ) " - 1 ^ [X(x - a)a]. (4.1.16)

In particular, the Cauchy type problem

(DlJ_?y)(x) - Xy(x) = f(x), (/a1(2J/)(a+) =b (b £ E) (4.1.17)

has the solution given by

(4.1.18)and the solution to the problem

(D1a/?y)(x)-Xy(x)=0, (7Q

1{ 2y)(a+) = 6 (b £ R) (4.1.19)

is given by

y(x) = b(x - arÊ^,, [x(x - a)1'2] . (4.1.20)

fJ a

Chapter 4 (§ 4.1) 225


(DZ+y)(x)-\y(x)=f(x), (D£1y)(a+) = b, (D^2y)(a+) = d, (4.1.21)

with 1 < a < 2 and A, b, d G M. has the form

y(x) = b(x - aÊcc [\{x - a)a] + d(x - a ^ E ^ [X(x - a)a]

(x- t)a~lEa,a [\{x - t)a\ f(t)dt. (4.1.22)

In particular, the solution to the problem

(D2+y)(x) - Xy(x) = 0, (Z^1y)(a+) = b, (Daa-

2y)(a+) = d (4.1.23)

is given by

y(x) = b(x - arÊ [\(x - a)a] + d(x - a)a-2£;a,a_i [X(x - a)a]. (4.1.24)

Next we consider the Cauchy type problem for the following more generalhomogeneous fractional differential equation than (4.1.11):

(D%+y)(x) - X{x - afy{x) =0 (a < x ^ b; a > 0; A G K), (4.1.25)

{D^-ky){a+) = bk {bk G 1; k = 1, , n; n = -[-a]), (4.1.26)

with /3 > - { a } . By Property 3.1 and Remark 3.18, the problem (4.1.25)-(4.1.26)is equivalent in the space Cn-a[a, b] to the Volterra integral equation of the secondkind

We again apply the method of successive approximations to solve this integralequation. We use the notation yo(x) in (4.1.4) and set

(m G N). (4.1.28)

Using the same arguments as above, we find for that

yi{x)=yo{x)+X(l°+yo)(x)

n n ,j (T

Z.r(a-j + i r

By the condition j3 > - { a } , all integrals (l"+{t - a)a+0~j) (x) (j = 1, , n) areconvergent, and applying the first relation in (2.2.16) yields the following result:


Similarly, using (4.1.28) with m = 2 and taking (4.1.29) into account, we derive

1/2(z) = 2/o(z)+A {m+yi) {x) = yo{x)+X {1%+yo) (x)

n ,

2 Y^ "ii=i

n ,

~ F(a — j + 1)

where_ r (a + /3 - j + 1) = r (a + 0 - j + 1) T(2Q + 20 - j + 1)

Cl ~ T(2a + 0 - j + 1)' °2 ~ T(2a + 0 - j + 1) T(3a + 20 - j + 1)'Continuing this process, we derive the following relation for ym(x) (m € N):

where

^ 1 [r(a + p) + a — j + 1J

Taking the limit as m — oo, we obtain the following explicit solution y(x) to theintegral equation (4.1.27) and hence to the Cauchy type problem (4.1.25)-(4.1.26):

y(x) = YlJ = I

(4.1.32)

According to the relations (1.9.19) and (1.9.20), we rewrite this solution in termsof the generalized Mittag-Leffler function Eatmj(z):

= E r(a-j

If (3 0, then f[x,y] = X(x — a)P satisfies the Lipschitz condition for anyx\,x2 £ (a, b] and any y e G, where G is any open set of C. If 7 ^ n — a, then,by Property 3-l(b) and Remark 3.18, there exists a unique solution to the Cauchytype problem (4.1.25)-(4.1.26) in the space C"_a[a, b], and thus this solution hasthe form (4.1.33). This leads to the following result.

Chapter 4 (§4.1) 227

Theorem 4.2 Let a > 0, n = - [ - a ] , A £ R and (3 0. Then the Cauchy typeproblem (4.1.25)-(4.1.26) has a unique solution y(x) in the spaceC"_a[a, b] :— C"_a0[a, b] and this solution is given by (4.1.33).

Remark 4.2 For a — 0, the solution (4.1.33) to the Cauchy type problem (4.1.25)-(4.1.26) in the space Iioc[0,d] of locally integrable functions on [0, d] (0 < d < oo)was first given by Kilbas and Saigo ([397], Equation 53).

Remark 4.3 If a > 1 and bn = 0, then, in accordance with Property 3.5, (4.1.10),and (4.1.12) and (4.1.33) are unique solutions to the respective Cauchy type prob-lems (4.1.1), (4.1.2), and (4.1.11), (4.1.2), and (4.1.25), (4.1.26) in the spaceC0%[a,6].

Remark 4.4 If f(x) £ L(a, b), then, by Theorem 3.3, y(x) in (4.1.10) and (4.1.33)would yield unique solutions to the Cauchy type problem (4.1.1),(4.1.2) and , by(4.1.25), (4.1.26) in the space La(a,b) defined by (3.2.1). In this case one mayformulate statements similar to those of Theorems 4.1 and 4.2.

Remark 4.5 Equation (4.1.33) yields the explicit solution to the Cauchy typeproblem (4.1.25), (4.1.26) for any p > -{a}, but, according to Theorem 4.2 andRemark 4.2, this solution is unique in spaces C"_a [a, b] and La(a, b) when /? 0.The problem of the uniqueness of this solution in the case —{a} < j3 < 0 remainsopen.


(DZ+y)(x)-\(x-afy(x) = 0; (^"12/)(a+) = b (b G E) (4.1.34)

with 0 < a < l, /3 € R (/? > -{a} ) and A e M is given by

Viz) = ~ ( Z - a)a-1â,l+â,l+(i8-l)/a [*(* " o)"+^] (4.1.35)


(D1a^y)(x)-X(x-afy(x) = 0; (D^/2

2/)(a+) = b (6 e R) (4.1.36)

has a unique solution given by

y(x) = -L(x - ayÊy,^ [\(x - a)^1'2] . (4.1.37)


{Daa+y){x) - \{x - afy{x) = 0; {Da

a^y){a+) = b, (D^2y)(a+) = d (4.1.38)

with b, d e R, 1 < a < 2, /? £ E (/? > - {a} ) and A £ E has the form

-2)/a [X(x - a)a+P] . (4.1.39)


Remark 4.6 When 0 — 0, then, Ea ltl_j/a(z) = Y{a - j + 1) Eata-j+i{z), by(1.9.23). Hence the solution (4.1.33) to the Cauchy type problem (4.1.26)-(4.1.27)yields the solution (4.1.10) to the Cauchy type problem (4.1.1)-(4.1.2). In particu-lar, solutions (4.1.35), (4.1.37) and (4.1.39) to the problems (4.1.34), (4.1.36) and(4.1.38) yield the solutions (4.1.16), (4.1.20) and (4.1.24) to the problems (4.1.15),(4.1.19) and (4.1.23), respectively.

4.1.2 The Cauchy Problems for Ordinary DifferentialEquations

In this section we use the results of Section 4.1.1 with a = n 6 N to derive theexplicit solutions to the Cauchy problems for ordinary differential equations oforder n on a finite interval [a, b]. First we consider the Cauchy problem

y(n){x)-Xy{x) = f{x); y{n-k){a) = bk (X,bkeR, n.fceN). (4.1.40)

By (2.1.7), this problem is a particular case of the problem (4.1.1)-(4.1.2) witha = n £ N. Therefore, from (4.1.10) we derive the solution to (4.1.40) in thefollowing form:

V(*) = J2 bj(x-a)n-iEn,n_j+1 [X(x - a)n}+ f (x-i)"" 1 En,n [\{x - t)n] f(t)dt.i=i a

(4.1.41)This is the unique explicit solution to the Cauchy problem (4.1.40) in the spaceCn[a,b\. In particular, when f(x) = 0, the solution to the problem

= 0; y(n-fc)(a) = 6* (A,6fc € R, n , * e N) (4.1.42)

is given byn

y(x) = ^ bj(x - a)n-iEn,n-j+1 [X(x - a)n]. (4.1.43)

Example 4.5 The solution to the Cauchy problem

y'(x)-Xy(x) = f(x); y(a) = b (b e E) (4.1.44)

is given by

y{x) = b Ehl[X(x -a)}+ / EM [A(a: - t)]f(t)dt,Ja

which, in accordance with (1.8.18) and (1.8.2), takes the well-known form

y(x) = b ex{x~a) + f ex^-^f(t)dt. (4.1.45)Ja


y"{x) - Xy(x) = f(x); y{a) = b, y'(x) =d(b,d&R) (4.1.46)

Chapter 4 (§ 4.1) 229

is given by

tx

y{x) = b{x-a)E2,2[X(x-a)2]+dE2,1[X(x-a)2] + / {x-t)E2<2[X{x-t)2]f{t)dt.J a

(4.1.47)In particular, for f(x) = 0 the solution to the problem

y"(x) - Xy(x) = 0; y(a) = b, y'(x) = d(b,deR) (4.1.48)

has the form

y(x) = b(x - a)E2,2[X(x - a)2] + dE2A[X{x - a)2]. (4.1.49)

If A > 0, then, according to (1.8.19), we can rewrite (1.8.18) and (1.8.2),(4.1.47) and (4.1.49) in the respective forms

y(x) = —j=(x - a)1/2 sinh[VA(:r - a)] + dcos\x[-JX{x - a)}V A

and

- ^ / (x - t)1/2 smh[V\(x - t)]f(t)dt (4.1.50)V A Ja

y(x) = -^=(x - a)112 s inhfv^z - a)] + dcosh[VA(a; - a)]. (4.1.51)vA

Next we consider the following Cauchy problem:

y^(x)-X(x-afy(x) = f(x); y("" fc)(a) = h, (A,bk G K, n,k G N) (4.1.52)

with real A,/3 0. By (2.1.7), this problem is a particular case of the problem(4.1.25)-(4.1.26) with a = n, and (4.1.33) yields the solution to (4.1.52):

& E (a; a ^ J E 0 / ( 0 ) / n [Hx - a)n+f] . (4.1.53)r ( n i + l)

I t is directly verified, on the basis of the formula (1.9.25), that (4.1.53) is thesolution to the problem (4.1.52) for any j3 > —n. By Property 3.1(b) (witha = n and 7 = 0) and Remark 3.18, the solution (4.1.53) is unique in the spaceCn[a,b]:=C$ tO[a,b].

Remark 4.7 The problem of the uniqueness of the solution (4.1.53) in the case—n — 1 has the form

y(x) = bEhl+Pt0 [X(x - a)1+0] . (4.1.55)



y"(x) - \{x - afy{x) = f(x), y(a) = b, y'(a) = d(b, d e l) (4.1.56)

with 0 > — 2 is given by

y(x) = b{x - a)£l2,i+/3/2,(/3+i)/2 [A(a; - a)2+0] + d£2,i+/3/2,/3/2 [Mx - a)2+/3](4.1.57)

4.1.3 The Cauchy Problems for Differential Equations withthe Caputo Fractional Derivatives

In this section we construct the explicit solutions to linear fractional differen-tial equations with the Caputo fractional derivative {°D"+y) (x) of order a > 0(a £ N) given by (2.4.1) in the space C^'â^b] (n = [a] + 1), defined in (3.5.3).

First we consider the Cauchy problem for the fractional differential equation(3.1.38) of order a > 0 with the initial conditions (3.1.37):

(cD«+y){x) - \y(x) = f(x) (a^x^b; n - 1 < a < n; n € N; A e R ), (4.1.58)

y (a) = bk {bk G R; k = 0, , n - 1). (4.1.59)

We suppose that f(x) G C7[a, b] with 0 ^ 7 < 1 and 7 ^ a. Then, by Theorem3.24, the problem (4.1.58)-(4.1.59) is equivalent in the space Cn~1[a,b] to theVolterra integral equation of the second kind of the form (3.5.4):

To solve (4.1.60), we apply the method of successive approximations by setting

3=0 J'

and

F Vm-l(t)dt 1 r f(t)dt(men).

Arguments similar to those in Section 4.1.1 yield the following expression for ym(x):

, t 9 - f [| £j=0 k=0

Taking the limi t as m —> oo, and taking into account (1.8.17), we obtain

n - l .x

y(x) = Y bj{x - a)jEaJ+1 [\{x - a)a] + (x - i ) " " 1 ^ ^ [A(z - t)a] f(t)dt.3=0 â

(4.1.62)

Chapter 4 (§ 4.1) 231

This yields an explicit solution y(x) of the Volterra integral equation (4.1.60) andhence of the Cauchy problem (4.1.58)-(4.1.59).

Since f[x,y] = Xy + f(x) satisfies the Lipschitz condition, by Theorem 3.25there exists a unique solution y(x) to the Cauchy problem (4.1.58)-(4.1.59) in thespace C"'™"1^, b] denned by (3.5.3). From here we obtain the following result.

Theorem 4.3 Let n — 1 < a < n (n 6 N) and let 0 _ 7 < 1 be such that 7 ^ a.Also let A G E. If f{x) € C7[a,b}, the Cauchy problem (4.1.58)-(4.1.59) has aunique solution y(x) e C"1"" 1 [a, 6] and this solution is given by (4.1.62).

In particular, 1/7 = 0 and f(x) G C[a, b], then the solution y(x) in (4.1.62)belongs to the space Ca'n~1[a,b] defined in (3.5.18).

The Cauchy problem (4.1.59) involving the homogeneous differential equation(4.1.58)

(cD%+y)(x) - Xy{x) = 0 {a x b; n - 1< a < n; n £N; X £R) (4.1.63)

has a unique solution y(x) G C"'™"1^, 6] of the form

n-l

y(x) = Y, bj(x - a)jEaJ+1 [X(x - a)a] . (4.1.64)3=0


(°DZ+y){x) - \y{x) = f(x), y(a)=b(b£R) (4.1.65)

with 0 < a < 1 and A G R has the form

y(x) = bEa [X{x - a)a] + f (x - t)a~lEata [X(x - t)a] f(t)dt, (4.1.66)J a


(cD%+y)(x) - Xy(x) = 0, y(a) = b { b £ R ) (4.1.67)

is given byy(x) = bEa [X{x - a)a]. (4.1.68)


(cDl /2y)(x)-Xy(x)=f(x), y{a) = b{b£R) (4.1.69)


y{x) = bE1/2 [X(x - o)1/2] + Az- * ) - 1 / 2£ i / 2 , i / 2 [A( I - i)1/2] f(t)dt, (4.1.70)

and the solution to the problem

(cD1a/*y)(x)-Xy(x)=0, y(a) = b(beR) (4.1.71)

is given by

y(x) = bE1/2 [X(x - a)1'2] . (4.1.72)



(cD%+y)(x) - \y(x) = f(x), y(a)=b, y'(a) = d (b,d G R) (4.1.73)

with 1 < Q < 2 and A G IR has the form

y(x) = bEa [X(x - a)a] + d(x - a)Ea,2 [X(x - a)a]

+ I (x - i ) " " 1 ^ [X(x - t)a] f(t)dt. (4.1.74)J a


{cD«+y){x) - Xy(x) = 0 (1< a < 2), y(a) = b, y'(a) =d{b,d£R) (4.1.75)

is given byy{x) = bEa [\{x - a)a] + d{x - a)Ea,2 [X[x - a)a]. (4.1.76)

Now we consider the Cauchy problem for the following more general homoge-neous fractional differential equation than (4.1.63):

(cD%+y){x) - \(x - afy(x) = 0 (a ^ x ^ 6; n - 1< a < n; n G N; A G R),(4.1.77)

V{k) (0) = bk (bk G 1; k = 0, , n - 1), (4.1.78)

with /3 > —a. Note again that, in accordance with Theorem 3.24, the problem(4.1.77)-(4.1.78) is equivalent in the space Cn~1[a,b] to the following Volterraintegral equation of the second kind:

y(x) = g hj{x - ay + jA_ £ ( ^ ) » . (4.L79)

We again apply the method of successive approximations to solve this integralequation. We use the notation yo{x) in (4.1.61) and set

(m G N).

The same arguments as in Section 4.1.1 lead to the following expression for ym(x):

k (\(x - a)a+0)k , (4.1.80)3=0

Jk=l

where— a

r = l

Taking the limi t as m -> oo, and taking (1.9.19) and (1.9.20) into account, weobtain the explicit solution y(x) of the integral equation (4.1.79) and hence of

Chapter 4 (§ 4.1) 233

the Cauchy problem (4.1.77)-(4.1.78) in terms of the generalized Mittag-Lefflerfunction (1.9.19):

— -blit™ _ »\3 I? . , . . . .. \\(r _ a)a+P~\ (A 1 82*1

j= 0 J'

If /3 ^ 0, then f[x,y] = X(x — a)^y satisfies the Lipschitz condition. Hence,by Theorem 3.25, (4.1.28) is a unique explicit solution to the Cauchy problem(4.1.77)-(4.1.78) in the space C^'^b].

Theorem 4.4 Let n — 1 < a < n ( n £ N) and let 0 ^ 7 < 1 be such that 7 ^ a.Also let X e R and /? ^ 0. If f £ Cy[a,b], then the Cauchy problem (4.1.77)-(4.1.78) has a unique solution y(x) & C"'n~1[a, b] and this solution is given by(4.1.82).

Remark 4.8 Equation (4.1.82) yields the explicit solution to the Cauchy prob-lem (4.1.77)-(4.1.78) for any (3 > -{a}. But, in accordance with Theorem 4.4,this solution is unique in spaces C"'n~1[a,b] when ft ^ 0. The problem of theuniqueness of this solution in the case —{a} < /3 < 0 remains open.


(cDZ+y)(x)-\{x-a)ey{x)=0, y{a) = b(beR) (4.1.83)

with 0 < a < 1, /? > -a and A e R has the form:

y(x) = bEaA+p/Qt0/a [\{x - a)a+P] . (4.1.84)

In particular, the solution to the Cauchy problem

(cD1a/_?y)(x)-\(x-afy(x) = 0, y(a) = b(beR) (4.1.85)

with /? > —1/2 is given by

y(x) = bE1/2t2p+lj2P [\(x - af+1l2] . (4.1.86)


(°DZ+y)(x)-\{x-a)('y{x)=0, y(a) = b, y'(a) = d (b,d £ E) (4.1.87)

with 1 < a < 2, /? > - a and A € IR has the form

y(x) = bEail+f)/at()/a [X(x - a)a+fi} + d(x - a)Eatl+f)/a,w+1)/a [X{x - o)a+/3] .(4.1.88)

Remark 4.9 When /? = 0, in accordance with (1.9.23) we have

Thus the solution (4.1.82) to the Cauchy problem (4.1.77)-(4.1.78) yields the so-lution (4.1.64) to the Cauchy problem (4.1.58)-(4.1.59). In particular, solutions(4.1.84), (4.1.86) and (4.1.88) to problems (4.1.83), (4.1.83) and (4.1.87) yield so-lutions (4.1.68), (4.1.72) and (4.1.76) to problems (4.1.67), (4.1.71) and (4.1.75),respectively.


4.1.4 The Cauchy Type Problems for Differential Equationswith Hadamard Fractional Derivatives

In this section we construct the explicit solutions to linear fractional differentialequations with the Riemann-Liouville fractional derivative (T>"+y) (x) of ordera > 0 given by (2.7.7) in the space Cf.n_a 7[a, b] (n = - [ - a ] ; 0 7 < 1), definedin (3.6.1).

First we consider the following Cauchy type problem:

(V2+y)(x) - Xy(x) = f(x) (a < x b; a > 0; A G M), (4.1.89)

{Vaa-

ky){a+) = bk (bk G K; k = 1, , n; n = - [ -a ] ) , (4.1.90)

with f(x) G C7jiog[a,6] (0 <; 7 < 1) [see (1.1.27)]. Then, by Theorem 3.28,(4.1.89)-(4.1.90) is equivalent in the space Cn_ajiog[a, b] to the following Volterraintegral equation of the second kind:

Applying the method of successive approximations, we set

(4.1.92)

/ ( )(4.1.93)

with m G N; and we derive the relation for ym(x) (m G N):

71 m +l

g - ) /(*)? (4.1.94)

Taking the limit a s m ^ o o, we obtain the solution to the integral equation (4.1.91)and hence to the problem (4.1.89)-(4.1.90) in terms of the function (1.8.17):

y(x) = J2 bj(x - a)a-jEa,a_j+1 [A (log -Y]3=1 a

+ J* (log?)0"1 Ea,a [A((log|)a] /(t) j - (4-1.95)

Chapter 4 (§ 4.1) 235

I t is clear that f[x, y] = Xy + f(x) satisfies the Lipschitz condition. If 7 ^ n - a,then, by Theorem 3.29, there exists a unique solution to the problem (4.1.89)-(4.1.90) in the space C%un_a^[a,b]. Equation (4.1.95) yields this solution. Thuswe derive the following result.

Theorem 4.5 Let a > 0, n = -[-a] and let 7 (0 7 < 1) be such that 7 ^ n-a.Also let A e l . If f & C7)iog[a,6], then the Cauchy type problem (4.1.89)-(4.1.90)has a unique solution y(x) G C".n_ [a, 6] and this solution is given by (4.1.95).

In particular, the Cauchy type problem involving the homogeneous differentialequation (4.1.89)

(V%+y){x) - Xy(x) = 0 (a < x ^ b; a > 0; A G E), (4.1.96)

with the initial conditions (4.1.90), has a unique solution y(x) in the spaceCl,n-a\

aA n - a , o M] of the form

y(x) = J2 bj (log I)"' 3 Ea,a.j+1 [A (log )a] . (4.1.97)

Remark 4.10 The explicit solution (4.1.95) to the Cauchy type problem (4.1.89)-(4.1.90) was given by Kilbas et al. [383].


(VZ+y)(x) - \y(x) = f(x), (Vaa^y)(a+) = b (beR) (4.1.98)

with 0 < a < 1 and A G M. has the form

-, (4.1.99)L \ I / J I


{T%+y)(x) - Xy(x) = 0; (K+1y){a+) = 6 (6eR) (4.1.100)

is given byy(x) = b (log -~) Ea,a \x (log -) 1 . (4.1.101)


{VlJ*y){x)-Xy{x)= f{x); (2?«:+/22/)(a+) = b {b G M) (4.1.102)



J* (log|) " 1 / 2E1 / 2,1 /2 [A (logf)1/2] f(t)j, (4.1.103)

and the solution to the problem

(V1J^y)(x)-Xy(x)=0; (V~l/2y)(a+) = b (6 G E) (4.1.104)

is given by

y(x) = b (log-) i?i/2,i/2 [A ( log-) J. (4.1.105)

Example 4.14 Let b, d £ K. The solution to the Cauchy type problem

(V +y)(x)-Xy(x)=f(x); (V^ly)(a+) = b, {Vaa~

2y){a+) = d (4.1.106)

w i t h 1 < a < 2 a nd A g l has t he form

y(x) = b (log ^ ) ° " 1 Ea,a [X(x - a)a] + d(x - o)a-2Ba>Q_i [x (log ^

+ J' (log I ) " " 1 Ea,a [A (log j)a] f(t)dt. (4.1.107)


(VZ+y)(x) - \y(x) = 0; {Vaa^y){a+) = b, (V^2y)(a+) = d (4.1.108)

with 1 < a < 2 and b, d e M., is given by

y(x) = b (log | ) °"1 E«,a [A (log | ) "] + d(a; - a ) * - 2 ^ - ! [A (log |a(

Next we consider the Cauchy type problem for the following more generalhomogeneous fractional differential equation than (4.1.89):

(V%+y){x) - \{x - afy(x) =0 (a < x b; a > 0; X e E)), (4.1.110)

{Vaa+

ky){a+) = bk (bk G E; k = 1, , n; n = - [ -a] ), (4.1.111)

with /? > - {a } . Again, by Theorem 3.28, the problem (4.1.110)-(4.1.11) is equiv-alent in the space Cn-a,iog[a> b] *° *ê following Volterra integral equation:

Chapter 4 (§ 4.1) 237

We again apply the method of successive approximations to solve this integralequation. We use the notation yo{x) in (4.1.92) and set

~\t-afym.l{t)j (m e N). (4.1.113)

Using the same arguments as above, we derive the following relation for ym(x)(m e N):

k~

, (4.1.114)

where c are given by (4.1.31). Taking the limit a s m -) oo, we obtain the solu-tion y(x) to the integral equation (4.1.112) and hence to the Cauchy type prob-lem 4.1.110)-(4.1.111) in terms of the generalized Mittag-Lefffler function (1.9.19)-(1.9.20):

x\<*-i \ ( x\<*+0]a) ^.^/-.i+W-j)/- [A O a ) \VW = L r(a-j + l

(4.1.115)

If j3 0, then f[x,y] = \(x — a)P satisfies the Lipschitz condition. If 7 ^ n — a,then, by Theorem 3.29, there exists a unique solution to the Cauchy type problem(4.1.110)-(4.1.111) in the space C".n_a 7[a, b], and thus this solution has the form(4.1.115). This leads to the following result.

Theorem 4.6 Let a > 0, n = - [ - a ] , A G K and /? ^ 0. Then the Cauchy typeproblem (4.1.110)-(4.1.111) has a unique solution y(x) in the spaceCf.n_a[a,b] := C".n_afi[a,b] and this solution is given by (4.1.115).

Remark 4.11 If a > 1 and bn = 0, then, in accordance with Theorem 3.31,(4.1.95), and (4.1.97) and (4.1.115) are unique solutions to the respective Cauchytype problems (4.1.89), (4.1.90), and (4.1.96), (4.1.90), and (4.1.110), (4.1.111) inthe space Cf[a,b]:=C%Ofi[a,b.

Remark 4.12 Equation (4.1.115) yields the explicit solution to the Cauchy typeproblem (4.1.110), (4.1.111) for any /3 > -{a}. But, in accordance with Theorem3.29, this solution is unique in the space C".n_a [a, b] when /3 0. The problemof the uniqueness of this solution in the case —{a} < (3 < 0 remains open.


(V2+y)(x) - A (log ^)P y(x) = f(x); (P^12/)(a+) = b (4.1.116)

with 0 < a < 1, b,/3 6 E (j3 > -{a} ) and A £ K is given by



(Vla/iy)(x)-\(x-afy(x) = f(x); (2?^/2j/)(a+) = b (4.1.118)

with b £ E, has a unique solution given by

h / r \ - V 2

() ( l ) E(l0g) E1/2<1+2p,2p_1^\{l0g) j . (4.1.119)


{Vaa+y){x)-\{x-afy{x) = f{x); (^+1t/)(a+) = b, {Da

a-2y){a+) = d

(4.1.120)with 1 < a < 2, b,d,/3 £ l (/? > -{a}) and A G E, has the form

b / x\a~^V{X) = )Y{a) V°* a) ^.i+/V-,i+O-i)/« [A (l0S a J

r ( a - l ) ( l 0 g a) ^.i+i8/°.i+O-2)/a[A(log-) j . (4.1.121)

Remark 4.13 When /? = 0, in accordance with (1.9.23), the solutions (4.1.115),(4.1.117), (4.1.119) and (4.1.121) to the problems (4.1.110)-(4.1.111), (4.1.116),(4.1.118) and (4.1.120) yield the solutions (4.1.97), (4.1.101), (4.1.105) and (4.1.107)to the problems (4.1.96), (4.1.100), (4.1.104) and (4.1.106), respectively.

4.2 Compositional MethodIn this section we present the method for solving, in closed form, some linearfractional differential equations and boundary value problems for these equationsbased on compositions of fractional integrals and derivatives with special functionsof the Mittag-Leffler and Bessel type. For simplicity, we present our results forequations with fractional derivatives of order a > 0. The corresponding results forequations with fractional derivatives of complex order are derived similarly.

4.2.1 Preliminaries

We illustrate the idea of the compositional method with the following formula forthe Riemann-Liouville fractional derivative (2.1.6) given in (2.1.17):

. yjb uf \JJ s* (-x s* u j . y^.î.A.j

From this equality we immediately derive that the simplest differential equation

(DZ+y)(x) = (x- af-°-1 (4.2.2)

has the explicit solution

T{Pa)(x-af-1 ((3>a>0). (4.2.3)

Chapter 4 (§ 4.2) 239

Moreover, (4.2.1) means that the composition of the Riemann-Liouville frac-tional derivative D"+ with the power function (x — a)@~1 leads to the same functionexcept for a certain factor. It follows that

y(x) = {x- af^1 (4.2.4)

is a solution to the homogeneous differential equation

(Daa+y)(x) = r ( ^ I ° p % ( » ) (4-2-5)

These arguments lead us to the conjecture that compositions of fractionalderivatives with elementary functions can give solutions to fractional differentialequations. Moreover, from here we derive another assumption about the possi-bilit y of such results for compositions of fractional calculus operators with specialfunctions of the type Mittag-Leffler and Bessel. Further in this section we developsuch an approach for solving, in closed form, certain classes of linear fractionaldifferential equations with the Riemann-Liouville and Liouvill e fractional deriva-tives.

4.2.2 Compositional Relations

In this section we present compositions of the Riemann-Liouville and Liouvillefractional derivatives D"+ (a G K) and Dt with the generalized Mittag-Lefflerfunction EatTnti(z) defined in (1.9.19)-(1.9.20), and also compositions of D°L withthe Bessel type functions Zv

p{z) and \^v,a{z) denned in (1.7.42) and (1.7.51),respectively.

Compositions of D"+, D°_ with Ea<rn< i(z) are given by the following statements.

Proposition 4.1 Let a > 0 and m > 0 such that

l > m - l - - , a{jm + I) <£ Z~ (j e No). (4.2.6)a

Then there holds the formula

(Vaa+ [it ~ a)a«-m+VEa,mj (X(t - a)am)]) (x)

= r r [ a ( 7 - m) + l ] 1 ] ( a; " ^ ^ " ^ + X{X ~ a)a'Ea'm'1 {X(X ~ a)am) (4-2-7)

In particular, if a(l — m) = —j for some j = 1, , —[—a], then

(Vaa+ [(t - a)^l-m+Êa,m!l (X(t - a)am)] ) (x) = \(x - a)alEa>m,, (\(x - a)am).

(4.2.8)


Corollar y 4.1 If a > 0, /? > 0 and AeE, then

r(x - a)0-"-1 + X{x - afÊ^p {X{x - a)a). (4.2.9)

In particular, if /? — a G ZQ , then

{Vaa+ [(t - af~xEa3 (X(t - a)a)]) (x) = X(x - afÊ (X(x - a)a). (4.2.10)

Corollar y 4.2 If a > 0 and AeK, i/ien

>£+ [Ea (X(t - a)a)}) (x) = / (a; - a)" a + XEa (X(x - a)a). (4.2.11)i {i a)

In particular, ifO<a<l, then

(Vaa+ [Ea (X(t - a)a)}) (x) = (x- a)-aEaâ (X(x - a)a). (4.2.12)

Proposition 4.2 Let X € M. and let a > 0 and m > 0 such that I > m — {a}/a,where {a} is the fractional part of a.


{va_ [^( ro-;)£a,mi,(_ Y[a{l - m + 1) + 1] a(m_t_i)_i . 4 2 1 3

" r[a(J-m) + l] i J o'™ ' i ( A l j - ( 4'2- l t 3 )

Corollar y 4.3 If a > 0 and /3>[a] + 1, then

{VI [ta~0Eatp (At-)] ) (x) = T{l3l_a)x~0 + Xx-a-pEa,0 (Xx~a) . (4.2.14)

By (2.1.7), whena = n G N, (D%+y)(x) =yW(x). Therefore, from Proposition4.1 and Corollaries 4.1 and 4.2, we derive the following assertions.

Proposit ion 4.3 Let X, a G E. Also let n G N, m > 0 and I G R be such that

n(jm + l) ,-n (j G No). (4.2.15)

Suppose that D = d/dx.


Dn [{x - a) "( ' -m + 1 )En,m, ; (A(a; - a)nm)]n

l -m)+ j](x - a)"('-m) + X(x - a )n^ n ,m ; i (X(x - a)nm). (4.2.16)

/n particular, if n(l - m) = -j for j = 1, , n, iften

= X(x - a)n i£

(4.2.17)

Chapter 4 (§ 4.2) 241

Corollar y 4.4 If n £ N, (3 > 0 and A £ R, tfien

0" [(* - a ) ^ "1 ! ^ (A(z - a)")]

i

= (x _ ay~n~1 + X(x — a)P~lE « (A(:r — a)n) (4 2 18)

/n particular, if ft = 1, , n, iften

Dn [(x - af-1En,0 {X{x - a)n)] = X{x - afÊ {X(x - a)n), (4.2.19)

Dn [En (X(x - a)")] = XEn (X{x - a)n). (4.2.20)

Remark 4.14 The condition / > m — 1 — I/a in Proposition 4.1, needed for theexistence of the left-hand side of (4.2.7), is omitted in Proposition 4.3.

The Proposition 4.3 can be proved directly by using the definition (1.9.24) forEn,m,i(z)- This definition is directly used to establish the next assertion.

Proposition 4.4 Let X,l £ R and let n £ N and m > 0 such that the condition(4.2.14) is satisfied. Also let D = d/dx.


L) \X v ' tin.m.l \XX

'™ _ ]\ _ ol™«(™—I — 1) — 1 i I 1 \n \_—n(l+l) — 1 p (\ —nm\ (191)11/ ' * V jj 1'' ' \ x / / ^ a ; -C'njrriji \AX ) . yi.L.LL)

In particular, if n(m — I) = j for some j = 1, , n, then

\X ^71,171,1 (AX ) —- y i-j AX ^n m I \^X J . \Q..2i.L2i}

Corollar y 4.5 If n £ N, (3 > 0 and X £ R, then

Dn \xn~Ên3 {Xx~n)\ + (-l)nXx-n-0Ent0 (Xx~n) . (4.2.23)

7n particular, if (3 = 1, , n, iften

Dn [^"^^, 0 (As—)] = (-l)nAa;-ri-^£;n,ig (Ax"" ) , (4.2.24)

Z?n [ / - ^ n (Ai- n)] = (- l) narn-1.En (Aa;"") . (4.2.25)

Remark 4.15 By (2.2.5), (D^y)(x) = {-l)ny^(x), and thus (4.2.13) witha — n yields (4.2.21). The condition I > m — {a}/a in Proposition 4.2, neededfor the existence of the left-hand side of (4.2.13), is omitted in Proposition 4.4.


Remark 4.16 Results presented in Propositions 4.1 and 4.4 and Corollaries 4.1and 4.5 were established by Kilbas and Saigo [397] and Saigo and Kilbas [725] (seealso [391] and [724]).

Finally, we give the compositions of the operator of Liouville fractional differ-entiation D°_ with the Bessel type functions Zv{z) and X],J(z).

Proposition 4.5 If a > 0, v G M. and p > 0, then there holds the formula

{Da_Z;)(x) = Z;-a(x). (4.2.26)

In particular, if a = m G N, then

DmZvp(x) = {-l)mZu

p-m(x). (4.2.27)

Proposition 4.6 Let a > 0 and c £ l . Also let (3 > 0 and a G M. fee such thata > (1/(3) - 1.


(x)=\W+a(x). (4.2.28)

In particular, if a — m G N, then

t l ) . (4.2.29)Remark 4.17 Results presented in Propositions 4.5 and 4.6 were proved by [373]and by Glaeske et al. [283], respectively.

4.2.3 Homogeneous Differential Equations of FractionalOrder with Riemann-Liouville Fractional Derivatives

We consider the homogeneous differential equation (4.1.25) of order a > 0:

(D°+y){x) = X(x - afy(x) (a<x<bôo). (4.2.30)

In what follows I\OC{£1) and Bioc(Cl) denote spaces of locally integrable functionson a subset 0 of R.

There holds the following result.

Theorem 4.7 Let n - 1 < a < n (n G N), /3 > -a and AeK.

(i) // 0 < a < 1, then equation (4.2.29) has the solution given by

y(x) = (x- a)Q-1Ea,1 + / 3 / o > 1 + ( ( 0_1 ) /a (X(x - a)a+0) G Iloc(a, b). (4.2.31)

(ii ) If a > 1 and

(a + p){k + 1) ^ 1, , [a] (he No), (4.2.32)

then the equation (4.2.29) has n solutions given by

yj(x) = (x- a)a- 'EQ ; 1 + / 3 / a,1 + ( / 3_j ) / Q {X(x - a)a+0) (j = 1, , n). (4.2.33)

Moreover, yj(x) G Bloc(a,b) (j = 1, , n - 1) and yn{x) G /ioc(o,b).

Chapter 4 (§ 4.2) 243

Proof. We apply Proposition 4.1 with m = 1 + /3/a and Z = / = 1 + (/? — j ) / a(j = 1, , n). Condition (4.2.6) acquires the form

l + - > 0 , {a + P)(k + l)-j<£Z- (JfeGNo). (4.2.34)

If 0 < a < 1, then j = 1, and condition (4.2.34) is satisfied. If a > 1 (a £ N\{1}) ,then (4.2.34) is satisfied for ( a+ /?)(& + 1) _ j - 1; hence we obtain (4.2.32). Since

t = tj = 1 H = 1"! > =771 — 1a a a a a a a

for any j = 1, , n, then the hypotheses of Proposition 4.1 are satisfied.

Substituting y(x) — Vj{x) given by (4.2.33) into (D"+y)(x), and using (4.2.8)with m = 1 + (3/a and Z = lj = 1 + (j5 — j)/a (lj = m — j /a), we have

(x)

= X(x - a)a-Êa<m<l. {X(x - a)a+P) = \(x - afVj{x)

for j = 1, , [a] + 1. This means that yj(x) in (4.2.33) are solutions to (4.2.30).

Now we prove that the solutions yj{x) in (4.2.33) are linearly independent for/3 > -{a}. For this we introduce the function Wa(x) denned by

Wa(x)=det{{D£kyj)(x))n

kJ=1 (n = [a] + l; a = x = b). (4.2.35)

This function is an analog of the Wronskian suitable to ordinary linear n-orderdifferential equations [see, for example, Erugin ([251], p. 225)]. We have thefollowing statement, whose proof is similar to that of the corresponding theoremfor ordinary differential equations [see, for example, Erugin ([251], p. 226)].

L e m m a 4.1 The solutions y\{x), y2{x),---, yn{x) in (4.2.33) are linearly inde-pendent if, and only if, Wa(xo) 0 at some point xo £ [a,b].

Now we prove the following uniqueness result.

Theorem 4.8 Let n - 1 < a < n (n G N) and A e E. If /3 > -a, then thesolutions yn{x) in (4.2.33) are linearly independent.

Proof. Let k, j = 1, , n. By (4.2.8) and (1.9.19), we find that

K S , ) (*) = (D [(t - a)a-iEa,l+0/atl+(fl _j)/a {X(t - ar+P)]) (x)

( ) (x),

where c0 = 1 and cp (p G N) are given in (1.9.20) with m = 1 + fi/a andI = I + (P - j)/a. Using (2.1.17) for 1 ^ p < n and (2.1.16) for p = n, wehave

{Daa+

kyj) (x) = f dp(k,j)^(x - a^+fip+i'-i (j, k = 1, , n), (4.2.36)

p=0


where

In particular, when k = j = 1, , n, we get

) = r(l + a-k), dp(k,k) = c / [ 1 + ° ^ ++

( ^ y ) P ] (p G N). (4.2.38)

If fc < j , then do(k,j) = 0 and (4.2.36) takes the form

{Daa^ yj) (x) = Y/dP(k,j)V(x - a)^+^+k^ (k,j = l,-.., n; k < j ) .

P=i

(4.2.39)

If k ^ j , then, taking the limi t as x — a+ in (4.2.36), we have

(££+fcy,-) (a) := xHm+ ) (*) = 0 (*, j = 1, , n; fc > j), (4.2.40)

(££+*?*) (a) := alim+ (l>°+fcy*) (x) = T(l + a - fc) (fc = 1, , n). (4.2.41)

If fe < j , then (a + /3)p + k - j ^ {a} + /3 > 0 for any p £N and j , k = 1, , n,in accordance with the condition (3 > —{a} . Then, taking the limi t as x —> a+ in(4.2.39), we derive

{Daa+

kVj) ( a) : = B l m + { D ^ k

y j ) ( x ) = 0 ( k , j = l , - - - , n ; k < j ) . ( 4 . 2 . 4 2)

It follows from (4.2.40)-(4.2.42) that

k=l

for an analog of the Wronskian (4.2.35). Therefore, by Lemma 4.1, the solutionsyi(x), , yn(x) in (4.2.33) are linearly independent.

The next assertion follows from Theorems 4.7 and 4.8 when /? = 0.

Corollar y 4.6 If n — 1 < a < n ( n£ N), then the equation

{Daa+y){x) = Xy(x) (a<x^b<oo) (4.2.43)

has n = [a] + 1 linearly independent solutions given by

yj{x) = (x- a)a~iEa,a+1-j (\{x - a)a) (j = 1, , n). (4.2.44)

Moreover, yj(x) £ J3ioc(a, 6) (j = 1, , n - 1] and yn(x) e 7ioc(o,6).

7n particular, for 0 < a < 1, i/ie unique solution has the form

y(x) = (x- aÊca (X(x - a)a) 6 / l oc(o, b). (4.2.45)

Chapter 4 (§ 4.2) 245

Remark 4.18 If we consider the equation (4.2.30) on a finite interval (a, b)(b < oo), the spaces of the solutions (4.2.31), (4.2.45) and (4.2.33), (4.2.44) can becharacterized more precisely. Thus the solutions y(x) in (4.2.31) and (4.2.45) be-long to the weighted space Ci-a[a, b], while in (4.2.33) and (4.2.44) the solutionsVj(x) (j = ) ft — 1) G C[a,b] are continuous and yn{x) G Cn-a[a,b] withn = - [ - a ] .

4.2.4 Nonhomogeneous Differential Equations of FractionalOrder with Riemann-Liouville and Liouville FractionalDerivatives with a Quasi-Polynomial Free Term

We consider the nonhomogeneous differential equation of fractional order a > 0with a quasi-polynomial free term f{x) = X^r=i fr{x ~ o)^7"-

k

(D°+y){x) = X(x - afy{x) + £ fr(x - a)"' (a < x -a, fir G M. (r = 1, , k) besuch that

A * r > - l - a , M r T ^ - i (j . [ a ]+ 1; r = l , , (4.2.47)

(j + l ) ( a + 0) + nr i Z - (r = 1, , k; j G No) . (4.2.48)

(i ) Equation (4.2.46) is solvable in the space 7;oc(o, 6) if there exists aj G {1, , k} such that fj,j < —a and in the space Bioc(a,b) if /j,r ^ —a forall r G {1, , k}. Furthermore, there is a particular solution of the form

k

(4.2.49)

(ii ) If ft > —{a} , i/ien ifte general solution to the equation (4.2.46) is given by

n

y(x) = Y, cj(x - a)a-jEa,1+0/aA+(0_j)/a (\(x - a)a+l3) + yo(x), (4.2.50)

where Cj (j = 1, , n) are arbitrary real constants.

Proof. We use Proposition 4.1 with m = 1 + /3/a and J = Zr = 1 + (/? + fir)/<x(r = 1, , £;). Condition (4.2.6) acquires the form (4.2.47)-(4.2.48). Substitutingy(x) given by (4.2.49) into (D"+y)(x) and using the formula (4.2.7), we have

k

(Daa+y)(x) = X : '


1 [Ur i Oi ~T~ 1Jr=l r=l x '

h

r=l

Thus y{x) in (4.2.49) is a particular solution to the equation (4.2.46), and (i) isproved. The second assertion (ii) of Theorem 4.9 follows from Theorem 4.7.

The next assertion contains sufficient conditions for the validity of (4.2.47) and(4.2.48).

Corollar y 4.7 Let a > 0 (a £ N), /? > -a, and fir G 1 (r = 1, , k). Also leteither of the following conditions be satisfied:

( i ) Hr > - 1 - a , fir > -1-a-P, fj,r^-j(r = l,---,k;j = l,---,n = - [ - a ] ) .(4.2.51)

(ii ) fj,r> -1 (r — 1,--- , n).Then the first assertion of Theorem 4.9 is valid. If, in addition, f3 > -{a},

then the second assertion of Theorem 4.9 holds.

Setting /? = 0, from Theorem 4.9 we derive the corresponding result for thedifferential equation (4.2.46) with ft = 0:

k

(D%+y)(x) = Xy(x) + J 2 f A x ~ ° )M " ( a < x < b ^ o o ) . (4.2.52)r=l

Corol lary 4.8 Let n - 1 < a < n (n G N) and fir £ R (r = 1, , k) be suchthat the conditions (4.2.47)-(4.2.48) are satisfied. Then the differential equation(4.2.52) is solvable in the space Iioc(a,b) and its general solution has the form

n

(* - a)a-iEa-j+1 (X(x - a)a)

J2 ir+1 (X(x - a)a), (4.2.53)r=l


Example 4.17 Let n - 1 < a < n (n £ N), 0 G E and p. > - 1 - a be such that

fi>-l-a, MT^-J (j = l , - - - , n ), (r + l ) (a + /3) + /x f 1~{r G No).

If /3 > —a, then the equation

{D°+y){x) = X{x - afy(x) + f(x - a)" (a < x — {a} , then the general solution to equation (4.2.54) is given by

)/« (A(z - a)a+/3) + j/0(a;), (4.2.56)


Example 4.18 If n —1 < a < n (n G N), y, > — I — a and (M —j (j = 1, , n),then the differential equation (4.2.54) with /? = 0

{D°+y)(x) = Xy(x) + f{x - a)" {a < x a-j+l (X(x - a)a)

(A(a; - a)a), (4.2.58)


In particular, for 0 < a < 1 and \i > - 1 - a (/J, ^ -1 ), the equation (4.2.57)has the explicit solution given by

y(x) = c(x - a)aÊa<a (X(x -a)a) + +T(p +l)f(x- a )Q +^ a, a + M +1 (X(x - a)a),(4.2.59)

where c is an arbitrary real constant.

Now we consider the following differential equation with the Liouvill e fractionalderivative (D°Ly){x) defined in (2.2.4):

i

(,D°Ly)[x) = Xx0y{x) + ^ frX- (0 ^ d < x < oo), (4.2.60)r= l

where A,/3 G E and / r , ^ r G K (r = 1, , k) are given real constants.

Theorem 4.10 Letn — 1 < a < n (n G N), /? < - a and fir > n for r = 1,- , fc.Equation (4.2.60) is solvable in the space Iioc(d,oo) and has a particular solution

r = 1


Proof. We use Proposition 4.2 with m = —1 — (3/a and, for r = I,--- , k,I = lr = —2 + (/j,r — P — l ) / a. The condition I > m — a / {a} acquires theform fj,r > n. Substituting y(x) in (4.2.61) into {Dty){x) and using the formula(4.2.13), we have

(DZy)(x) = £ r ( A < r ( ^) / r (^ - [^ (m^ )" 1^,m, (Ar Qm)]) (a;)

= £ frx-"' + A £r= l r= l

Thus y(x) in (4.2.61) is a particular solution to the equation (4.2.60).

Corollar y 4.9 Let a > 0 (a £ N) and //,. > [a] + 1 (r = 1, , jfc). Then theequation

k

{Da_y){x) = Xx-2ay{x) + ] T /ra;-"' (0 d < a; < oo; A, /P G M), (4.2.62)

r= l

is solvable in the space Iioc{d, oo) and ftas a particular solution given by

k

y(x) = J2 r(nT - a)frxa-ÊQilir (\x~a) . (4.2.63)

r= l

Example 4.19 Let a > 0 (a ^ N), /3 < —a and /i > [a] + 1. Then the equation

(D°^y){x) = \{x-afy(x) +f(x-a)-» (0 d< x < oo) (4.2.64)

with A, / G M. has a particular solution given by

3 . (4.2.65)

4.2.5 Differential Equations of Order 1/2

In this section we present solutions to the differential equations of order 1/2. Firstwe consider the differential equations (4.2.30) and (4.2.46) with a = 0:

(Dl^y)(x) = \XPy{x) (0 < x y)(x) = \xPy{x) + fr^r (0<x<bôo), (4.2.67)r = l

with given A, fr, fj,r € M. (r = 1, , /). Theorems 4.7 and 4.9 with a = 0 implythe following result.

Chapter 4 (§ 4.2) 249

Theorem 4.11 There hold the following assertions:

(i) If /3 > —1/2, then the equation (4.2.66) has the solution given by

y(x) = z-1/2£i/2,2/3+i,2/3-i (Aa^1/ 2) G Iioc{a,b). (4.2.68)

(ii ) If fi > —1/2 and fir > - 3 / 2, fir ^ - 1 (r = 1, , /), then the equation(4.2.67) is solvable in the space I\oc(a,b). Its particular solution is given by

^~ >Jr Im \Mf + l / 2 jp (\lA- 1 r ' l/2>2/3+1,2((/3+Mr)+l I /^\a' ~~ u)

(4.2.69)«is general solution has the form

y(x) = ca;-1/2£;1/2,2/3+l i2/3_1 (A ;E / 3 + 1/ 2) + yo(a;), (4.2.70)


Corollar y 4.10 If fir > - 3 /2 (fir ^ -1) (r = 1, , /), then the equation

i

(Do'+ y)(x) = \y(x) + y^ frX (0 < x -1 /2 and fi > -3 /2 (fi ^ -1), then the equation

(D\l2y)(x) = \xpy(x) + fx" (0 < x < b oo; A, / G M) (4.2.73)


( ) (4.2.74)


In particular, the equation (4.2.73) with /3 = 0

(Dl /2y)(x) = \y(x) + fx" (0 < x < b ^ oo) (4.2.75)


y(x) = ca;-1/2i51/2,1/2 (A^1/2) + T(M + l)fx^+1l2E1/2 +z/2 (Xx1'2) , (4.2.76)



Equations of the form (4.2.73) arise in applications. For example, the voltmeterequation in electrochemistry has the form

x1/2(D10/?y)(x)+xwy(x) = l (0<x^b<<x>; 0 < w 1) (4.2.77)

[see Oldham and Spanier ([643], p. 159 and [644])] and the equation of the po-larography theory is given by

{Dl' 2y){x) = Xx0y{x)+x~1/2 [x>0; - - < /3 0; A G 1 , (4 .2 .78)

[see Wiener [881]]. From the above considerations, we derive the solutions to thesetwo equations as follows.

Corollar y 4.11 A particular solution to the equation (4.2.77) has the form

yo(x) = VÊi/2,2^2™-! {-xw) (4.2.79)

and the general solution to such a voltmeter equation is given by

y{x) = cx-1l2E1/2t2w,2w_2 {-xw) + VÊ1/2fiwfiw_1 {-xw), (4.2.80)


Corollar y 4.12 A particular solution to the equation (4.2.78) has the form

j/o(z) = v^S1 / 2 i 2 / 3 + 1 > 2 /3 (Ax" +1/2) (4.2.81)

and its general solution is given by

y(x) = cx-Ê^w+w-! (A^+1/2) + M/2,2/3+1,2/3 ( A ^ + 1 / 2 ) , (4.2.82)


Remark 4.19 Using the definition (1.9.19)-(1.9.20) of the generalized Mittag-Leffler function Eatmti(z) and the relation F(l/2) = y/w (see the second formula in(1.5.8)), we can rewrite the particular solution 2/0(2:) given in (4.2.79) in the form

r 7 + ( f {keNo)- (4-2-83)fc=0 r=0 ^Wr '

Such a form for a particular solution to the voltmeter equation (4.2.77) was ob-tained by Oldham and Spanier ([643], p. 160) by using the formal representationof 2/0(2;) in the form of a quasi-power series.

From Theorem 4.10 we obtain the following solution to the equation (4.2.60)with a = 1/2:

1

{D1J2y){x) = Xx0y(x) + frx~ {0^d<x<oo), (4.2.84)r=l

where X,j3eR and fr, /xr G M (r = 1, , /) are given real constants.

Chapter 4 (§ 4.2) 251.

Theorem 4.12 If (3 < - 1 /2 and fj,r > 1 (r = 1, , I), then the equation (4.2.84)is solvable in the space IiOc{d, oo) and its particular solution has the form

, (4.2.85)

Corollar y 4.13 If fir > 1 (r = 1, , 1), the equation (4.2.84) with /3 = - 1

(D1J2y)(x) = A — + ^frx' (0^d<x<oo) (4.2.86)r= l


i

r= l

Example 4.21 If/3 < -1/2 and > 1, then the equation

(D1_/2y)(x) = Xxpy(x) + fx~» (0 d < x < oo). (4.2.88)


(4.2.89)

Moreover, the equation (4.2.88) with 0 = - 1

(£>i/2y)(a;) = A ^ + /a;-" (0 g d < i < oo) (4.2.90)


y[x) = r (M - l /2) /^^+ 1/2JE;1 / 2 i / i ( A Z " 1 / 2 ) . (4.2.91)

4.2.6 Cauchy Type Problem for Nonhomogeneous Differen-tial Equations with Riemann-Liouville FractionalDerivatives and with a Quasi-Polynomial Free Term

We consider the Cauchy type problem for the differential equation (4.2.46):

i

{D°+y){x) = X(x - a)py{x) +Y,fr(x~ a ) " r (a<x<bôo; A e i ) , (4.2.92)

(D«+ky)(a+) = bk (bk G K; k = 1, , n; n = -[-a]), (4.2.93)

where A,/3 G R and / r , yur G R (r = 1, , I), and 6fc € R (A; = 1, ,n).


Theorem 4.13 Let n - 1 < a < n (n 6 N), /? > -{a} and \iT > - 1 (r =I , - - - , I).

Then the solution to the Cauchy type problem (4.2.92)-(4.2.93) exists inI\oc(a,b), is unique and given by the formula

T ( i + a j )

(4.2.94)where

(4.2.95)

Proof. By Corollary 4.7, (4.2.94) is a solution to equation (4.2.92). By Theorem4.2 (in Section 4.1.1), the first term in (4.2.94) is the solution to the Cauchy typeproblem (4.2.30), (4.2.93) for the corresponding homogeneous equation, and thus

- a) = bk

(4.2.96)for k = 1, , n. By (1.9.19) and (2.1.17), we have

(Z?°+fc [t - ar+ÊaA+0/Q,1+((p+llr)/a {X(t - a)a+e)}) (x)

~ h r[fc + Mr + (a + /3)n + l ] ^ a)

for r = 1, , / and k = I,--- , n. By the condition ji r > —1, we have/nr + (a + /3)m + ft > 0 for any r = 1, , /; k = 1, , n and m G No. Thus

[t - a)a+Êatl+l3/a,1+w+llr)/a (X(t - a)a+e)}) (o+) = 0

and hence (D" jT*j/o ) (a+) = 0 for any fe = I,--- , n. Therefore, in accordancewith (4.2.96), the solution (4.2.94) satisfies the relations (4.2.93). The uniquenessof the solution follows from (4.2.94)-(4.2.95). This completes the proof of Theorem4.13

When (a, b) is a finite interval and fir > - 1 (r = 1, , I), the space I\oc{a, b)of the solution y(x) in (4.2.94) can be characterized more precisely in terms of thespace C£_a)7[a,6] defined in (3.3.24).

Theorem 4.14 Let a > 0 (a £ N), n - [a] + 1, /3 > - { a } , fir > - 1(r = 1, , I), and let (a, b) be a finite interval.

Ifbn yi- 0, then the unique solution (4.2.94) of the Cauchy type problem (4.2.92)-(4.2.93) belongs to one of the spaces C"_Q7[a,b] as follows:

Chapter 4 (§ 4.2) 253

(i ) Let [i r ^ 0 (r = 1, , / ). Then y(x) G C£_a0[a,&] for /3 ^ 0, whileV(x) e Cn-a,-p[ b] for -{a} < /? < 0.

( i i ) L et there e x i st r i , - - - , r * G { I , - - - , 0 (1 = ^ = 0 SMC^ a^ Mr3- < 0(j = 1, , k) and let \i be a minimum of such /j,rj:

H= min fir.. (4.2.97)Mr-,-<O; n , - , r f c 6 { l , - , J}i ! ^ * ^ '

Tften y(a;) G C £ _Q j _ M [ a , 6] / or 0 ^ 0 , » M e </(* ) G C ° _a >_ m i n [ / l i f l ] [ a , 6] for-{a}</3 <0.

Proof. By Theorem 4.13, the unique solution j/(a;) given by (4.2.94) has the form

y(x) =yi(x)+yo(x),

where

and j/o(a;) is given by (4.2.95). Since bn 0, yi(a;) G Cn-a[a,b]. Since /zr > - 1 ,Hr + a > - 1 (r = 1, , /). If ^ r + a ^ 0, then yi(x) G C[o, 6], and hencey(x) G Cn_a[a,6]. If there exist ro , rjfe G {I,-- - , Z} (0 ^ fc /) suchthat fj.rj + a < 0 (j = 0 ,1 , - -- , k) and /i* is a minimum of such /j,rj, thenyi{x) G C_(M .+a)[a,6]. Hence y(a;) G C7[a,b], where 7 = max[n - a,—ft* — a].Since n = [a] + 1 1 > — /U*, 7 = n — a, and hence again 2/(2;) G Cn-a[a, b].

Further, since for any y e G CM. the first term \(x-a)Py in the right-hand sideof (4.2.92) belongs to the space C[a, b] for /? 0, and to the space C-p[a, b] when—{a} < /? < 0. If nr ^ 0 (r = 1, , I), then the second term in the right-handside of (4.2.92) belongs to C[a,b], but if there exist ro,r1,--- , ru G {1 , , 1}(0 ^ k ^ I) such that //rj. < 0 (j = 0,1, , k) and yu is a minimum of such /j,rj,then the second term of the right-hand side of (4.2.92) belongs to C-^[a, b].

Thus, if fj,r ^ 0 (r = 0,1, , I), then the right-hand side of (4.2.92) and henceD°+y belong to C[a,b] for /? 0, and to C_0[a,6] when - { a } < /3 < 0. If thereexist ro, r i , , r* G {0,1, , /} (0 k ^ /) such that /xrj. < 0 (j = 0,1, , fe)and /x is a minimum of such /jrj, then the right-hand side of (4.2.92) and henceDl+y belong to C-^b] for ^ ^ 0, and to Cmax[_/ii_/3[a,fe] = C_ min[Mii8 ] when- { a } < /? < 0.

By the definition (3.3.4) of the space C"_Q 7[a,6], the above arguments provethe assertions (i) and (ii) , and Theorem 4.14 is proved.

Corollar y 4.14 Let n - l < a < n ( n G N) and /J, G M be such that either(i > - 1 - a, (i -j (j = 1, , n) or fi> - 1 .

T/ien £/ie solution to the Cauchy type problem

1

(D%+y){x) = \y(x) + fr(x - a)"" (a < x —I — a or n > —1, then the solutionto the Cauchy type problem

(D:+y)(x) = Xy(x) + f(x-ar; (Daa~

l y){a+) = d (4.2.101)

with a < x -{a} and / i € l b e such thateither

fi>-l-a, n>-l-a-0, / i#-j (j = l,-" » ») (4.2.103)

or /x > — 1. Then the solution to the following Cauchy type problem for theequation (4.2.54) with the initial conditions (4.2.93)

(D°+y){x) = X(x - afy(x) + f(x - a)" (a < x — a and fj, G R be such that either of theconditions (4.2.103) or /J, > —1 is satisfied, then the solution to the Cauchy typeproblem

{Daa+y){x) = \{x - afy(x) + f(x-ar; {Da

a+1y){a+) = d (4.2.107)

w i t h a < x - 1 /2 and p,r G E (r = 1, ) be such that either ofthe following conditions is satisfied:

(i ) A*r > - | , l*r>-\-P, / W - l (r = l , . - . , / ) , (4.2.109)

(ii ) Mr > - 1 (r = l , - - . , 0-

Then the solution to the Cauchy type problem (4.2.92)-(4.2.93) with a = 1/2

(D^y)(x) = X(x - afy(x) + £/«.(* - a)"'; (Z^/22/)(a+) = d, (4.2.110)

with a < x - 1 / 2, / i r > - 1 (r = 1, , I), and let (a, b) (-oo < a <b < oo) be a finite interval.

Ifb 7 0, then the unique solution (4.2.111) of the Cauchy type problem (4.2.110)belongs to one of the spaces C " _a 7[ a, b] as follows:

(i ) Let firÔ (r = l,--- , I).

Then y{x) G Cf_a 0[a, b] for 0^0, while y{x) e C«_Q _0[a, b] for - 1 /2 <0<0.

(ii ) Let there exist , ru G { I , - - - , 1} (1 ^ k ^ I) such that firj < 0(j = 0 ,1, , k) and let be a minimum of such fj,rj given in (4.2.97).

Then y(x) G C?_a>_M[a,6] for 0 Z 0, while y(x) G C«_Q_min[M/3] [a,6] for

Example 4.23 Ii 0 > - 1 /2 and p. > - 3 /2 (fi ^ -1 ), then the Cauchy typeproblem

x» (0 < x < b £ oo) (D^ / 2j / ) (o+) = d e R (4.2.112)


= £

r ( M + i ) / ^ ^T / 3+ iM M , i n l

r(/ i + a + l ) £V2,2i9+i,2(i8+^)+i â ;p J . (4.2.113)


In particular, the Cauchy type problem for equation (4.2.112) with /? = 0

{Dy*y){x) = Xy(x) + fx»; (D^/2y)(a+) = d (4.2.114)

with d 6 l and 0 < x < b oo, has the solution given by

y(x) = A z

(4.2.115)

Corollar y 4.15 The following Cauchy type problem for the voltmeter equation(4.2.77) in electrochemistry

x^2(D^y)(x) + x™y(x) = 1; (JDa^/2y)(a+) = d (4.2.116)

with 0<x^b<oo,0<w 1/2 and d £ l , has the solution given by

y(x) = ^X-^2E1/2I2WI2W_2 (-xw) + v^Si/2,2U,,2U,-i i-xw), (4.2.117)

Corollar y 4.16 The following Cauchy type problem for the polarography equation(4.2.78)

+ x~1/2; (D^/2y)(a+) = d, (4.2.118)

with x > 0, - | < / 3 ^ 0, and A , r fe t, has the solution given by

y(x) = - j - X v ( ) / (

(4.2.119)

Remark 4.20 Results presented in Sections 4.2.3-4.2.6 for differential equationsinvolving the fractional derivatives D +y and Dty in spaces of integrable functions/ioc(0)d) (0 < d ^ oo) and /ioc(d, oo) (0 ^ d < oo), respectively, were establishedby Kilbas and Saigo [397] [see also Kilbas and Saigo [393], [394] and Saigo andKilbas [724])].

Remark 4.21 Statements of Propositions 4.3 and 4.4 and Corollaries 4.4 and4.5 can be used to find solutions to homogeneous and nonhomogeneous ordinarynth-order differential equations of the form

y(n){x) = X(x - afy(x), (4.2.120)

!/(">(* ) = X(x - afy{x) + Y,M* ~ Y" (4.2.121)

and Cauchy problems for them. Such results were given in Saigo and Kilbas [725].

Chapter 4 (§ 4.2) 257

4.2.7 Solutions to Homogeneous Fractional DifferentialEquations with Liouvill e Fractional Derivatives inTerms of Bessel-Type Functions

In this section we apply the relations (4.2.26) and (4.2.28) to find explicit solutionsto certain homogeneous differential equations of fractional order in terms of theBessel-type functions Zv

p(z) and \(£l{z) defined in (1.7.42) and (1.7.51). Using theusual notation D = d/dx, first we consider the fractional differential operators:

Lvpy := xv-p+1Dxp-vDp_y = -xDp_+1y + {p - v)Dp_y, (4.2.122)

Mvpy := xv-p+1Dxl-pDx2p-vDp_y

= x2D2_p+2y + (2v - 3p - l)xD2_p+1y + (v - p{v - 2p)D2_py. (4.2.123)

When p = 1, the operator in (4.2.122) is a differential operator of second order:

The Bessel-Clifford operator [see Kiryakova [417]]

B-»!=s*'- r s*"»=*0 +«1+")s - (i2-125)and the operator of axisymmetric potential theory [see Gilbert [281]]

LvV : = x x2u+i^l = f]L+ 2v+ldy (4.2.126)dx dx dx2 x dx'

are expressed in terms of the operator (4.2.124) as follows:

Bvy = -x-vL\xvy, Lvy =--L~2vy. (4.2.127)x

A function Zvp(x) is invariant with respect to the operators Lp(z) and Mp,

apart from a constant multiplier factor.

Lemma 4.2 IfueR and p > 0, then

{L;Z;) (X) = -Pz;(x), (4.2.128)

(M;Z ; ) (X) = -P2z;{x). (4.2.129)

In particular, if p = 1, then

{L\ZD (x) = [MIZD (x) = -Z?(x), (4.2.130)

xv {Bvx-"ZVX) (x) = Zl (4.2.131)

andx (LvZ~2v) (x) = Z~2v. (4.2.132)


Remark 4.22 Formula (4.2.128) provides a more precise version of the result inRodriguez et al. [713]. Also Lemma 4.2 provides the explicit solutions to twofractional differential equations.

Theorem 4.17 // v G M. and p > 0, then the function y(x) = Zvp(x) is a solution

to the following differential equation of order p + 1:

(X) + {y-p) (Dp_y) (x) - py(x) = 0 (4.2.133)

and to the following differential equation of order 2p + 2:

x2 (D2_p+2y) (X) + {2v - 3p - 1)3 (D2_p+1y) (X)

+{y - P){v - 2p) (D2_py) (x) + p2y(x) = 0. (4.2.134)

In particular, y(x) — Zy{x) is the solution to the following differential equationsof the second and fourth orders:

xy"{x) + (1 - v)y'[x) - y(x) = 0, (4.2.135)

x2y^(x) - 2{v - 2)xy(-3\x) + (v - \)v - 2)y"{x) + y{x) = 0. (4.2.136)

Corollar y 4.17 // Kv{x) {v G E) is the Macdonald function (1.7.25), then thefunction y{x) = x"Ku(x) is a solution to the differential equations

xy"(x) + (1 - 2v)y'(x) - xy{x) = 0, (4.2.137)

xzy(i\x) - 2(2 - l)y(3)(a;) + (4^2 - l)xy"{x) - [\v2 - l)y'(x) + x3y{x) = 0.(4.2.138)

Corol lar y 4.18 The functions y(x) = x"v Z{{x) and y(x) = Z^2u{x) with i / £ lare solutions to the Bessel-Clifford equation

xy"{x) + {v + l)y'(x) - y{x) = 0, (4.2.139)

and to the following equation of axisymmetric potential theory:

xy"{x) + (2v + l)y'(x) - y{x) = 0. (4.2.140)

Now we consider the fractional differentiation operator L"^ denned by

La/ := x (Da_+P+1 - D1+1) + (<r + a + 1 - 0)D°, (4.2.141)

with a ^ 0, (3 > 0 and 7, a G K. There holds the following assertion.

Lemma 4.3 Let a ^ 0, /3 > 0 and u,a e l be such that v > (1//3) - 1. Then,for x > 0, the relation

*) = ( ^ + a + a) (Dl+f} \W_0) (x) (4.2.142)

holds for the Bessel-type function \fl_p{x) defined in (1.7.51).

Chapter 4 (§ 4.2) 259

From Lemma 4.3 we derive solutions to two fractional differential equations.

Theorem 4.18 If a ^ 0, /3 > 0 and a, v G R are such that v > (1//3) - 1, theny(x) = Xfa_p(x) is a solution to the following differential equation of fractionalorder:

[( (x) - (D^y) (x)]

-iyfi + a + a) (r>Z+pyj (x) + (a + a + 1 - /3) (D°y) (x) (4.2.143)

and, in particular, to the following differential equation:

x (Dt+1y) (x) - (v/3 + a) (Dty) (x) - xy'(x) + (a + 1 - /3)y{x). (4.2.144)

It is known that, if /? = 1, v G C (<H(i/) > 0), a e K and z € C (<R{z) > 0),then

Xi]l(z) = e~z^{v, v + cr + l;z), (4.2.145)

where ^(v, v + a + 1; z) is the Tricomi confluent hypergeometric function definedin (1.6.25). In particular

*) = *-"*- E tj^lt}f (m € No). (4.2.146)

(See, in this regard, Glaeske et al. [283], Kilbas et al. [406], Bonilla et al. [93] andKilbas et al. [390]). Then, from Theorem 4.18, we obtain the following assertions.

Corollar y 4.19 If a ^ 0, v > 0 and a G E, then the following differential equa-tion of fractional order

x {D°:+2y) (x)-(v + o- + a + x) (D1+1y) (x) + (a + a) (D°Ly) (x) = 0 (4.2.147)

has the solution y(x) = e~x^(i/, v + o;x).

In particular, if a = rn + 1 (m G No), then

y(x) = x-"e- £ ^ f ^ = A& (*) (4-2.148)fc=O

is the solution to the following fractional differential equation:

x {Da_+2y) {x)-{v + m + \ + a + x) {D°L+ly) {x) + (m + l + a) (DZy) {x) = 0.(4.2.149)

Corollar y 4.20 If a ^ 0, v > —1/2 and m G No, then the following differentialequation

£>°+2t/) {x)-x (Da_+1y) (x) + {m+a) (Da_y) {x) = 0(4.2.150)

has the solution y(x) = \v,m{x).


Example 4.24 Taking m = 0 in Corollary 4.19, we see that y(x) = x~ve^x isthe solution to the fractional differential equation of fractional order

x [Da_+2y) (x) - [y + 1 + a + x) {Da_+1y) {x) + (1 + a) (D°Ly) {x) = 0. (4.2.151)

Example 4.25 It is known (see, for example, Bonilla et al. [93]) that A 0{z) a nd

X\, l(z) can be expressed in terms of the Macdonald function (1.7.25):

A SM = 2"+17r-1/2z-"tf-_I,(*) ) \<$(z) = 2^1n-1/2z-K_{l/+1) (z), (4.2.152)

provided that z,v G C (SR(z) > 0; 9t(i/) > -1/2). Therefore, taking m = 0and m = 1 in Corollary 4.20, we derive that the following differential equation offractional order

x {Di+3y) (x) - (2i/ + a + 1) (D°L+2y) {x) - x {Da_+1y) (x) + a (Da_y) (x) = 0(4.2.153)

has the solution y(x) = x~^K l/(x), while y(x) = z~"K_(u+i}(x) is the solutionto the following differential equation of fractional order:

x (D°+3y) (x)-(2v + 2 + a) {Da_+2y) (x)-x {Da_+1y) (x) + (l + a) (Da_y) {x) = 0.(4.2.154)

Remark 4.23 The results in Theorem 4.18, Corollaries 4.19-4.20 and Examples4.24-4.25 give solutions to ordinary differential equations, obtained from (4.2.143),(4.2.144), (4.2.147), (4.2.149)-(4.2.151) and (4.2.153)-(4.2.154) for a = m G No

and p = I £ N. For example, y(x) = x~ve~x is a solution to the equation

xy"(x) + (v + l + x)y'(x) + y{x) = 0, (4.2.155)

being the equation of the form (4.2.151) with a = 0.

Remark 4.24 Lemma 4.2, Theorem 4.17 and Corollaries 4.17-4.18 were estab-lished by Kilbas et al. [373] for a G C (SR(a) > 0), j3 > 0 and v G C, while Lemma4.3, Theorem 4.18 and Corollaries 4.19-4.20 were proved by Bonilla et al. [93] for

- 1).

4.3 Operational Method

The usefulness of operational calculus in solving ordinary differential equations iswell known (see, for example, Ditkin and Prudnikov [195],[196],[197]). The basisof such an operational calculus for the operators of differentiation was developedby Mikusiriski [599]. It is based on the interpretation of the Laplace convolution(1.4.10)

U*9)(x)= ! f(x-t)g(t)dt (4.3.1)./o

as a multiplication of elements / and g in the ring of functions continuous on thehalf-axis E+. Mikusinski [599] applied his operational calculus to solve ordinary

Chapter 4 (§ 4.3) 261

differential equations with constant coefficients. We also mention that Mikusiriski'sscheme was used by Ditkin [193], Ditkin and Prudnikov [194], Meller [572] andRodriguez [712], to develop the operational calculus for Bessel-type differentialoperators with nonconstant coefficients. The transform approach to the develop-ment of operational calculus was considered by Dimovski [191],[192]. Rodriguezet al. [713] were probably the first to apply operational calculus for a Kratzeltransform defined in [451] to solve the partial differential equation of fractionalorder in the form (6.2.4).

A series of papers was devoted to developing operational calculus for the frac-tional calculus operators so as to solve differential equations of fractional order.In Sections 4.3.1-4.3.2 we shall present an operational approach based on the con-struction of the operational calculus for the Riemann-Liouville fractional derivativeDQ+U in a special space C_i of functions y(x) such as x~p(DQ+)ky(x) G C[0, oo)(k = 1, , m) for some p > — 1.

4.3.1 Liouville Fractional Integration and DifferentiationOperators in Special Function Spaces on the Half-Axis

For / j f i w e denote by CM the space of functions f(x) given on the half-axis M.+and represented in the form f(x) = xpfi(x) for some p > /J, where the functionfi(x) is continuous on [0, oo):

CM = {f(x) : f(x) = xph{x), fi{x) G C[0,oo), for some p > /x G K} . (4.3.2)

Let / Q +/ and D$+y t>e the Liouvill e fractional integral and fractional derivativeof order a > 0 defined on K+ by (2.2.1) and (2.2.3), respectively. Using (4.3.2),we can directly prove the following properties of these operators.

Propert y 4.1 If a > 0, then the operator IQ+ is a linear map of the space C-\into the space Ca-i> that is, IQ+ : C_i —> Ca-\. In particular, IQ+ is a linear mapofC-x into itself: J£+ : C_i -> C_i.

Propert y 4.2 If a > 0 and /3 > 0, then the semigroup property (2.2.25) holds:

(JoVoVX*) = (%+*/)(*) (fix) e C-i). (4.3.3)Propert y 4.3 For a > 0 and f(x) € C_i, then relation (2.2.26) holds:

{D%+I%+f){x) = /Or). (4.3.4)

For k £ N we denote by 7* and 7J>* the compositions of k operators I$+ and

(/*/ ) (x) = (IS+ /?+/) (x), {Dkay) (x) = {D%+ D%+y) (x). (4.3.5)

For a > 0 and m G N we introduce the space Q™(C_i) of functions f(x) G C_isuch that (Dk

af) (x) G C_i for k = 1, , m:

I O C - ! ) = {/(* ) e C_x : (I?*/) (x) G C_! (fc = l , - - - , m ) } . (4.3.6)

The following assertion follows from Property 4.3.


Property 4.4 The space Cl™(C-i) (a > 0; m e N) contains the functionsy(x) 6 C-i which are representable in the form

/)(^) (VW e C-i). (4.3.7)

The next assertion gives sufficient conditions for the operator DQ+ to be a rightinverse of the operator Iff+.

Lemma 4.4 Let a > 0, m G N, f(x) e C_i and y(x) = (I^f)(x). Then

(D%y)(x) = (DX?y)(x) (4.3.8)

and

(IZ+D5+y)(x)=y(x). (4.3.9)

Remark 4.25 It follows from (4.3.9) that the operator D%+ is a right inverse ofthe operator Ift+, but only in the subspace of fi^(C_i) which consists of functionsy{x) 6 Q^(C_i) that are representable in the form

y(x) = (/0V)(z) (/WeCO.

In the case of the whole space f2^(C_i), there holds the following statementfor the composition (Io+DQ+y)(x).

Lemma 4.5 Let a > 0 be such that m - 1 < a ^ m (m £ N).

i{C-X), then

= y(x) - £ ^t !^L*-« . (4.3.10)fe=i

Remark 4.26 If we denote by i£ an identity operator in the space n^(C_i), then(4.3.10) can be rewritten in the form

—' 1 y(X — k -\- 1)

The operator E — IQ+DQ+ is called a projector of the operator Ig+.

Remark 4.27 Lemma 4.5 yields the formula for the composition (/g+i?o+j/)(a;)in the space f^(C_i) of functions y(x) denned on the half-axis K+. It is analo-gous to the formula (2.1.39) in Lemmas 2.5(b) and 2.9(d) giving the composition(I%+D"+y)(x) of the Riemann-Liouville fractional integrals and derivatives (2.1.1)and (2.1.5) in special spaces of functions y(x) denned on a finite interval [a,b].

Chapter 4 (§ 4.3) 263

4.3.2 Operational Calculus for the Liouville FractionalCalculus Operators

To construct the operational calculus for the Liouvill e fractional integral andderivative operators Ia and Da, denned by (2.2.1) and (2.2.3), we construct ananalog of the Laplace convolution (4.3.1). For this we use the general definitionof the convolution of a linear operator A in a linear space X given by Dimovski[192]. Thus, a bilinear, commutative, and associative operation o : X x X Xis said to be a convolution of the linear operator A if

A(f o g) = {Af o g) for any f,g G X.

Lemma 4.6 For A 1 the operation ox given by

(/ ox g)(x) = (/0A+ 7 * g) (x) := /Vo+V)(* " t)g(t)dt (4.3.12)

Jo

is the convolution (without divisors of zero) of the linear operator IQ+ (a > 0) inthe space C-\.

The following statement plays an important role in the further development ofthe operational calculus.

Lemma 4.7 If a > 0 and 1 A < a + 1, then the Liouville fractional integraloperator IQ+ has the following convolutional representation:

xa-x(IZ+f)(x) = (h ox f)(x), h(x)=ria_x_iy (4.3.13)

Corollar y 4.21 Let a > 0, n G N, and 1 £ A < na + 1. Also let

Then

(12/) (a:) = (hox xhox f)(x) = (hn ox f) (x). (4.3.15)

I t is easily verified that the operations ox and + possess the property of dis-tributivity in the space C_i, that is, that

( / ox (g + h)) (x) = (/ ox g)) (x) + (f ox h) (x), for any f,g,h£ C_i. (4.3.16)

By the property (4.3.16) and Lemma 4.6, the space C_i with the operationsox and + becomes a commutative ring without divisors of zero. This ring can beextended to the quotient field V, along the lines of Mikusinski [599]:

V = C_i x (C_i\{0}) / ~, (4.3.17)


where the equivalence relation (~) is denned, as usual, such that (/,g) ~ (fi,gi)if, and only if, (/ o\ gi){x) = (g o\ fi)(x). Thus, we can consider the elements ofthe field V as convolution quotients f/g and define the operations in V as follows:

( 4 - 3 > 1 8)

9 9i {9 ox gi) \9J \9iJ (9 o\ gi)

It is clearly seen that the ring C_i can be embedded in the field V by the

map: f[x) -> {h °^)(x), where h{x) is given in (4.3.13). Moreover, the field C of

complex numbers can also be embedded in V by the map: a —> \^\

On the basis of these facts we define the algebraic inverse of Ia as an elementS of the field V, which is reciprocal to the element h(x) in the field V; that is,

s= s! = <*£/&=£ (4-3-19)where / = h/h denotes the identity element of the field V with respect to theoperation of multiplication.

The relation between the Liouvill e fractional derivative operator Da and theaforementioned algebraic inverse of Ia is contained in the following assertion.

Theorem 4.19 Let a > 0, m £ N and f{x) £ fi™(C_i). Then there holds thefollowing relation in the field V of convolution quotients:

m - l/ nm f\ (n\ nm f \ ^ om — k 771 r-jA; r (A Q on\

{l' aJ)(x) = b f - 2^/ bDafi (4.3.20)fc=0

where F = E — IaDa is the projector of the operator Ia given by (4.3.11). Thisresult means that the Liouville fractional derivative operator Da is reduced to theoperator of multiplication in the field V.

Proof. We represent the identity operator E in fi™(C_i) in the following fashion:

m - l

fc=0

Then, for f(x) e fi™(C_i), we have

m - l

i. (4.3.21)

fc=0

Multiplying both sides of (4.3.21) by Sm and applying the relations (4.3.15) and(4.3.19), we obtain (4.3.20). This completes the proof of Theorem 4.19.

For applications it is important to know the functions of 5 in V which can berepresented by means of the elements of the ring C-\. One useful class of suchfunctions is given in the following assertion.

Chapter 4 (§ 4.3) 265

Property 4.5 // the power series

kZk (z,ckeC)k=0

is convergent at a point ZQ 0, then the power series

k=0 k=0

where hk(x) (k £ No) are given in (4.3.14), defines an element of the ring C-\.

Remark 4.28 Applying Property 4.5, we may obtain various functions of 5 inthe field V, which can be represented by using the elements of the ring C_i. Inparticular, we have the following corollaries.

Corollar y 4.22 If a > 0, 1 ^ A < a + l andw £ C, then there holds the followingoperational relation:

^ uh + cu2h2 + . . . )= xa~xEa^x+1 (wxa), (4.3.22)o — ui hi — uih

where Eata-\jri(z) is the Mittag-Leffler function (1.8.17).

Furthermore, if m £ N, then

_ _ L _ = x™-*E?>ma_x+1 (UJX") , (4.3.23)

where E™ma_x+1(z) is the generalized Mittag-Leffler function (1.9.1).

Corollar y 4.23 If a > 0, l ^ A < a + l and w £ C, then there hold the followingoperational relations:

L ?lA (4.3.24)

andg

— T = xa~x cos\a (ujxa), (4.3.25)where

(_l)kzkz2k+l

and

cosXa(z) = JT ^ ^ = E2a 2a.x+1 (-z2). (4.3.27)k=0

Remark 4.29 Other operational relations of the types contained in Corollaries4.22 and 4.23 can be derived similarly from Property 4.5, especially for thosefunctions of S which can be represented as a finite sum of partial fractions.


4.3.3 Solutions to Cauchy Type Problems for FractionalDifferential Equations with Liouville FractionalDerivatives

In this section we apply the results of Section 4.3.2 to derive the explicit solutionto the Cauchy type problem for differential equations with the Liouville fractionalderivative DQ+.

Let Pm(z) be a polynomial of degree m £ N in z £ C with complex coefficients:

r = 0

We consider the following boundary-value problem involving the Liouville frac-tional derivative operator DQ+ of order a > 0 (n — 1 < a n; n £ N):

(Pm(D%+)y) (x) = f(x), (FDray) (x) = lr{x) (4.3.28)

(r = 0,1,--- , m - 1; 7r(z) = ker Da),

where F = E—IQ+DQJ_ is the projector of the operator IQ+, the function f{x) £ C-\is given, and the unknown function y(x) is to be determined in the space $7™(C_i).

Making use of (4.3.11), we rewrite the Cauchy type problem (4.3.28) in thefollowing more familiar form:

(Pm(D%+)y) (x) = f(x), lim ((Da-kDT

a)y) (x) = brk (4.3.29)

(r = 0,1, m - 1; k = 1, , n; drk G C)

Since y(x) £ 0™(C_i), by using the formula (4.3.20), we reduce the Cauchytype problem (4.3.28) to the following algebraic equation in the field V:

m - l

Pm(S)y = f + Y, p*( shv> (4-3-30)9=0

where

daE r(a Z= l ^

- l ) . (4.3.31)

Equation (4.3.30) has a unique solution in the field V given by

This leads us to the following result.

Chapter 4 (§ 4.3) 267

Theorem 4.20 / / a > 0 (n - 1 < a 5j n; n G N) and m G N, iAen ifte Cauchytype problem (4.3.29) has a unique solution y(x) in the space fi™(C_i) and thissolution is represented in the form

I mi px n

= E Ec ^ y o Eiua{Zj{X-tr)f{t)dt+Y.

n m —1 I "» j ,

+ E E E E Ai«* T(aJr + 1fUa+a~rElua+a-r+ i &**) , (4-3.33)

r= l 9=0 j= l u=l ^ '

where the constants Zj, Cju and AjUq are given by

1 1 ' mj

,1juq (g-I ... m-l)W - l , ,m I).

P(z) fri£ri(z-zj)u' P(*) ^ ^ i (*-*,) "(4.3.35)

Proof. Problem (4.3.29) is equivalent to (4.3.28). Above we have reduced (4.3.28)to the algebraic equation (4.3.30) in the field V, which has a unique solution(4.3.32). Using (4.3.22) and (4.3.23), we express the rational functions 1/P(z)and Pqj'P(z) (q = 0,1, , m — 1) as the sums of partial fractions, and then from(4.3.32) we derive, taking (4.3.12) (with 1 A < a + 1) into account, the solutionto (4.3.28) in the form

S^X^r- (f(f\n,tua-xKu . . (~ +a\\ <-r\ i Y^ d°r

! _ / / y

= 1 u=l

E E c i » ( / (* ) °A tK,ua-x+i (Zjt)) (x) + E T(a

n m - l / rrij

+ E E E E A T((y_

9rr,^

ua+a~rElua+a-r+ i iâ), (4-3.36)r= l 9=0 j= l u=l M "^

where 1 A < a + 1. According to (4.3.12), we have

A - A ^ , u a _ A + 1 (î") ) (X) = (<"a-A^,UQ_A+ 1 (^^)

o

From (2.2.1) and (1.9.1), by term-by-term fractional integration using (2.2.10), weget

\10+ I1 - ^ a . u a - A +l \z3l )\)\x)

— (TX~1 \fua-\pu (y .ia\]\ ( \ _ a-1 j?u f a\— ^0+ [ l B a , « o - A +1 1ZJE )\)\x)— x â,ua\zJx )


and hence

(Zjta)) (x) = f Elua (Zj(x - t)

a) f(t)dt.Jo

Substituting this relation into (4.3.36), we obtain (4.3.33).

Setting jr(x) = 0 (r = 0,1, , m - 1) in (4.3.28) and applying Lemma 4.4,we obtain the following corollary.

Corollar y 4.24 If a > 0 (n — 1 < a ^ n; n € N) and m £ N, then the uniquesolution to the Cauchy type problem

+ 0 W = 0 (4.3.37)

(r = 0,1, , m - 1; k = 1, , n)

m i/ie space Jl™(C_i) may 6e represented in the form

y(x) = (I™g)(x (g(x)€C1). (4.3.38)

Example 4.26 We consider the Cauchy type problem

(D%+y)(x) - coy(x) = f(x), Vm(D^ky)(x) = bk e C (k = 1, , n), (4.3.39)

where w eC and (D%+y)(x) is the Liouvill e fractional derivative (2.2.3) of ordera > 0 such that n - 1 < a ^ n (n G N). Using the relations (4.3.20) and (4.3.11),we reduce the problem (4.3.39) to the following algebraic equation:

n ,

Sy - wy = f + 57o, 7o = r ( a _) + rf"^' (4-3-40)

whose solution in the field V of convolution quotients has the form

V = -^—f + ^ — 7 o- (4.3.41)b - LU b - to

In order to evaluate the first term on the right-hand side of (4.3.41) by meansof (4.3.22) with A = 1, we observe that

xa-lEa>a (ujxa). (4.3.42)o —

On the other hand, using (4.3.40) and taking (4.3.22) with A = 1 and (1.9.1) intoaccount, we have, in accordance with (1.8.17),

^ xa-rEa r+1 (ujxa) (4.3.43)7o— LU

r=l

Chapter 4 (§ 4.3) 269

Substituting the expressions (4.3.42) and (4.3.43) into the right-hand side of(4.3.41) and taking into account the convolution (4.3.12), we obtain the solutionto the Cauchy type problem (4.3.39) in the form

y(x)= [ (x-tr-'Eîujix-t^fitdt+TbrXÊ^^+.iLux0). (4.3.44)Jo jfrj

By Theorem 4.21, this solution is unique in the space fi^(C_i), provided thatf(x) e C-i.

Remark 4.30 By Theorem 4.20, the relation (4.3.44) yields the explicit solutiony(x) of the Cauchy type problem (4.3.39) with f(x) G C_i in the space O^(C_i):

J £ ( C _ I) = {y(x) e C_i : (Dg+y)(x) 6 C_i} . (4.3.45)

By (4.3.38), (4.3.39) is the Cauchy type problem (4.1.1)-(4.1.2) with o = 0 andA = to, and its solution (4.3.43) coincides with (4.1.10) (when a = 0 and A = co).Thus, by Theorem (4.1), (4.3.44) gives the explicit solution to the problem (4.3.38)with f(x) G C7[0,6] (0 7 < 1) in the space C^_Q)7[0,6] of functions defined onany finite interval [0, i i ] c l and having, in accordance with (3.3.24), the form

C£_Q,7[0, b] = {y(x) e Cn-a[0, b] : (D%+y)(x) £ C7[0, b]}. (4.3.46)

Example 4.27 We consider the following Cauchy type problem:

(Dl!?y)(x) - uy'(x) + \2(D$y)(x) - oj\y(x) = f(x), (4.3.47)

^y)(x) = lim y(x) = Vm(I$y)(x) = 0. (4.3.48)

This problem in the field V of convolution quotients (with a = 1/2) is reduced tothe algebraic equation

S3y - UJS2 + X2Sy - uj\2y = f, (4.3.49)

so that

Now, using (4.3.22), (4.3.24) and (4.3.25) with a = 1/2 and u> = 1, we obtain

_ 1 / S UJ__ II\ \X) . I „ , , , Oo . \oto2 + A2 V S2 + A2 S2 + A2 5 - u

(4.3.51)Then, from (4.3.12), we obtain the solution to (4.3.47)-(4.3.47)

y(x) = K(x- t)f(t)dt, (4.3.52)Jo

where ^(a;) is given by (4.3.50). By Theorem 4.20, this solution is unique in thespace fi|/2(C_i), provided that f(x) G C_i.


Remark 4.31 Results presented in Sections 4.1-4.4 were established by Luchkoand Srivastava [508].

Remark 4.32 Theorem 4.20 yields conditions for the unique solution y(x) ofthe Cauchy type problem for the differential equation in (4.3.28), involving integerpowers of the Liouvill e fractional derivative operator Day = D +y. Such construc-tions are known as sequential (see Section 3.1) and therefore, such an equation canbe considered as an equation with sequential fractional derivatives.

Remark 4.33 Elizarrazaz and Verde-Star [237] obtained the explicit general solu-tion to the differential equation in (4.3.29) and the explicit solution to the Cauchytype problem (4.3.29) by using linear algebraic construction and classical meth-ods of operational calculus. Their approach was based on introducing divideddifferences of fractional order, coinciding with the Riemann-Liouville fractionalderivative operator in a certain space of functions, and some generalized exponen-tial polynomials, which are related to Mittag-Leffler type functions.

4.3.4 Other Results

In Section 4.3 we presented the operational method for solving, in closed form, dif-ferential equations of fractional order of the form (4.3.52) involving integer powersof the Liouvill e fractional derivative Day := Dft+ on the positive half-axis K+.Such an approach was also applied to solving Cauchy type and Cauchy problemsinvolving other differential equations of fractional other.

Luchko and Yakubovich [509], [510] and Al-Basssam and Luchko [24] devel-oped operational calculus for the Erdelyi-Kober type fractional derivative oper-ator D%+w defined in (2.6.29) (with a = 0) on the half-axis E+ and appliedthe results obtained to solve, in closed form and in a special function space, theCauchy type problems for fractional differential equations of the form (4.3.52) inwhich the Liouvill e fractional derivative Dg+y is replaced by the Erdelyi-Kobertype fractional derivative DQ+;I7 vy. The explicit solutions to the problems consid-ered were expressed via the generalized Mittag-Leffler function Ee((aj,[3j)m;z)denned in (1.9.14).

Hadid and Luchko [323] have applied the operational calculus of Section 4.2 tosolve the Cauchy type problem for the linear fractional differential equation

m

(Dg+y) (x) - Y, Ar {Dfoy) (x) + \oy{x) = f(x) (x > 0) (4.3.53)r= l

(D°+ky) (a+) = bk£R (k = 1, , n; n = -[-a]). (4.3.54)

involving the Liouville fractional derivatives Dfi+y, D^y (r = 1, , m)(0 < ai < < am < a) and constant coefficients Ar € M.; r = 0,1, , m.They suggested that f(x) G C_i and established a unique solution y(x) of theCauchy type (4.3.53), (4.3.54) in the space fi£(c-i) (see (4.3.6)), in terms of themultivariable Mittag-Leffler function E(ai... an)tt,(zi, , zn) defined by (1.9.27).

Chapter 4 (§ 4.3) 271

Luchko and Gorenflo [507] developed the operational calculus for the Caputoderivative (CDQII){X) defined by (2.4.1) with o = 0 in the space C-\ and gave anapplication to solve the following Cauchy problem:

(cD$+y)(x) -Xy(x) = f{x) (z ^ 0; n - 1< a ^ n; n G N; A G R), (4.3.55)

y( fe)(0)=6f c (bk e l ; fc = 0 , l , - - - n - l ) . (4.3.56)

They introduced the space C™ (m G No; a > 0) of functions in the following space:

Ca = iv(x) y(m)(x) e C-i, m G No; a > 0} (4.3.57)

and established a unique solution y(x) G C"x of (4.3.55)-(4.3.56) in the form

y{x) = J2 bkxkEa,k+1 {Xxa) + / ta-xEa,a {Xta) f{x - t)dt,

fc=0 Jo

(4.3.58)

provided that f(x) G C for a £ N and /(a;) G C_i for a G N.

Remark 4.34 When a £ N, (4.3.58) yields the explicit solution y(z) of theCauchy problem (4.3.55)-(4.3.56) with f(x) G C\ in the space C"LX defined by(4.3.57). (4.3.55)-(4.3.56) is the Cauchy problem (4.1.58)-(4.1.59) with a = 0, andits solution (4.3.58) coincides with the solution (4.1.62) (when a = 0). Thus, byTheorem (4.3), (4.3.58) gives the explicit solution to the problem (4.3.55)-(4.3.56)with f(x) G C7[0,6] (0 ^ 7 < 1) in the space C^"-1^,*) ] of functions defined onany finite interval [0, i i jc M and having, in accordance with (3.5.3), the form

C^-MO. t] = {y(x) G Cn_i[0,6] : (CD%+y)(x) G C7{0,b}}. (4.3.59)

Luchko and Gorenflo [507] also investigated the Cauchy problem for the follow-ing more general fractional differential equation of the form (4.3.53) than (4.3.55):

m

(cD?y)(x)-Y,K(CD^y)(x)=f(x) (xÔ) (4.3.60)r = l

with the initial conditions (4.3.56). Making the same assumptions for f(x) as inthe above Cauchy problem (4.3.55)-(4.3.56), they proved that a unique solutiony(x) of the Cauchy problem (4.3.60), (4.3.56) in the space C"x has the form

y{x) = yo(x) + Y] bkyk{x), yo(x) = / ta-1£7(.))O(a: - t)f{t)dt. (4.3.fc=o Jo

k

61)

, n - l ) . (4.3.62)r=l k

Here yo(x) is a solution to the problem (4.3.60), (4.3.56) with zero initial conditionsbk = 0 (k = 0,1, , n — 1), and the function

a\---, a-am) (4.3.63)


is a particular case of the multivariable Mittag-Leffler function (1.9.27), in whichthe natural numbers Ik {k = 0,1, , n — 1) are determined from the condition

mh ^ k + 1, mh+i ^ k. (4.3.64)

In the case when mr ^ k (r = 0,1, , n - 1) we set Ik = 0, and if mr ^ k + 1(r — 0,1, , n — 1), then Z = m.

The above results of Luchko and Gorenflo [507] were also presented in a surveypaper by Luchko [502] where historical remarks and an extensive bibliography maybe found.

4.4 Numerical Treatment

In this section we discuss the numerical treatment of fractional differential equa-tions. We note that many authors have discussed numerical methods for Abel-Volterra type integral equations of the first and second kind, called fractionalintegral equations, which contain the Riemann-Liouville fractional integral (2.1.1)[see the book by Gorenflo and Vessella [312] and the paper by Gorenflo [291]].However, such investigations into fractional differential equations began only re-cently. Here we discuss the numerical treatment of fractional differential equationsbased mainly on the available good approximations of the fractional derivatives,analogous to the approximations of fractional integrals for fractional integral equa-tions.

Shkhanukov [762] first applied the difference methods to study the Dirichletproblem for the differential equation

Ly = A ffc(z) J l - r(x)(Dg+y)(x) - q(x)y(x) = -f(x) (0 < x < 1; 0 < a < 1),

(4.4.1)y(0) = i/(l ) = 0; k{x) ^ c0 > 0, r(x) 0, q{x) 0 (4.4.2)

with the Riemann-Liouville fractional derivative (2.1.5). His approach was basedon the following approximation of (DQ+y)(x) by its discrete analog:

O(h), A%Xiy = * £ (x\z°+1 - x\l%) yik (4.4.3)r(2 - a)

fc=i

in a uniform grid {XJ = jh : j = 0,1, , N — 1} of the interval (0,1) withstep h = 1/N. Using (4.4.3), Shkhanukov obtained a one-parameter family ofdifference schemes for the problem (4.4.1)-(4.4.2) and proved the stability andconvergence of these difference schemes in the uniform metric. Using the approx-imation (4.4.3), Shkhanukov [762] also constructed the difference scheme for thefirst initial-boundary problem for the partial differential equation

(D%+.tu)(x,t) = ^ ^ + fix,t) (0 < x < 1; 0 < t < T; 0 < a < 1), (4.4.4)

Chapter 4 (§ 4.4) 273

u(0, t) = u(l, t) = 0 (0 ^ t ^ T); u(a;, 0) = 0,

(X?S+;tu)(z,i)|t=o=0 ( O ^ ^ l ) , (4.4.5)

with the partial Riemann-Liouville fractional derivative (2.9.9), and proved thestability and convergence of this difference scheme in the uniform metric.

Blank [91] applied the collocation method with polynomial splines in the nu-merical treatment of the ordinary fractional differential equation

k=0k\

(x) = - \ a y ( x ) + f(x) ( 0 < x < 1 ), ( 4 .4 .6)

with the Caputo derivative (2.4.1) of order a > 0 ( i V < a ^ J V + l; N £ NO),where A > 0, y^k(0) (k = 0,1, , N) are given initial data and f(x) is a givenfunction on [0,1]. Blank's method is based on replacing (4.4.6) by

N

s = 0

(4.4.7)

in the collocation points xnj = nh + Cjh (j = 1, , k = m — (r + 1); n G N) andon evaluating the Riemann-Liouville fractional derivative {D%+y)(xn^) at thesecollocation points. Here ci,--- , Ck (0 < c\ < < Ck ^ 1) are collocationparameters, that are chosen for the r times continuously differentiate numericalsolution y(x) and for the spline y defined on any interval \XJ,XJ+\\ (j £ No) bythe piecewise polynomial

m — 1

y(Xj+vh)= (4.4.8);=o

with constants a\ (I = 0,1, ,m — 1; j 6 No). In this way the systems ofequations characterizing the numerical solution to (4.4.6) were determined. Sucha system was simplified for the case of an TV-times differentiable spline (i.e. r = N),and numerical results were given for N = 1 and N = 2.

Diethelm [170] suggested another approach for the numerical treatment of thefractional differential equation (4.4.6) with 0 < a < 1:

f(x) (4.4.9)

For a = 1/2 such an equation describes the behavior of a damping model inmechanics [see Gaul et al. [275]]. The algorithm suggested by Diethelm is basedon the interpretation of the Riemann-Liouville fractional derivative (2.1.8) as aHadamard finite part integral

P.f. (4.4.10)

in the sense of a finite part of Hadamard [see Samko et al. ([729], Section 5.1)], andon the algorithm for the quadrature of finite-part integrals developed by Diethelm


in [171]. Using (4.4.10) the relation (4.4.9) is discretized at the equispaced gridXj — j/N (j = 0,1, , N — 1) with a given N, and then the obtained integral

i p [y(t)-y(Q)]dt = _£L , /* ^ - ^ ) - , ( 0 )J- \—ot) Jo (Xj - t j T i (—a) Jo T ^

is replaced by a first degree compound quadrature formula with the equispacednodes 0,1/j, , 1, which yields the resulting equation for the approximate valuesyj of y(xj) (j = 1, , n) in the form

oi0 - ( £ y3 = Uty T(-a)f(Xj) - . (4.4.12)

The error of this approximation method was found for sufficiently smooth involvedfunctions, and two numerical examples for /3 = — 1 and f3 = — 2, with the choiceof f(x) such that the exact solutions to (4.4.9) have the form y(x) = x2 andy(x) = cos(7r;r), respectively, confirmed the obtained error bounds for a = 0.5,a = 0.75 and a = 0.25.

The relation (4.4.12) was used by Diethelm and Walz [189] to obtain the asymp-totic expansion for the sequence of approximate values {yn} in the form

M1 M2

yn = y(xn) + Y, ain'~a + J2 hin~2j + ° (*~ A M ) (n -> °°)> (4-4-13)1=2 j=l

where the natural numbers M\ and M2 are defined by the smoothness of f(x)(and therefore of y{x)), a* (fc = 2, ,Mi ) and bj (j = I,--- ,M2) are certainconstants depending on k — a and 2j and M = min[a — M\,2M2\. They appliedthe asymptotic estimate in (4.4.13) to state an extrapolation algorithm for thenumerical solution to the problem (4.4.9) and illustrated the results by numericalexamples.

Diethelm and Freed [186] interpreted the nonlinear differential equation

- y(0)]) (x) = f[x,y(x)} (0 < x < 1; 0 < a < 1) (4.4.14)

as the Volterra integral equation of the second kind

and implemented a product integration method for the latter by using a producttrapezoidal quadrature formula with nodes tj (j = 0,1, , n + 1) taken withrespect to the weight function (tn+i — t)a~l.

For the numerical solution to fractional differential equations, Podlubny ([681]and [682], Chapter 8) has used the approach based on the approximation of theRiemann-Liouville fractional derivative (DQ+y)(x) of order a > 0 by the relation

h-a(A%y)(x), (4.4.16)

Chapter 4 (§ 4.4) 275

where

— 1 ) "^ - fix — 7 h) (4.4 17)

is the "truncated" difference of fractional order a > 0 of 2/(2:) with a step /i > 0and with a center at the point x [see Section 2.10 and Samko et al. ([729], Sections5.6 and 20.1)]. Such a construction in (4.4.16) for x = nh approximates (D$+y)(x)with an accuracy O(h). It should be noted that Zheludev [926] first applied sucha method to numerically solve the Abel integral equation (Ig+y)(x) = f(x), andalso a more general convolution integral equation.

Assuming that the step h and the number n £ N of nodes are related by x = nh,Podlubny ([681] and [682], Chapter 8) presented a numerical solution with orderof approximation O(h) for the following Cauchy type-problems:

(Dg+y)(x) + Ay(x) = f(x) (x > 0), i/(*>(0) = 0 (k = 0,1, , n - 1), (4.4.18)

with the Riemann-Liouville fractional derivative (1.1) of order a (n — 1 < a ^n; n € N); the problem

ay"(x) + b(D30^y)(x) + cy(x) = f(x) (x > 0; a > 0), y(0) = j/'(0) = 0, (4.4.19)

with b, c £ M, first considered by Torvik and Bagley [820] while studying thebehavior of certain viscoelastic materials; also for the mathematical model whichdescribes the dynamics of certain gases dissolved in a fluid,

^ (f(x)[y(x) + 1]) + \{Dl^y){x) = 0 (0 < x < 1), t/(0) = 0, (4.4.20)

considered by Babenko [44]; and finally for the nonlinear problem of the form

(Dl0^y)(x) - a[u0 - y(x)}4 = 0 {x > 0; a G E; u0 G E), 2/(0) = 0, (4.4.21)

which models the dynamics of the cooling process of a semi-infinite body by radi-ation. It should be noted that Podlubny claims that the order of approximationof his algorithm is O(h), but he does not give a proof.

Numerical treatment of the equation (4.4.19) was studied by Diethelm andFord [178] on the basis of the reformulation of this equation as a system of frac-tional differential equations of order 1/2. This allowed them to propose numericalmethods for its solution which are consistent and stable and have an arbitrary highorder. In this way they especially looked at fractional linear multistep methodsand Adams type predictor-corrector methods. An Adams type predictor-correctormethod for the numerical solution of fractional differential equations, used for bothlinear and nonlinear problems, was discussed by Diethelm et al. [182].

Edwards et al. [218] showed how the numerical approximation of the solutionto a linear multi-term fractional differential equation

m

ck (D^y) {x) + coy(x) = f(x) {x > 0; am < am > > ai > 0) (4.4.22)


can be obtained by reducing the problem to a system of ordinary and fractional dif-ferential equations each of order unity, at most. They solved the problem (4.4.19)as an example, and showed how the method can be applied to a general linear equa-tion (4.4.22). Note that numerical solutions to multi-order fractional differentialequations were also discussed by Diethelm and Ford [179].

Ford and Simpson [261] used the fixed memory principle described by Podlubny([682],Chapter 8) to numerically solve the Cauchy problem of the form (3.5.1)-(3.5.2):

(cD%+y) (x) = f[x,y(x)} (x > 0); j/(*>(0) = bk (k = 0,1, , n - 1) (4.4.23)

with the Caputo derivative CDQ+D of order n — 1 < a < n (n 6 N) given by(2.4.15). Diethelm et al. [183] suggested an algorithm for the numerical solutionto (4.4.23), which is a generalization of the classical one-step Adams-Bashforth-Moulton scheme for first-order equations.

We also mention Boyjadziev et al. [106] who have applied the tau-method ofapproximation to obtain the numerical solution of the fractional integro-differentialequation with the Liouville fractional derivative (2.2.4) of order a > 0

(Diy) (x) = [ y{x- t)teivtdt {a G C; v e M), (4.4.24)Jo

with the initial conditions (3.1.2), and Boyadziev and Dobner [102] who computednumerical values of the solution y(x) of this problem.

In concluding this section, we indicate that from our point of view the approachsuggested by Lubich ([499]-[501]) for the numerical treatment of the Abel-Volterraintegral equations and more general weakly singular Volterra integral equations,can be also used to obtain numerical solutions for both ordinary and partial frac-tional differential equations. This approach is based on the numerical approxima-tion of the Riemann-Liouville fractional integral (Io+f)(x) (0 < a < 1), definedby (2.1.1), for x = nh (n £ N; 0 ^ nh ^ T < oo; h > 0) by a discrete convolutionquadrature:

(4.4.25)1=0

Here the weights w" are the coefficients of (l in the expansions of coa(Q, wherew(C) = <7(l/C)/p(l/C) with polynomials a(z) and p(z). In particular, whena(z) = 1 and p(z) = 1 - z, u(z) = 1/(1 - z) and u>(()a = (1 - z)~a and (4.4.25)takes the form (4.4.16) with a replaced by —a. Such an approach was applied bySanz-Serna [732] to numerically solve the homogeneous partial integro-differentialequation

^11 = [\t- 8)-V2uxx(x, t)dt (0g ig l ;O0) (4.4.26)at Jo

with the boundary conditions

u(0,t) = u(l,t) = 0 (t ^ 0), u{x,0) = f{x) (0 ^ x ^ 1). (4.4.27)

Chapter 4 (§ 4.4) 277

We note that the above approach by Lubich as well as other known numer-ical methods for solving Abel-Volterra integral equations will be useful for thenumerical treatment of initial value problems for fractional differential equationswhich are equivalent to these integral equations. The results presented in Chap-ter 3, which established an equivalence for Cauchy type problems for differentialequations with Riemann-Liouiville, Caputo, and Hadamard fractional derivatives,and the corresponding Volterra integral equation, permit us to apply the abovementioned numerical methods. For example, we could use successful the corre-sponding routines of the known numerical mathematical libraries called NAG andIMSL. In this regard we refer the reader to Kilbas and Marzan [379] and Marzan[559], where approximate solutions were obtained to the Cauchy type problem(3.1.1)-(3.1.2) and to the Cauchy problem (4.4.31), respectively. In the bibliog-raphy of this book, the reader can find more references on numerical methods tosolve fractional differential equation.


Chapter 5

INTEGRA L TRANSFORMMETHO D FOR EXPLICI TSOLUTION S TOFRACTIONA LDIFFERENTIA LEQUATION S

The present chapter is devoted to the application of Laplace, Mellin, and Fourierintegral transforms to construct explicit solutions to linear differential equationsinvolving Liouville, Caputo, and Riesz fractional derivatives.

5.1 Introduction and a Brief Survey of Results

First we present a scheme for solving the one-dimensional fractional nonhomoge-neous differential equation with constant coefficients of the form

Y,Ak(D^y)(x) + Aoy(x) = f(x) (x > 0) (5.1.1)fc=i

with m £ N; 0 < £H(ai) < < 9i(a:TO); ô, -Ai, , Am £ R, and involving theLiouvill e fractional derivatives D^y (k = 1, , m), given by (2.2.3). By (2.2.38),for suitable functions y, the Laplace transform (1.4.1) of -Do+2/ is g iv en by

(£D%+y)(s) = sa(Cy)(s). (5.1.2)

279


Applying the Laplace transform to (5.1.1) and taking (5.1.2) into account, we have

(Cy)(s) = (Cf)(s). (5.1.3)

Using the inverse Laplace transform C x, given by (1.4.2), from here we obtain aparticular solution to the equation (5.1.1) in the form

We note that the Laplace transform method was applied earlier by Hill e and

Tamarkin [349] to solve the integral equation

with a > 0 and A G E, in terms of theMittag-Leffler function (1.8.1):

<P(X) = J Ea[\(x-t)a]f(t)dt. (5.1.6)

Maravall [549], [550],[553], suggested a formal approach based on the Laplacetransform for obtaining the explicit solution to a particular case of the fractionaldifferential equation (5.1.1). Probably he was first to present applied fractionalmodels, connecting with anomalous oscillatory phenomena (see, also, [551],[552],[554], and [555]).

The Laplace transform was used by Oldham and Spanier ([643], Section 8.5)to solve the homogeneous case of the following equation (f(x) = 0):

y'{x) + b{D%+y){x) + cy{x) = f(x) (0 < a < 1; b, c £ ffi), (5.1.7)

with a = 1/2, 6 =1 and c = —2, by Dorta [200] to obtain solutions to some simplefractional differential equations (4.1.11) with rational a > 0, and by Seitkazieva[756] to construct the solution to the equation (5.1.7).

Mille r and Ross ([603], Sections V.5-V.9) applied the Laplace transform methodto give explicit representations of solutions for the particular case of the equation(5.1.1) with derivatives of orders ak = ka (a = ^;n £ N) for the homogeneousequation

m /I \

J 2 M D o + y ) ( x ) + A0 y ( x ) = 0 f - e N j , (5.1.8)fc=i ^ a '

and for the nonhomogeneous equation

m /1 \

YJAk{Dk£y){x)+ Aoy(x) = f(x) - £ N . (5.1.9)

Chapter 5 (§ 5.1) 281

It should be noted that Miller and Ross ([603], Section V.6) practically foundlinear independent solutions to the homogeneous equation (5.1.8). This fact andthe results of Section 4.2.3 mean that the homogeneous linear differential equationof the form (5.1.1)

Aoy(x)=0 (0<&t (a i )< . - -<3 t (am) ) , (5.1.10)k=i

can have nontrivial linearly independent solutions by analogy with the ordinarydifferential equation of order m obtained from (5.1.10) for au = k (k = 1, , m).We note that Leskovskij [477] constructed linearly independent solutions to theequation (5.1.10) with specific distinct real exponents 0 < a& < 1 (k = 1, , m)in terms of the Mittag-Leffler function (1.8.17).

Miller [see Miller and Ross ([603], Sections V.7-V.9)] introduced a fractionalanalog of the Green function Ga(x) defined, via the inverse Laplace transform(1.4.2), by

Ga(x) = (c-1Lt-V-)}/ fc=1

represented a particular solution to the nonhomogeneous equation (5.1.9) in theform of the convolution of Ga{x) and f(x):

y(x)= [ Ga(x-t)f(t)dt, (5.1.12)Jo

and proved that this formula yields a unique solution y(x) to the Cauchy problemfor the equation (5.1.9) with the following initial conditions:

j/(0) = y'(0) = = y(m-1] (0) = 0. (5.1.13)

Miller and Ross ([603], Section VI.3) indicated that the Laplace transform canbe applied to solve homogeneous differential equations of the form (5.1.8):

(5.1.14)

with polynomial coefficients Ak(x), and illustrated such an approach by solvingthe following equation:

{Dy*y){x) = V- {x>Q). (5.1.15)

Miller [600] obtained linearly independent solutions to the equation (5.1.14) interms of an analog of the Green function (5.1.11).

It should be noted that the relation (5.1.15) was the first known differentialequation of fractional order discussed by O'Shaughnessay [650] and Post [686] [seein this regard Samko et al. ([729], Section 42.1) and Kilbas and Trujllo ([407],Section 3)].


Miller and Ross [603] also solved the so-called sequential fractional differentialequations of the form (5.1.9):

{Pm(DS+)y) (x) = f > * i(Do+)ky) (z) + cOy(x) = f(x) ^ N ) , (5.1.16)

by using the roots A = A.,- of the characteristic polynomial Pm(X) = X fcLo Ak^k-In the case k = 2 the solution to such an equation was given by Miller [600].

The above investigations of Miller and Ross [603] were developed by Podlubny[679], [681] [see also ([682], Chapter 5)], who defined a fractional analog of theGreen function Ga(x,t) for the equation (3.1.17) and showed that a particularsolution y(x) of the Cauchy type problem (3.1.22) with bk = 0 (k = 0, , n - 1)

(Dcry)(x)=f(x), (D^y)(x)\x=o = 0 (fc = 0, , n - 1) (5.1.17)

can be expressed in terms of Ga(x,t) as follows:

y(x) = f Ga(x,t)f{t)dt. (5.1.18)Jo

When a,k(x) = ak G C are constants, Ga(x,t) = Ga(x — t), and the explicitsolution to the Cauchy type problem (3.1.22) has the form

y(x) = £ hVk(x) + fX Ga(x - t)f(t)dt, yk{x) = (D^D^T1 D«k+Ga){x).

(5.1.19)In particular, for the equation (5.1.1), Podlubny ([682], Section 5.6) constructedthe explicit formula for Ga(x) as a multiple series containing the Mittag-LefRerfunctions (1.8.17).

Examples of linear fractional differential equations of the forms (5.1.8), (5.1.9),(5.1.10), (5.1.14) and (5.1.16), solved by using the Laplace transform method andthe analogs of the Green function, were given by Miller and Ross ([603], ChaptersV and VI) and Podlubny ([682], Sections 4.1.1 and 4.2.1).

Gorenflo and Mainardi [304] applied the Laplace transform to solve the Cauchyproblem for the fractional differential equation with the Caputo derivative (2.4.52)

(cD%+y)(x) + y{x) = f(x) {x > 0; m - 1< a ^ m; m G N), (5.1.20)

with the initial conditions (5.1.13), and the Cauchy problems

y'(x) + a(cD$+y)(x) + y{x) = /(re), y(0) = c0 G R (a; > 0; 0 < a < 1; a > 0)(5.1.21)

y"{x) + a(cD$+y)(x) + y(x) = f(x), y(0) = c0 G R, y'(0) = cx G R (5.1.22)

(a; > 0; 1 < a < 2).

See also Gorenflo and Rutman [311], Gorenflo and Mainardi [303], Gorenflo et al.[310] and Debnath [162].

Chapter 5 (§ 5.2) 283

Podlubny ([682], Section 6.1) indicated that the one-dimensional Mellin trans-form (1.4.23) can be applied to solve the Cauchy problem for the fractional differ-ential equation

xa+l (£,a+ly) + xa ( £«y ) = f > Q) (5.1.23)

with the Liouville Da+ky = D%+ky or the Caputo Da+ky = cD%+ky (k = 0,1)fractional derivatives (2.2.6) and (2.4.52) of order 0 < a < 1. Stankovic [800]

studied the Cauchy problem

y"(x) + a{D%+y)(x) + g{x)y(x) = f(x) (x > 0; 0 < a < 1) (5.1.24)

y(0) = 6o, y'{0) = bi (61,63 GR), (5.1.25)where a > 0, f(x) £ Lioc(M.+ ) and g(x) is a real-analytic function on an open setcontaining E+ . He reduced the above problem to a differential equation of thesecond order in a special space of hyperfunctions with support on E+ and usedthe Laplace transform to prove the existence of a unique solution for the Cauchyproblem (5.1.24)-(5.1.25) in Lioc(R+). Stankovic showed that, in the case j/(0) = 0,this solution y(x) belongs to C2(E+) and is a classical solution, while for y(0) ^ 0,y(x) e C1(K+) and is a solution in the sense of distributions. He also proved thatif f(x) is a continuous function such that |/(a:)| ^ KeXx for x ^ 0 with constantsK and A, then the solution y(x) will have the same estimate.

5.2 Laplace Transform Method for Solving Or-dinary Differential Equations with Liouvill eFractional Derivatives

Our discussion involving the Laplace transform method for the Liouville fractionalderivatives will be presented under several subsections as follows.

5.2.1 Homogeneous Equations with Constant Coefficients

In this section we apply the Laplace transform method to derive explicit solutionsto homogeneous equations of the form (5.1.10)

m

Ak (I>o+y) (x) + Aoy{x) = 0 (a: > 0; m e N; 0 < ax < < am), (5.2.1)fc=i

with the Liouville fractional derivatives Dg+y (k = 1, , m), given by (2.2.3).Here Ak € M. (k = 0, , m) are real constants, and, generally speaking, we cantake Am = 1. We give the conditions when the solutions y\{x), , yi(x) of theequation (5.2.1) with I — 1 < a := am ^ I (I £ N) will be linearly independent,and when these linearly independent solutions form the fundamental system ofsolutions, which (by analogy with the ordinary case) is denned by

k Vi) (0+) = 0 (k,j = I; k?j),


(D%+kyk) (0+) - 1 (* = 1, , 0- (5.2.2)

The Laplace transform method is based on the relation (2.2.37) which, in ac-cordance with (2.2.3), is equivalent to the following one:

/(CD%+y)(s)=sa(£y)(s)-Y,djs

j-1 (l-Ka^l; I G N), (5.2.3)

0'= , 0- (5-2.4)

First we derive explicit solutions to the equation (5.2.1) with m = 1 in theform (4.1.11)

{D$+y) (x) - Xy(x) = 0 (x > 0; I - 1< a I; I G N; A G R), (5.2.5)

in terms of the Mittag-Lefler functions (1.8.17). There holds the following state-ment.

Theorem 5.1 Let I - 1 < a ^ I (I G N) and X E R. Then the functions

yj{x) = xa-jEa,a+1_j (Xxa) (j = 1, , Z) (5.2.6)

yield the fundamental system of solutions to equation (5.2.5).

Proof. Applying the Laplace transform (1.4.1) to (5.2.5) and taking (5.2.3) intoaccount, we have

J 2 ^ (5.2.7)^

where dj (j = 1, ,1) are given by (5.2.4). Formula (1.10.9) with (3 — a + 1 - jyields

C [f-iE^+r-j (Xta)] (s) = ^ (|*-*A | < l ) . (5.2.8)

Thus, from (5.2.7), we derive the following solution to the equation (5.2.5):

y(x) = djyj(x), yj(x) = xa-iEa,a+1-j (Xxa). (5.2.9)

It is easily verified that the functions yj(x) are solutions to the equation (5.2.5)

and, moreover,

{Da-kyj) {x) = g A"B=0 ^ ' J)

Chapter 5 (§ 5 . 2 ) 2 8 5

I t follows from (5.2.11) that

{ D « - k y 3 ) ( 0 + ) = 0 ( k , j = l,---, I; k > j ) , { D ^ k y k ) ( 0 + ) = 1 (k = 1, , I).(5.2.12)

If k < j , then

<Da~kv) (r) - V^ - ran+k-j

(5.2.13)

and, since a + k—j^.a + l-l>Oiov any fc, j = 1, ,1, the following relationshold:

(D%-kyj) (0+) = 0 (A, j = 1, , I; k < j). (5.2.14)

By (5.2.12) and (5.2.14), Wa(0) = 1 for the analog of the Wronskian Wa(x)defined by (4.2.34). Then the result of the Theorem 5.1 follows from Lemma 4.1and (5.2.2).

Corollar y 5.1 The equation

[D%+y) (x) - \y(x) =0 (x > 0; 0 < a ^ 1; A G R) (5.2.15)

has its solution given by

y(x)=xa-1Ea,a(\xa), (5.2.16)

while the equation

{D%+y) {x) - \y{x) = 0 {x > 0; 1< a 2; A £ R) (5.2.17)

has the fundamental system of solutions given by

Vl(x) = xa~lEa,a (Xxa), y2(x) = xa~2Ea,a-X (Xxa). (5.2.18)

Example 5.1 The equation

(l>o+1/2j/ ) (a:) - Xy(x) =0 {x > 0; i G N . A e M ) , (5.2.19)


yj{x) = ^ ^ - 1 / 2JB ; _ 1 / 2 , ; _ J + 1 /2 (Xx1-1'2) (j = I,--- , /) (5-2.20)

Example 5.2 The following ordinary differential equation of order / € N

y { l ) { x ) - X y { x ) = 0 {x > 0; l e N ) (5.2.21)


y j { x ) = x l - * E l , , + 1 - j ( \ x t ) (j = I). (5.2.22)


In particular, the equations

y" - Xy{x) = 0 (x > 0; A > 0) (5.2.23)

andy" + Xy{x) =0 {x>0; A > 0) (5.2.24)

have their respective fundamental systems of solutions given by

1 f /- \ ( r~ \2/i (x) = —— sinh I v Aa;) , t/2 (%) = cosh I v Xx) (5.2.25)

VA ^ ' ^ 'and

2/i (a;) = -y= sin (%/Az V 2/2(2) = cos ( V L A . (5.2.26)

Next we derive the explicit solutions to the equation (5.2.1) with m = 2 of theform

{D%+y) (x)-X (-D^+2/) (x)-ny{x) = 0 (a; > 0; l-Ka^l; I G N; a > f3 > 0),

(5.2.27)with A , / J 6 K, in terms of the generalized Wright function (1.11.14) with p = q = 1of the form

(n + 1,1)

(an + /?, a)

The following assertion follows from Theorem 1.5

Lemma 5.1 The generalized Wright function (5.2.28) is an entire function ofz G C for a > 0.

Theorem 5.2 Let I - 1 < a ^ / (I G N), 0 < /? < a and A,/U G E. T/ien ifte/tmcfricms

M j.nn+tt-j

n =0

(n + 1,1)

(an + a + 1 — j , a — ,, 0

(5.2.29)are solutions to the equation (5.2.27), provided that the series in (5.2.29) is con-vergent.

In particular, the equation

(D%+y) {x) - A (D$+y) (x) = 0 (x>0; l - K a ^ l ; I G N; a > /3 > 0),

(5.2.30)has its solutions given by

yj{x) = xa-iEa-fi,a+l-j (j = 1, , / ). (5.2.31)

If a — I + 1 ^ /?, then yj(x) in (5.2.29) and (5.2.31) are linearly indepen-dent solutions to equations (5.2.27) and (5.2.30), respectively. In particular, fora — I + 1 > /3 they yield the fundamental systems of solutions.

Chapter 5 (§ 5.2) 287

Proof. Let m — 1 < fi m (m G N; m /). Applying the Laplace transform to(5.2.27) and using (5.2.3) as in (5.2.7), we obtain

'= !>...£_„ (5.2.32)

where, for all j = m + 1, , /

dj = (D%-jy) (0+) + (Dl-iy) (0+) (j = 1, , m), d,- = (z>oV'y) (0+).

For s G C and

1

< 1, we have

sa - - u n =0

(5.2.33)

and hence (5.2.32) has the following representation:

(5.2.34)j=l n=0

Using (1.10.10), with a replaced by a - f3 and /3 by a + fin + 1 — j , and taking(5.2.28) into account, for s g C and | As^3 " | < 1, we have

sJ-l-0-l3n s(a-0)-{a+Pn+l-j)

sa-p _ A) n+1

= -Ac tan+a-j ^ XtC (s). (5.2.35)(n + 1,1)

(an + a + 1 — j , a — fi)

From (5.2.34) and (5.2.35) we derive the solution to the equation (5.2.27)

(5.2.36)

where Vj(x) (j = 1, , /) are given by (5.2.29). It is readily verified that thesefunctions are solutions to the equation (5.2.27), which proves the first assertion ofTheorem 5.2.

For j , k = 1, , the direct evaluation yields

(D^yj) (x) = J2^n =0 n!

(n + 1,1)

[an + k + 1 - j , a - fi)(5.2.37)


It follows from (5.2.37) that the relations in (5.2.12) hold for k ^ j . If k < j , then,in accordance with (5.2.28), we rewrite (5.2.37) as follows:

(x) =

n=\ n\

(n + 1,1)

(an + k + 1 — j , a — 0)= h(x)

(5.2.38)If a - I + 1 0, then (a - 0)q + a - 0 + k - j ^ a - 0 + 1 - I ^ 0 for anyj,f e = I , - - - , I and q G No. Thus l im xô+-f i (^ ) = 0 (k,j = I , - - - , /; k < j)except for the case a — l + 1 = 0 with A; = 1 and j = I, for which limx_>o+ -TL (^) = A.Moreover, since an + fc — j ^ a + 1 — / > 0 for any j , A; = 1, , / and n G N,then lima:_>.o+ -^(z) = 0 (fc, j = 1, , 1; k < j). Thus (5.2.38) yields the relation(5.2.14) for any solution yj(x) in (5.2.29), except for the case a — I + 1 = 0 wi thk = 1 and j = I, for which

{D«-lVl) (0+) = A. (5.2.39)

According to (5.2.12), (5.2.14) and (5.2.39) for the analog of the Wronskian Wa{x),given by (4.2.34), we have 1^(0) = 1. Thus it follows from Lemma 4.1 thatthe functions yj(x) in (5.2.29) are linearly independent solutions to the equation(5.2.27). If a — I + 1 > 0, then the relations (5.2.2) are valid, and hence yj(x) in(5.2.29) yield the fundamental system of solutions to the equation (5.2.27). Thiscompletes the proof of the Theorem 5.2 for the equation (5.2.27).

Setting JJL = 0 and taking into account the formula

1 * 1

(1,1)= E0tb(z) (b e C; 0 G (5.2.40)

we obtain the assertion of the Theorem 5.2 for the equation (5.2.30).


(Dg+y) (x) - A


(x) - ny{x) =0 (x > 0; 0 < 0 < a g 1; A, n G R)(5.2.41)

2/1 (n = 0 n!

(n + 1,1)

(an + a, a — ,

In particular,

(5.2.42)

(5.2.43)

is the solution to the following equation:

(D%+y) (x) - A (p%+y) {x)=0 (x > 0; 0 < 0 < a g 1; A G E). (5.2.44)

Chapter 5 (§ 5.2) 289


(Dg+y) (x)-\(l>$+y) (x) - = 0 (x > 0; 1< a £ 2, 0 < /3 < a; A,

n = 0 n!

has two solutions yi{x), given by (5.2.42), and

(71+1,1)

(an + a — l,a — /3)


(D%+y) (x) - A (Do+y) {x)=0 (x > 0; 1 < a ^ 2, 0 < j3 < a; A G

ias two solutions yi(x), given by (5.2.43), and

y2\d>) — J-Ja—p,a — l \AJ/ )

(5.2.45)

(5.2.46)

(5.2.47)

(5.2.48)

If a ^ /3 + 1, then the above functions y\{x) andy2{x) are linearly independentsolutions of the equations (5.2.45) and (5.2.47), respectively. In particular, fora > j3 + 1 these functions provide the fundamental system of solutions.


y'(x) - A (D$+y) (x) - fiy(x) = 0 (x > 0; 0 < (3 < 1; A,/i G R) (5.2.49)


n = 0 n!

In particular,

n = 0

is the solution to the equation

(n + 1,1)

(n + 1 , 1 -/

(n+1,1)

(n + 1,1/2)

= 0 (x>0;

Xx1 - / (5.2.50)

(5.2.51)

(5.2.52)

Remark 5.1 Oldham and Spanier ([643], Section 8.5) expressed the solution(5.2.51) of the equation (5.2.52), with A = —1 and \i = 2, in terms of the comple-mentary error function erfc(,z) defined via the incomplete gamma function (1.5.27)by (Abramowitz and Stegun [1], formula 6.5.16)

erfc(s) := -7 (Lz2) = - f^ V2 / T Jo

e^'dt. (5.2.53)



y"(x) - X (£>£+j/) (a;) - ny{x) = 0 (a; > 0; 0 < /? < 2; A, /z e E) (5.2.54)

has its two solutions given by

(n + 1,1)

(2n + 2 , 2 -/

(n + 1,1)

n=0

(5.2.55)

XX2~P . (5.2.56)(2n +1,2- /3)

These solutions are linearly independent when /3 1 and form the fundamentalsystem of solutions when /? < 1.


2/2W = 2 ^ TTa

n=0

Ax3/2

-/ij/(a;)=O (x > 0; A,/i£K) (5.2.57)


oo „ (n + 1,1)

n=o n ! L (2n + 2,3/2)

oo ^n ' (n + 1,1)

«=o n ! (2n +1 ,3 /2)

Example 5.5 The following ordinary differential equation of order I g N

y{l) (x) - Ay(m) (a;) - ny{x) = 0 (a; > 0; I , m e N ; m < l; A, \i £ E) (5.2.60)

has / solutions given by

(n + 1,1)

(5.2.58)

(5.2.59)

n = 0 n!Aa;'

(5.2.61)When m = 1, these solutions are linearly independent.

In particular, the following ordinary second order differential equation

y"(x) - Xy'(x) - fiy(x) = 0 (x > 0; X,fi E E) (5.2.62)

has two linearly independent solutions given by

(n+1,1)~ 2 ^ -,? x x

n=0 (2n + 2,Aa; (5.2.63)

Chapter 5 (§ 5.2) 291

n = 0n\

(n + 1,1)(5.2.64)

Finally, we find the explicit solutions to the equation (5.2.1) with anym G N\{1 , 2} in terms of the generalized Wright function (5.2.28). It is convenientto rewrite (5.2.1) in the form

m - 2

(D%+y)(x) - \(D$+y)(x) - £ Ak(D^y)(x) =0 (x > 0) (5.2.65)k=0

( m G N \ { l , 2 } ; 0 = a0 < oi < < am_2 < /3 < a; A, Ao, ^2 G

T h e o r e m 5 . 3 L e t m G N \ { 1 , 2 } , l - l < a % l (I G N ) , and let/3 a n d ax , - - - , a m _ 2

be such that a > /? > am_2 > > ai > ao = 0, ana" /e£ A, Ao, , ^4m-2 G K.T/ien ifte functions

E 1

n = 0ko\---km-2\

((a -

fm-2n (

a

(n + 1,1)

(5.2.66)

with j = 1, , I, are solutions to the equation (5.2.65), provided that the series in(5.2.66) are convergent. The inner sum is taken over all ko, ,ftm_2 € No suchthat ko + + km-2 = n.

Ifa — l + 1 ^ 0, then yj{x) in (5.2.66) are linearly independent solutions to theequation (5.2.65). In particular, for a — I + 1 > /?, they provide the fundamentalsystem of solutions.

Proof. . Let

,m-2; 0 g h ^ /m- i S 0-

Applying the Laplace transform to (5.2.65) and using (5.2.3) as in (5.2.32), weobtain

i j _ i

(5.2.67)

where

j = (D^ jy) (0+) - A [Dfc'y) (0+) -m - 2

k=l

jy) (0+) (j = 1,


dj = (D«-iy) (0+)-A (Dl-iy) (0+)- £ Ak

m - 2

k=2

(0+) (j = Zi + 1, , l2),

,dj= (D«-iy) (0+) - A (/>£-Jy) (0+) (j = Zm_2 + 1, , lm_{),

dj = (D%-jy) (0+) (j = /m_! + 1, , 0;

h e re Y2=mAk 0 (m > n). For s G C and

(5.2.33), we have

< 1, just as in

s" - .

n =0j2

n!

n =0

rn—2

IK -, (5.2.68)

if we also take into account the following relation:

n\m - 2

J] 4% (5.2.69)n=0 VfcoH \-km-2=n. i/=0

where the summation is taken over all k0, , fcTO_2 6 No such thatko + + km-2 = n [see, for example, Abramowitz and Stegun ([1], p. 823)].

According to (1.10.10) and (5.2.28), just as in (5.2.35), for s G C and\\sP-a\ < 1, we have

- A)

= —n!

—n!

n + 1

(n+1 ,1)

kv, a -

(5.2.70)

Chapter (§ 5.2) 293

From (5.2.67), (5.2.68) and (5.2.70) we obtain the solution to the equation (5.2.65)in the form (5.2.36), where yj(x) (j = 1, ) are given by (5.2.66). It is easilyverified that these functions are solutions to (5.2.65), and thus the first assertionof the Theorem 5.3 is proved.

For j , k = 1, , I, the direct evaluation leads to the following equations:

(1,1)

m - 2

+ E Y[(A^\x(«-^

1

(n + 1,1)

((a - fin + k + 1 - j + Er=o2(/? - ^V)K, a-p). (5.2.71)

with j = 1, . If k ^ j , then the last formula yields the relations in (5.2.12).If k < j , then, by (5.2.28), the first term in the right-hand side of (5.2.71) takesthe form

(1,1)\xa~

EA"

_ xk-j+a-0 (5.2.72)

If a - / + 1 ^ /3, then, for any k < j , we have k-j + a-/3^.a-/3+l — l^.O and{a-P)n + k-j + Y,Z=a(P ~ av)K ^ a - j3 + 1 - I > 0 for any n G No. Then,from (5.2.71) and (5.2.72), we derive (5.2.14), except for the case a - Z + 1 = /?with k = 1 and j = I, for which the relation (5.2.39) holds. By (5.2.12), (5.2.14)and (5.2.39), we have Wa{0) = 1 for the analog of the Wronskian in (4.2.34).Thus it follows from Lemma 4.1 that the functions yj{x) in (5.2.66) are linearlyindependent solutions to the equation (5.2.65). When a — I + 1 > /?, then therelations in (5.2.2) are valid, and thus yj(x) in (5.2.66) yield the fundamentalsystem of solutions to the equation (5.2.65). This completes the proof of Theorem5.3.

Corollar y 5.4 If I G N\{1,2} and X,A0,-nary differential equation of order I

[-2 G M., then the following ordi-

fe=O

. .( f c ) l x) = 0 (a; > 0) (5.2.73)


has I solutions given by

En =0

1 Xx j = l,---,m). (5.2.74)

(n + 1,1)

. (n + l + l-j + E l=oC - ! - v)k», 1)


(D%+y) (x) - X (#o+2/) (x) ~ s iDo+y) (x) ~ M(x) = ° (x > °; ^ i S ^ e E)(5.2.75)

with I — 1 < a ^ / (/ G N) and 0 < 7 < /? < a, has Z solutions given by

n=0 \g+t/=n/ ^

(n + 1,1) _ "IXx"-*3 (i = 1, - ,0-

(5.2.76)If a — / + 1 ^ /?, then the functions yj(x) in (5.2.76) are linearly independentsolutions to the equation (5.2.75). In particular, for a - I + 1 > /? these functionsprovide the fundamental system of solutions.

Example 5.7 The ordinary differential equation of order / G N

y^l\x) — Xy(m\x) — 5y^k\x) — ny(x) = 0, (5.2.77)

where x > 0; /, m, k G N; k < m < I; X, S, /i G M, has I solutions given by

., (VI _ V^ / V^ 1 ^ " -{l-m)n+l-i+mq+(m-k)vyj\x> - z^ I Z^ ) q\ v\

n=0 \q+v=nj y"

1

(n + 1,1)

— m)n + Z + 1 — j + mg + (m — A;)F, / — m)Aa;' (j = , 0-

(5.2.78)

R e m a rk 5.2 (5.2.22) and (5.2.61) yield new systems of linearly independent so-lutions to the ordinary differential equation (5.2.21) and (5.2.60) (with m = 1) oforder I G N.

Chapter 5 (§ 5.2) 295

Remark 5.3 It was proved in Theorems 5.2 and 5.3 that, if a - I + 1 /?, thenyj{x) in (5.2.29) and (5.2.66) are linearly independent solutions to the equations(5.2.27) and (5.2.65), respectively. It would be interesting to see whether or notthey have the same property in the case a — l + l</3. In particular, it would alsobe interesting to see whether or not the functions yj(x) in (5.2.74) yield a systemof linearly independent solutions to the ordinary differential equation (5.2.73) oforder m £ N\{1,2} .

5.2.2 Nonhomogeneous Equations with Constant Coeffi-cients

In Section 5.2.1 we applied the Laplace transform method to derive explicit solu-tions to the homogeneous equations (5.2.1) with the Liouville fractional derivatives(2.1.10). Here we use this approach to find particular solutions to the correspond-ing nonhomogeneous equations

m

J2MDoly){x) + Aoy(x) = f(x) (a; > 0; 0< < am ; m € N) (5.2.79)

with real Ak G M. (k = 0, , m) and a given function f(x) on K+. Our argumentsare based on a scheme for deducing a particular solution (5.1.4) to the equation(5.2.79), presented in Section 5.1. Using the Laplace convolution formula (1.4.12)

k(x - t)f{t)dtj ) (s) = (Ck)(s)(Cf)(p), (5.2.80)

just as in (5.1.11) we can introduce the Laplace fractional analog of the Greenfunction as follows

(5.2.81)and express a particular solution (5.1.4) of the equation (5.2.79) in the form of theLaplace convolution of Gai,--, am(x) and f(x)

y{x) = f Gai,..., am{x - t)f{t)dt. (5.2.82)Jo

Generally speaking, we can consider the equation (5.2.79) with Am = 1. Firstwe derive a particular solution to the equation (5.2.79) with m = 1 in the form(4.1.1)

(D%+y) (x) - Xy(x) = f{x) (x > 0; a > 0) (5.2.83)

in terms of the Mittag-Lefler function (1.8.17).

Theorem 5.4 Let a > 0, X G M. and let f(x) be a given function defined on R+.Then the equation (5.2.83) is solvable, and its particular solution has the form

ry{x)= (x-tÊa^lXix-t^fitfdt, (5.2.84)


provided that the integral in the right-hand side of (5.2.84) is convergent.

Proof. (5.2.83) is the equation (5.2.79) with m = 1, a\ = a, A\ = 1, AQ = —A and(5.2.81) takes the form

^ ^ ) . (5.2.85)

By (1.10.9) with /? = a, we have

C [ta~lEata (Xta)] = — L - (W(s) > 0; |As"a| < l) . (5.2.86)

HenceGa(x)=xa-1Ea,a(\x

a), (5.2.87)

and thus (5.2.82), with GQl,-, am{x) = Ga(x), yields (5.2.84). Theorem 5.4 isproved.


(£>o+1/2y) (as) - Ay(a;) = f{x) {x > 0; i e N; A e M), (5.2.88)


y(x) = j\x - ty-Ê^/^/z [\(x - t)1-1'2] f(t)dt. (5.2.89)

Example 5.9 The following ordinary differential equation of order / £ N

y < - l ) { x ) - \ y { x ) = f { x ) {x > 0; l e N ) (5.2.90)


y(x) = f (x - t)l~lEhl [X(x - t)1] f(t)dt. (5.2.91)Jo

In particular, the following second order equation

y"{x) - Xy(x) = f{x) (x > 0; AeK) (5.2.92)

has different particular solutions for A > 0 and A < 0:

VA

(A < 0). (5.2.94)

Chapter 5 (§ 5.2) 297

Next we derive a particular solution to the equation (5.2.79) with m = 2 of theform

n = 0 n!

(D$+y) (a;) - A [Dg+y) (x) - ny{x) = f(x) (x > 0; a > /? > 0) (5.2.95)

in terms of the generalized Wright function (5.2.28).

Theorem 5.5 Let a > (3 > 0, A,/x € IR and let f(x) be a given function definedon R.-|-. Then the equation (5.2.95) is solvable, and its particular solution has theform

y(x) = f (x - t)a-1Ga,/3;A,M(a! - t)f(t)dt, (5.2.96)Jo

(n + 1,1)Xza'0 , (5.2.97)

(an + a, a — j3)

provided that the series in (5.2.97) and the integral in (5.2.96) are convergent.


(D$+y)(x)-x(D0S+yyx)=f(x) (x>0; a > /? > 0) (5.2.98)


(x - i ) " " 1 ^ - ^ [A(a; - t)a~0] f(t)dt. (5.2.99)

Proof. (5.2.95) is the same as the equation (5.2.79) with m = 2, a2 = a, ai = j3,A2 = 1, Ai = -A, Ao = —/i, and (5.2.81) is given by

According to (5.2.33) for s e C and ^ _ , < 1, we have

Ga,p(x) = C- l\n+l

(x). (5.2.100)

By (1.10.10) and (5.2.28), for s G C and (As'3"0! < 1, we have

*-/3_A )" H

1

n!

n!

(n + 1,1)

(an + a,a — ,Xtc («), (5.2.101)


and hence (5.2.100) takes the following form:

(n + 1,1)

n =0 (an + a, a — j3)(5.2.102)

Thus the result in (5.2.96) follows from (5.2.82) with Gaii..., am{x) = Ga,p(x).(5.2.96) with n = 0 yields (5.2.99) if we take (5.2.40) into account. Note that, inthe limiting case /? —> 0, the solution (5.2.99) of the equation (5.2.98) coincideswith the solution (5.2.84) of the equation (5.2.83).


y'(x)-X (/?o+y) (x)-w(x) = f(x) {x > 0; 0 < Re(/3) < 1; A , ^ e l ) (5.2.103)


y(x)= [ Gi,0iAlM (a;-t)/(t)di, (5.2.104)Jo

(n + 1,1)

(n + 1 ,1 - /!n =0Xzi-t (5.2.105)


y'{x) - A (pi1*y) (x) - w(x) = f(x) (x > 0; A,/i6


y(x)= / Ghl/2.xAx-t)f(t)dt,Jo

(n + 1,1)

(n +1,1/2)n = 0

(5.2.106)

(5.2.107)

(5.2.108)


y"(x) - X (#0+2/) (z) - M/(z) = fix) ix > 0; 0 < /? < 2; X,fj,£M) (5.2.109)


y(x) = f (x - t)G2,fi;\A* - t)f(t)dt, (5.2.110)Jo

(n + 1,1)

(2n + 2,2-/3)71=0

(5.2.111)

Chapter 5 (§ 5.2) 299

Example 5.12 The following ordinary differential equation of order I G N

y{l){x)-Xy{m){x)-ny{x) = f{x) (x > 0; Z,m€N, m < I : A , / i £ l ) (5.2.112)


fxy(x) = / {x-ty Gi<m-x^(xJo

(n + 1,1)

n=0 (In+ 1,1 - m)Xz'

(5.2.113)

(5.2.114)

In particular,

y(x) = / (x - t)G2,i;\,p(x - t)f(t)dt,Jo

(n+1,1)

n=0 (2n + 2,1)Xz

(5.2.115)

(5.2.116)

is a particular solution to the equation

y"(x)-Xy'(x)-W(x) = f(x) (x > 0). (5.2.117)

Finally, we find a particular solution to the equation (5.2.79) with anym G N\{1,2} in terms of the Wright function (5.2.28). It is convenient to rewrite(5.2.79), just as (5.2.65) in the form

m-2

(D%+y)(x)-X(D$+y)(x)-Y,MD^y)(x)-Aoy(x)=f(x) (x > 0), (5.2.118)

with m € N\{1, 2}, 0 < ai < < am_2 < /3 < a, and A, Ao, , Am_2 € E.

Theorem 5.6 Lei m € N\{1,2} , a > (3 > am_2 > > ai > a0 = 0, /eiA, Ao, , j4m-2 S K, and let f(x) be a given function defined on M+. Then theequation (5.2.118) is solvable, and its particular solution has the form

y(x) = f\x - tr^G^,.. ,am.2,0,a;x(x - t)f(t)dt,Jo

(5.2.119)

n = 0

[m-2n<i/=0

(n+1,1)

((a - /3)n + a(5.2.120)

provided that the series (5.2.120) and the integral in (5.2.119) are convergent. Theinner sum is taken over all ko, , km-2 G No such that ko + + km-2 = n.


Proof. (5.2.118) is the same equation as the equation (5.2.79) with am = a,am-i = /?, Am = 1, Am-i = -A , and with — Ak instead of Ak for k = 0, , m—2.Since cto = 0, (5.2.81) takes the form

Gai,-, am{x) = \ C1

(x).

For s G C and < 1, in accordance with (5.2.68), we have

= c- 1 n!

m-2n ( - A)n+1(X). (5.2.121)

Using (5.2.28) and (1.10.10) just as in (5.2.70), for s € C and \Xs0-a\ < 1, wehave

= ^!

1 / r ,— I r\f(a

~ n! \ I

Xt°

(n + 1,1)

/ v-^m-2

I t follows from (5.2.122) that GQl,..., Om(x) in (5.2.121) is given by

(s). (5.2.122)

n!Vm-2

(n + 1,1)

((a - (3)n + a + X Jo (fi ~ av)kv,a(5.2.123)

Hence (5.2.82) yields the result in (5.2.119), which completes the proof of Theorem5.6.

Chapter 5 (§ 5.2) 301

Coro l la r y 5.5 / / I € N \ {1 ,2} and X,A0,--- , M-2 £ R, then the ordinarydifferential equation of order I

1-2

0)k=0

is solvable, and its particular solution has the form

y(x)= [ (x-tY^GxAx-VfWdt,

(5.2.124)

(5.2.125)

E 1

ra=0 \ko-\

1-2

Li/=o

(n + 1,1)(5.2.126)

provided that the series (5.2.126) and the integral in (5.2.125) are convergent.


(D%+y) (x) - A (pe+y) (x) - 6 (DZ+y) (x) - m(x) = f(x) (x > 0; A, 6,» e R)

(5.2.127)wit h I — 1 < a^l (I E.N), 0 < < /3 < a, has a part icular solution given by

(5.2.128)

{(a-(3)

= /JO

r

n + a

(x -

s([n +

+ (3i

t)a~lC

\

\i+v=n)

1,1)

+ ( /3- l)v,a-

- t)f{t)dt,

/3)a (5.2.129)

Example 5.14 The following ordinary differential equation of order / £ N

y (x) - Aj/ (m) (x) - SyW (x) - /J,y(x) = f(x) {x > 0; I, m, k G N; k < m < I)(5.2.130)

with X,5,fi€ K, has a particular solution given by

y(x) = [ {x - t)1~1Gfc,m,iiA(a; - t)f(t)dt, (5.2.131)Jo

n =0

1

(n + 1,1)

((/ — m)n + I + mq + (m - k)v, a - (3)

A — m (5.2.132)


R e m a rk 5.4 The equation of the form (5.2.103) was considered by Seitkazieva[756], of the form (5.2.109) with 0 < /3 < 1 by Caputo [121], and of the form(5.2.109) with /? = 3/2 by Bagley and Torvic [52].

R e m a rk 5.5 The results obtained in Theorems 5.5 and 5.6 were presented inother forms by Podlubny ([682], Sections 5.4 and 5.6).

As in the case of ordinary differential equations, a general solution to thenonhomogeneous equation (5.2.79) is a sum of a particular solution to this equationand of the general solution to the corresponding homogeneous equation (5.2.1).Therefore, the results established in this section and in Section 5.2.1 can be usedto derive general solutions to the nonhomogeneous equations (5.2.83), (5.2.95) and(5.2.118). The following statements can thus be derived from Theorems (5.1), 5.4,Theorems 5.2, 5.5 and Theorems 5.3, 5.6, respectively.

Theorem 5.7 Let I - 1 < a ^ I (I G N), A £ R and let f(x) be a given realfunction defined on R+. Then the equation (5.2.83) is solvable, and its generalsolution is given by

y(x) = J (x - tyÊcc [X(x - t)a] f(t)dt + ^cjxa-'Ea>a+1-i (Xxa),

(5.2.133)where Cj (j — 1, , I) are arbitrary real constants.

Theorem 5.8 Let I - 1 < a I (Z G N), 0 < /3 < a be such that a -1 + 1 /?, letX, fj, G R and let f(x) be a given real function defined on K+. Then the equation(5.2.95) is solvable, and its general solution has the form

y(x)= [ {x-t^Ga^xAx-^mdtJo

(n + 1,1)

(an + a + 1 - j , a - /?)(5.2.134)

where Ga^\^(z) is given by (5.2.97) and Cj (j = 1, , Z) are arbitrary realconstants.

In particular, the general solution to the equation (5.2.98) has the form

rx i

y(X) = / (x-t)a-lEa-fi ta [X(X - t)a-P] f(t)dt + Y,Cixa~:'Ea-â+l-i {Xxa~$) .(5.2.135)

T h e o r em 5.9 Letm e N \ { 1 , 2 } , Z - l < a I (I <S N), and let /? andcti,--- , am _2

be such that a > f3 > am_2 > > ai > ao = 0 and a — Z + 1 f3, and let

Chapter 5 (§ 5.2) 303

X,AQ,--- , j4m_2 £ K-, and fe£ /(a;) &e a given real function onequation (5.2.118) is solvable, and its general solution is given by

y(x)= [ (x-t)a-1Gai,..,am_2iP,Q.,x(x-t)f(t)dtJo

.. Then the

, (5.2.136)

where Gai,-- ,am-2,i3,a;\{z) is given by (5.2.120) and Cj (j = 1, , I) are arbitraryreal constants.

Remark 5.6 Using the formulas obtained in Examples 5.8-5.14 and Corollary5.5, the general solutions to the fractional and ordinary differential equations con-sidered there can be found.

5.2.3 Equations with Nonconstant Coefficients

In Sections 5.2.1 and 5.2.2 we used the Laplace transform method to derive explicitsolutions to fractional homogeneous and nonhomogeneous equations of the forms(5.2.1) and (5.2.79) with real constant coefficients Au (k = , m). Such amethod can be also applied to solve linear fractional differential equations withnonconstant coefficients. We illustrate such an approach to derive the explicitsolution to the following fractional differential equation

{ X > 0 ; (5.2.137)

of order a>0(l — l<a^l, I £ N; / 1). First we consider the case 0 < a < 1and give the solution in terms of the generalized Wright function (1.11.14) withp = 0 and q = 1 of the form

0 * 1 ; pe (5.2.138)

The following assertion is derivable from Theorem 1.5.

Lemma 5.2 (5.2.138) is an entire function of z G C for (3 > —1.

The following assertion holds.


Theorem 5.10 The differential equation (5.2.137) with 0 < a < 1 and A e R issolvable, and its solution is given by

y(x) = = cxa A „«- !

1-a(5.2.139)


Proof. Rewrite the equation (5.2.137) in the form

(x) = xv(x) (x>0; 0 < a < 1, A G 1).

Applying the Laplace transform to (5.2.140), and using the relation

Dk (Cy) (s) = (-l) f c {Ctky(t)) (s) = £; kwith k = 1 and formula (5.2.3) with / = 1, we have

D {Cy) (s) = - (as-1 + Xs~a) {Cy) (s).

(5.2.140)

(5.2.141)

(5.2.142)

From (5.2.142) we derive the solution to this ordinary differential equation of thefirst order with respect to (Cy) (s):

(5.2.143)(Cy)(s) = cs exp\ 1 — Q

where c is an arbitrary real constant. Expanding the exponential function in aseries, we obtain

j = 0

( 5.2.144)

By the formula

we have

and from (5.2.143) we derive the following solution

y(x) = ex01'1

:xP-1)(s)=r((3)s-e (/?>-l), (5.2.145)

.-[a-Ka-lW] = , r — ) {s) {j 6 N o) ,

which, in accordance with (5.2.138), yields the result in (5.2.139). It is readilyverified that the function yi(x) in (5.2.139) is the solution to the equation (5.2.137),thus proving Theorem 5.10.

Chapter 5 (§ 5.2) 305



Xy(x)

V{X) = ^

(x > 0; A G

where c is an arbitrary real constant. In particular, the equations

y)(x) =

and

have the same solution given by

* ) ( * ) = -

eyJX

); AG I

0; A G

(5.2.146)

(5.2.147)

(5.2.148)

(5.2.149)

(5.2.150)

Note that (5.2.147) follows from (5.2.139), if we take into account the relation

1 _.2IA

0 * 1

(I _I)V2' 2/

(5.2.151)

Remark 5.7 Solution (5.2.150) of the equation (5.2.148) was obtained by Millerand Ross ([603], Chapter VI, Section 3). As mentioned in Section 5.1, this is thefirst differential equation of fractional order 1/2 discussed by O'Shaughnessay [650]and Post [686].

Next we consider the equation (5.2.137) with I — 1 < a I (I £ N\{1} ) andexpress its solution in terms of the generalized Wright function (5.2.138) and thegeneralized Wright function (1.1.16) with p = 1 and q = 2:

(0,61,62 GC; a , f t , f te (5.2.152)

(61,ft), (62,ft

The following assertion can be derived from Theorem 1.5.

Lemma 5.3 (5.2.152) is an entire function of z € C for j3\ + f t — ai > —1.

We also need the numbers Ck{a,m) defined, for a > 0, m = 2, ,1(a

{i , q e N) and k e No, by

(5.2.153)q,j=O, — k; q+j=k Lv

These numbers have the following property.


Lemma 5.4 The numbers Ck(a,m), with a > 0, m = 2, , I (aq £ N) and k £ No satisfy the following recurrence relations:

+k) ck+i(a,m) = ck(a,m),

a — 1 /

which yield the explicit expression for Ck(a,m) in the form

p I a — m\

ck(a,m) = \^l2 (fcGNo).

? +m 1

(5.2.154)

(5.2.155)

Now we can derive the explicit solution to the equation (5.2.137).

Theorem 5.11 The differential equation (5.2.137) with I-1 < a I (I £ N\{1} )and A £ l is solvable, and its solution is given by

y{x) = cxxa 1

0* i

m = 2

{a, a- 1)

(1,1)

A , . " - 1

a - 1"

a - 1, (5.2.156)

where ci, , cm are arbitrary real constants.In particular, the solution to the equation (5.2.137) with 1 < a < 2 is given by

y{x) =(a,a-1)

+ C2Xa-2 i * 2

(1,1)

A

a - 1

„"- !

(5.2.157)

where c\ and c2 are arbitrary real constants.

Proof. Applying the Laplace transform to (5.2.140), and using (5.2.141) with k = 1and (5.2.3), we have

D (Cy) (s) + (as-1 + \s~a) (Cy) (s) = ^ dm(m - l)snm~2 — a

m = 2

where dm = (D +my) (0+) (m = 2, ,/). From here we derive the solution to

this ordinary differential equation of the first order with respect to (Cy) (s):

(Cy) (s) = s~a exp (- (m - 1) Jsm~2 exp ds '

Chapter 5 (§ 5.2) 307

where C\ is an arbitrary real constant. Expanding the exponential function in theintegrand in a series and using term-by-term integration, we obtain

(Cy) (s) =c(Cyi) (s) + ^ dm{m - 1) (Cy'J (s),m=2

where

By the proof of Theorem 5.10,

(a, a- 1) 1 -a

— ^ 0 ^1A

a - 1

(5.2.158)

(5.2.159)(a, a- 1)

Expanding the exponential term exp I x^-i ) m Po w er series, multiplying the re-sulting two series, and taking (5.2.153) into account, we have

A Vs( 1-a)>

U=0

From here, by using the formula (5.2.145) withthe following expression for yîx):

= ( a-

or, in accordance with (5.2.155) and (5.2.152),

ym{x),

- m, we derive

(a — l)k+a —

k\

(5.2.160)


where

ym(x) = xa(1,1)

a- 1-a-l . (5.2.161)

It follows from (5.2.158) that the solution to the equation (5.2.137) has the form

y(x) = dm{m - l)y*m{x),m=2

where yi(x) and y^n(x) (m = 2, , I) are given by (5.2.159) and (5.2.160), re-spectively.

It is easily verified that the functions yi(x) and ym(x) (m = 2,--- , I) in(5.2.159) and (5.2.161) are solutions to the equation (5.2.137), and thus Theorem5.11 is proved.

Corollar y 5.6 The following ordinary differential equation of order I 6 N\{1 }

(5.2.162)

is solvable,

i

m-2

and

cmxl

its solution

y(x) -— C

_ (

xy(l\x)

is given

ix1'1 o*

f + l - r r

= Xy(x)

by

_ (1,1-

(1,1)

(A6

-1)

< l-m

1)

A/ - 1X

I

. - i

Axl 1 (5.2.163)

where Ci, , cTO are arbitrary real constants.


Xy(x)(D3J+2y) (x) =


(x > 0; A G

y{x) = 2A* 1/2

(1 I)

where c\ and c2 are arbitrary real constants.

1 * 2

(1,1)

L ( | , I ) , ( - 1 , 1)

(5.2.164)

2XX1/2

(5.2.165)

Remark 5.8 The generalized Wright functions I ' ^ - Z ) in (5.2.156) coincide withspecial cases of the generalized Mittag-Leffler function Ea>m^(z) defined in (1.9.19)

Chapter 5 (§ 5.2) 309

e x c e pt f or t h e c o n s t a nt m u l t i p l i e r . I n d e e d, i f A € R, t h e n, f or a ny m = 2 , , 1 ,w e h a ve

a — m(1,1)

A

a -1-x0-1

= S a , i - i / a , i - ( i + m ) /a ( A a ; " - 1 ) ( m = 2 ,- . ( 5 . 2 . 1 6 6)

Therefore the explicit solutions ym(x) (m = 1,- , I (I € N\{1} ) in (5.2.156)can be expressed in terms of the generalized Mittag-Leffler functions presented inthe right-hand sides of (5.2.16). Thus the solution (5.2.156) is equivalent to thefollowing one:

i

y(x) = Y, cmxa-mEaA_1/a (1+m)/a (Ax0" 1). (5.2.167)

Note that this solution can be derived from the solution (4.2.33) of the equation(4.2.30) with j5 = - 1 .

5.2.4 Cauchy Type Problems for Fractional DifferentialEquations

Explicit solutions to fractional differential equations, established in Sections 5.2.1-5.2.3, can be applied to obtain solutions to initial and boundary problems for suchequations. Here we illustrate such an application to Cauchy type problems forfractional differential equations of order a (I — 1 < a ^ Z; Z £ N) with initialconditions of the form (4.1.2)

(£>o+ kv) (0+) = hk G K {k = 1, ,Z; Z - K a ^ Z ) ) . (5.2.168)

The first results for the homogeneous equations (5.2.5), (5.2.27) and (5.2.65) followfrom Theorems 5.1-5.3 if we take into account the relations (5.2.12), (5.2.14) and(5.2.39).

Proposit ion 5.1 Let I — 1 < a ^ I (I £ N) and A g i . Then the Cauchy typeproblem (5.2.168) for the equation (5.2.5) is solvable, and its solution is given by

i

y(x) = J2 bJxa~JEa,a+i-j (\xa). (5.2.169)i=i

Proposit ion 5.2 Let I - 1 < a ^ I (I £ N) and 0 < /3 < a be such that a-l + 1 ^(3, and let A,/U G M.. Then the Cauchy type problem (5.2.168) for the equation(5.2.27) is solvable, and its solution has the form

i

(a-l + l>0) (5.2.170)


orI

< {a-l + 1 =P), (5.2.171)J=2

where yj(x) (j = 1, ) are given by (5.2.29).

Propos i t i on 5.3 Let m G N\ {1 ,2 } , I - 1 < a ^ I (I G N), and ie£ /? anda ii " > am-2 be such that a > P > a.m-2 > > oti > ao = 0 and a — Z + 1 ^ /3,and /ei X,A0,--- , Am_2 £ R- Then the Cauchy type problem (5.2.168) for theequation (5.2.65) is solvable, and its solution has the forms (5.2.170) and (5.2.171)for the cases a-l + 1 > (3 and a — l + 1 — 0 respectively, where yj(x) (j = 1, )are given by (5.2.66).

Our next results for nonhomogeneous equations (5.2.83), (5.2.95) and (5.2.118)follow from Theorems 5.7-5.9.

Proposition 5.4 Let I — 1 < a I (I £ N), A g i and let f(x) be a given realfunction defined on E+. Then the Cauchy type problem (5.2.168) for the equation(5.2.83) is solvable, and its solution is given by

rx iy(x) = / (x - trÊ [X(x - t)a] f(t)dt + V bjxa~jEa,a+1_j (Xxa).

Jo j=i(5.2.172)

Proposition 5.5 Let I - 1 < a ^ I (I G N) and 0 < P < a be such thata — I + 1 ^ P, let A, /i €E R and let f(x) be a given real function defined onIR+. Then the Cauchy type problem (5.2.168) for the equation (5.2.95) is solvable,and its solution has the form

r

y(x)= / (x-tr-'Ga^x^x-^f^dt + Y^bjyjix) (a-l + 1 >p), (5.2.173)J

or

y(x) = f\x - ty^Gcpîx - t)f(t)dtJo

(a-l + l = p), (5.2.174)

where Ga,p-,\^{z) and yj(x) (j = I , - - - ,1) are given by (5 .2.97) and (5 .2 .29),respectively.

Proposit ion 5.6 Let m G N\{1,2} , / - 1 < a ^ I (I G N), and let p andoil,-" , ctm-2 be such that a > P > am_2 > > ai > a0 = 0 and a - / + 1 ^ p,and let X,AQ,- , Am-2 G M., and let f(x) be a given real function defined on R+.

Chapter 5 (§ 5.3) 311

Then the Cauchy type problem (5.2.168) for the equation (5.2.118) is solvable, andits solution has the form

rx i

y(x) = / {x-t)a-1Gai,..,am ,a,x{x-t)f{t)dt + YJbjyM (a-l + l>p),Jo j=i

(5.2.175)or

y(x)= / (x-t)a-1Gai,...,am_a,p,a.,x{x-t)f{t)dtJo

i

Y (a-1 + 1=0), (5.2.176)3=2

where Gait...,arn-2,p,a;\(z) and yj(x) (j = I,--- ,1) are given by ( 5 .2 .120) and( 5 . 2 . 6 6 ), respectively.

Proof. The proofs of Propositions 5.4-5.6 follow from Theorems 5.7-5.9, on thebasis of the respective formulas (5.2.133), (5.2.134) and (5.2.136) and the directlyverified relations

fck [J (t - ry-'E [X(t - r)a] /(r)dr] ) (0+) = 0 (k = 1, , I),

| j f (t - 7 - ) a - 1 G a , i 8 i A , M ( t - T)f{r)dr]^ ( 0 + ) = 0 (k = 1, ,

~k [J*(t - T ) " - 1 ^ , . . ,Qm.2,0,a-At ~ r ) / ( r ) d r ] ) ( 0 +) = 0 (k = 1, , I),

if we also take the relations (5.2.12), (5.2.14) and (5.2.39) into account.

Remark 5.9 Results presented in Sections 5.2.1-5.2.4 were established in Kilbasand Trujillo [411], [410]. Using the explicit formulas obtained in Examples 5.8-5.14and Corollary 5.5, one may derive the solutions to the Cauchy type and Cauchyproblems for the corresponding fractional and ordinary differential the equations.

Remark 5.10 In Sections 5.2.1 and 5.2.2, we applied the Laplace transform toderive explicit solutions to homogeneous and nonhomogeneous equations of theforms (5.2.1) and (5.2.79) considered on the the positive half-axis E+. It is easilyverified that all obtained formulas remain true for these equations considered ona finite interval [0,6] (0 < b < oo). Moreover, these formulas also apply on anyfinite interval (a, b] (0 ^ a < b ^ oo), if the Liouville fractional derivative D +yis replaced by the Riemann-Liouville derivative D"+y, and (in the correspondingformulas for explicit solutions) xa is replaced by (x — a)a, and the integral f* byf \ The same applies to the ordinary differential equations and Cauchy type andCauchy problems for fractional and ordinary differential equations. In particular,in this way explicit solutions can be obtained for homogeneous and nonhomoge-neous fractional and ordinary differential equations with constant coefficients, andfor the corresponding Cauchy type and Cauchy problems, established in Sections4.1-4.3 by other methods. For example, from (5.2.6) we can derive the solution(4.2.44) to the equation (4.2.43).

3 12 Theory and Applications of Fractional Differential Equations

5.3 Laplace Transform Method for SolvingOrdinary Differential Equations with CaputoFractional Derivatives

We present our development of the Laplace transform method for the Caputofractional derivatives in the following subsections.

5.3.1 Homogeneous Equations with Constant Coefficients

In this section we apply the Laplace transform method to derive explicit solutionsto homogeneous equations of the form (5.2.1)

m

Y, Ak(CD^y)(x) + Aoy(x) =0 (x > 0; m G N; 0 < ax < < am), (5.3.1)

k=i

involving the Caputo fractional derivatives cDQ^y (k = 1, , m), given by (2.4.1),with real constants Ak G M (k = 0, , m — 1) and Am = 1. We give the conditionswhen the solutions y\ (x), , yi(x) of the equation (5.3.1) with l-l < a := am <I will be linearly independent, and when these solutions form the fundamentalsystem of solutions

yf(0) = 0 (fc.j = 0, ; fc#j), 2 / f ( 0) = l (k = 0, - 1). (5.3.2)

T he Laplace t ransform m e t h od is based on the formula (2.4.67):

djsa-i-1 (l-Ka^l, ieN), (5.3.3)

j=o

dj=y{j)(0) (j = 0, , Z - 1). (5.3.4)

First we derive the explicit solutions to the equation (5.3.1) with m = 1 in theform (5.2.5)

{cD%+y) {x) - \y{x) =0 (x > 0; l-Ka^l; I G N; A G K), (5.3.5)

in terms of the Mittag-Lefler functions (1.8.17). The following statement holds.

Theorem 5.12 Let I - 1 < a / (I G N) and A G M. Then the functions

yj(x) = x>Ead+1 (Xxa) (j = 0, , / - 1) (5.3.6)

yield the fundamental system of solutions to the equation (5.3.5).

Proof. Applying the Laplace transform (1.4.1) to (5.3.5) and taking (5.3.3) intoaccount just as in (5.2.7), we have

^ 4 ^ 4 , (5.3.7)

Chapter 5 (§ 5.3) 313

where dj (j = 0, , I - 1) are given by (5.3.4). Formula (1.10.9) with /? = j + 1yields

i + 1 ( A f ) ] ( S) = ^ ( |S- QA | < 1 ). (5.3.8)

Thus, from (5.3.7), we derive the following solution to the equation (5.3.5):

y(x) = J2 djVjix), yj{x) = x>Ea,j+ i (Xxa). (5.3.9)

It is directly verified that the functions yj(x) are solutions to the equation

(5.3.5):

(D%+ [ t j E a J + 1 (Xta)]) (x) = \ x > E a , j + 1 (Xxa) ( i = - 1 ), ( 5 . 3 . 1 0)

a ndyf\x) = xi-kEaJ-k+1 (Xxa). (5.3.11)

It follows from here that

I / i - f t > ( 0 ) = 0 ( k , j = O , - - - , l - l ; j > k ) , y{k)(0) = l ( k = 0 , - - - , l - l ) . ( 5 . 3 . 1 2 )

If j < k, then, on the basis of (1.8.17), the relation (5.3.11) takes the formyf] (x) = Xxa+i-kEatOl+j_k+l {Xxa), (5.3.13)

and, since a + j — k > 0 for any k, j = 0, , / — 1,

y( f c)(0) = 0 (k,j=O,---,l-l;j<k). (5.3.14)

By (5.3.12) and (5.3.14), the Wronskian

W(x)=det(yf)(x))l~1 (5.3.15)\ J ) k,j=O

at the point 0 is equal 1: W(0) = 1. Then the result of the Theorem 5.12 followsfrom the known theorem of classical analysis and (5.3.2).


{CD%+y) (x) - Xy(x) = 0 {x > 0; 0 < a ^ 1; A £ E) (5.3.16)

has its solution given byy(x)=Ea(Xxa), (5.3.17)

while the equation

(CD%+y) (x) - Xy{x) = 0 {x > 0; 1 < a ^ 2; A G E) (5.3.18)

has its fundamental system of solutions given by

yi(x)=Ea(Xxa), y2{x) = xEa,2{Xxa). (5.3.19)



{x) - Xy{x) = 0 (x > 0; Z G N; A e i )

has its fundamental system of solutions given by

yj(x) = xjEt_1/2j+1 ( A ^ " 1 / 2 ) (i = 0, - 1)

(5.3.20)

(5.3.21)

Next we derive the explicit solutions to the equation (5.3.1) with m = 2 of theform (5.2.27):

{°DS+y) (x)-X (CD%+y) {x)-ny{x) = 0 (x > 0; l-Ka^l; I e N; a > /3 > 0),

(5.3.22)with complex A, JJ, 6 C, in terms of the generalized Wright function (5.2.28).

Theorem 5.13 Let a > 0 and j3 > 0 be such that

0 < /? < a, Z - l < a ^ Z , m - 1 < /? ^ m ( I . m e N; m^l), (5.3.23)

and Zei A , / i £ t . Tften i/ie functions

(n + 1,1)

(an + j + l , a - /?)

(n + 1,1)

(an + j + 1 + a - /3, a - /3)

n =0

n =0 n!(5.3.24)

n =0

(n + 1,1)( j = m, , Z - 1)

(5.3.25)ore solutions to the equation (5.3.22), provided that the series in (5.3.24) and(5.3.25) are convergent.


{cD%+y)(x)-\(cD0o+y)(x)=O (x > 0)

has its solutions given by

Vj(x) = xtEc-M+i (\xa-?) - \xa-P+iEa_0,

(j = 0 , , m - 1 ) ,

(5.3.26)

(5.3.27)

Vj{x) = xjEa-P,j+1 (\xa~P) (j = m, , I - 1). (5.3.28)

Chapter 5 (§ 5.3) 315

Ifa-l + 1 fi, then the functions yj{x) in (5.3.24), (5.3.25) and (5.3.27),(5.3.28) are linearly independent solutions to the equations (5.3.22) and (5.3.26),respectively. In particular, when a — I + 1 > /?, these functions provide the funda-mental system of solutions.

Proof. Applying the Laplace transform to (5.3.22) and using (5.3.3) as in (5.3.7),we obtain

i-i

(Cy) (s) = V dj—a-j-l

5=0 - n- A E ^ —

J

5=o

where dj (j = 0, ,1 — 1) are given by (5.3.4). For s £ C and

accordance with (5.2.33), we have

~ (a-0)-{p

di E I? {sa-P _n =0 j=0 n=0

Using (1.10.10) and taking (5.2.28) into account, for s G C andhave

s(a-P)-(0n+j+l)

1n!

(n + 1,1)(s),

< 1, in

(5.3.29)< 1, we

(5.3.30)

1

n!tan+j+a

(n + 1,1)

(an + j + 1 + a - (3, a - j3)Xt° [s .

(5.3.31)

From (5.3.29)-(5.3.31) we derive the following solution to the equation (5.3.22):

i-i i-i

y(x) = (5.3.32)

5=o 5=0

where yj(x) (j — 1, , I) are given by (5.3.24) for j = 0, , m - 1 and by (5.3.25)for j — m, ,1 — 1. It is directly verified that these functions are solutions to theequation (5.3.22) which proves the first assertion of the Theorem 5.13.


For j , k = 0, ,1-1, the direct evaluation yields

vr(*) = Y,f7Txan+i~k^n=0

nw = Y,^n=0 n\

i * ]

(an +

= o,--.

(an +

(n

j +

,rn

(n

+ 1,1)

1 — k,a — p)Xxa p - XI(x)

(5.3.33)

- 1 ) ,

+ 1,1)

1 - k, a — (3)\xa~P (5.3.34)

with (j = m, , I — 1), and where

n = 0

(n + 1,1)

(an + j + 1 - k + a - (3, a - (3)Xxc

(5.3.35)It follows from (5.3.33)-(5.3.35) that the relations in (5.3.12) hold for j ^ k. Ifj < k, then, in accordance with (5.2.28), we have

(x) - V

n=l n!

(n + 1,1)

(an+j + 1- k,a- (3)

I f a - I + 1 ^ / 3, t h en ( a - /3)q + a - (3 + j - k ^ a - P + 1 - 1 ^ O f o rany j , k = 0, , I — 1 and g £ No. Moreover, an + j - ^ a + 1 — I > 0 for anyj , k = 0, , Z —1 and n £ N. Then linxô J(z) = 0 except for the case a—l+1 = Pwith k = I — 1 and j = 0, for which linxE_+o J(x) = A. If a — I + 1 > p, such that,for any n G No and j , k = 0, ,1 — 1, an+j — k + a — p > a — P - 1 + I > 0,then limx^0I(x) — 0. Thus (5.3.33)-(5.3.35) yield the relation (5.3.14) except forthe case a - I + 1 = /? with k = I — 1 and j = 0, for which

(' — 1) /r\\ \ / r o Q£?\

VO \y) ZZ \0.O.Ot)J

According to (5.3.12), (5.3.14) and (5.3.36) W(0) = 1 for the Wronskian (5.3.15).Thus it follows from known theorems in mathematical analysis that the func-tions yj(x) in (5.3.24), (5.3.25) are linearly independent solutions to the equation(5.3.22). If a - I + 1 > P, then the relations in (5.3.2) are valid, and hence yj(x)in (5.3.24), (5.3.25) yield the fundamental system of solutions. This completes theproof of Theorem 5.13 for the equation (5.3.22). Setting fi = 0 and taking theformula (5.2.40) into account, we arrive at the assertion of Theorem 5.13 for theequation (5.3.26).

Chapter 5 (§ 5.3) 317


(cDg+y) (x) - A


(a;) - w{x) = 0 (a; > 0; 0 < (3 < a g 1; A, p G R)

(5.3.37)

(n + 1,1)

n =0 n!

n!9

(n + 1,1)(5.3.38)

«=o " [ (an + 1 + a - /3, a - /?)

/n particular,

x) = kja-p y\x HJ — Aa; ^ f i a - ^ a - î (Aa; ^j (0.0.o9j

is a solution to the equation

{cD%+y) (x) - A (c£>o+2/) (a) = 0 (a; > 0; 0 < /3 < a ^ 1; A G M). (5.3.40)


(cD%+y) (x) - A (CD$+y) (x) - ^y(x) = 0 (5.3.41)

where x > 0; 1 < a ^ 2, 0 < /3 < a; A,|j 6 I , /ias one solution yi(x), given by(5.3.38), and a second solution 2/2(2;) given by

[ (n + 1,1)

n=0 (an + 2, a - /?)

n =0 n!n+l+a-,3 ^ i (5.3.42)

(5.3.43)

(an + 2 + a - /3, a - /?)

/or j = 0, , m — 1 and 1 < /? < a, w/ii/e, /or 0 < /3 1, by

°° n [ (n + 1,1)

^ y n!n=o |_ (an + 2, a - /3)

/n particular, the equation

{CDo+y) (x)-X (CD%+y) (x) = 0 (x > 0; 1< a ^ 2, 0 < f3 < a; A £ M) (5.3.44)

/ia« one solution yi(x) given by (5.3.39), and a second solution 2/2(2;) ^wen &«/

a—/?,2+a—/3 1^2; I , yo.o.îoj


for 1 < 0 < a, while, /or 0 < /? ^ 1, by

y2(x) = xEa (5.3.46)

If a ^ 0 + 1, then (5.3.38), (5.3.43) and (5.3.39), (5.3.46) are linearly indepen-dent solutions to the equations (5.3.41) and (5.3.44) respectively. When a > 0 + 1,they provide the fundamental systems of solutions.


y'(x) - X (cD%+y} (x) - fiy(x) =0 (x > 0; 0 < 0 < 1; A, /x e R). (5.3.47)


(n + 1,1)

(n + 1,1 - 0)

(n+1,1)

(n +2 - /3 ,1 - /3)

n =0 n\Xx

n=0

In particular,

n!

n =0 n!

n =0 n!

(n + 1,1)

(n +1,1/2)

(n + 1,1)

(n +3/2,1/2)

is a solution to the equation

y'(x) - X (cDl^y) (x) - fiy(x) = 0 (x > 0; A, \i e M).


y"(x) - A (cDQ+yJ (x) - ny(x) = 0 (a; > 0; 0 < 0 < 2; A, /J, G

has two solutions given by

(n + 1,1)

(2n +1,2- /3)

(n + 1,1)

n=0 '

n =0n!

1 * 1

(2n + 3 - 0,2 -Aar2-t

(5.3.48)

(5.3.49)

(5.3.50)

(5.3.51)

(5.3.52)

Chapter 5 (§ 5.3) 319

and

n =0 n!

(n + 1,1)

(2n + 2 , 2 - / 3)

(n + 1,1)Xx2-0

Ji-t

(j = 0, , m - l ) ,

(5.3.53)(2n + 4 - /?,2 - /3

for 1 < 0 < 2, while, for 0 < 0 ^ 1, the second solution 2/2(2) is given by

(n + 1,1)

n=0n!

(5.3.54)

The solutions (5.3.52) and (5.3.54) are linearly independent, and when 0 < 0 < 1they provide the fundamental system of solutions to the equation (5.3.51).


y"(x)-\(cD1o^y)(x)-fiy(x)=0 (x > 0; A, (5.3.55)

has the fundamental system of solutions given by (5.3.52) and (5.3.54) with 0 =1/2.

Finally, we find explicit solutions to the equation (5.3.1) with any m € N\{1,2}in terms of the generalized Wright function (5.2.28). It is convenient to rewrite(5.3.1) in the form (5.2.65)

m - 2

(cD%+y)(x) - \{cD%+y){x) - £ Ak(cD^y)(x) = 0 (z > 0) (5.3.56)

fc=O

(m G N\{1,2} ; 0 = a0 < ai < < am_2 < /? < a; A, A), , An-2 G R).

Theorem 5.14 Lei a, /3, aTO_2, , a0 and Z, Zm_i, , Zo G No (m G N\{1,2} )fee suc/i i/iai

0 = ao < ai < am_2 < 0 < a, 0 = Zo Zi ^ Zm_i ^ Z,

I - K a ^ l , Z r o _ i - K / 3 ^ Z m _ 1 , lk-Kak lk (k = , m - 2 ), (5.3.57)

and Zei A, i4o,--- , ^4ra-2 G M. T/ien i/ie solutions to the equation (5.3.56) aregiven by the formulas

Vm-2 1

TTn =0 *o!"-* m-2! lv=0 j

(n + 1,1)

((a — ^)n + 1 + j + El^=o (^ ~ av)kv,a — 0)


(n + 1,1)

( ( a - / 3 ) (n + l ) + l + j + £™=c - <**)k v, a - fi)

m - 2

k=i

(n + 1,1)

((a - P)n + a - ak + 1 + j + Y,™=o{P ~ av)kv, a - /?)

w/ien 0 ^ j ^m-2 ~ 1 with lq ^ j ^ /9+i — 1 /or g = 0, , m — 3;

Tm-2

E

(5.3.58)

n = 0 .i/=0

((a -

-Xx0

(n + 1,1)

(n + 1,1)

/or /TO_2 i ^ ' m -l — 1,

n = 0

1

(n + 1,1)

m - 2

((a - fin + 1 + j +(5.3.60)

for Im-x fL j fL I -1, provided that the series in (5.3.58), (5.3.59) and (5.3.60)are convergent. The inner sums are taken over all ko,--- ,km-2 € No such thatk0 H h km-2 = n.

If a — I + 1 ^ /?, then the functions yj(x) in (5.3.58)-(5.3.60) are linearlyindependent solutions to the equation (5.3.56). In particular, for a — I + 1 > /?,these functions provide the fundamental system of solutions.

Proof. The proof of Theorem 5.14 is carried out along the lines of the proof ofTheorems 5.12 and 5.13 using relations (5.3.3), (5.2.68), (1.10.10) and (5.2.28),which lead to the explicit solutions (5.3.58)-(5.3.60) of the equation (5.3.56). When

Chapter 5 (§ 5.3) 321

a — I + 1 p, then it is directly verified that the relations in (5.3.12) hold in thecase j ^ k, while, for j < k, the formula (5.3.14) is valid for any j , k = 1, , Iexcept for the case a — I + 1 = 0, for which (5.3.36) is true. This means that theWronskian W(0) = 1, which implies the linear independence of the solutions yj{x)in (5.3.58)-(5.3.60). If a - I + 1 > /?, then the relations in (5.3.2) hold, so thatthese functions form the fundamental system of solutions.


{cD%+y) (x) - A ( C^ + y ) (x) - 5 (cDl+y) (x) - »y(x) =0 (x > 0; A . ^ e l )(5.3.61)

with 0 < 7 < / 3 < Q ; ZI - 1 < 7 ^ ZI, l2 - 1 < P l2, I - 1 < a I

(Zi,Z2,Z G N; h ^ Z2 ^ I), has I solutions given by

En=0 \i-\-v=n/

(n + 1,1)

((a - p)n + 1 + j + pi + {fi - 7)1/, a-p)

(n + 1,1)

((a - P)n

(n+1,1)

+ l+j+P

(n + 1,1)

- 7)1/, a -

--y)v,a- P)

' -j)u,a-p)

\x°

when 0 j ^l\ - 1, by(5.3.62)

00 /

= E En=0 \i+i/=n/'

(n + 1,1)

((a - P)n + 1 + j + Pi + (/3 - 7)1/, a - /3)

(n + 1,1)

((a - p)(n + 1) + 1 + j + pi + 09 - 7)1/, a - /3)Aa;0

(5.3.63)


for l\ j I2 — 1, and by

!/;(* ) = E(n + 1,1)

\xa-P , (5.3.64)((a-/9)n + l+ j+ / 9i + (/8-7)i/,a-y8)

/or Z2 J ^ ' - 1-


/ C T Î 5 / 2 \ / \ \ / C T ~\ 3 / 2 \ / \ r / C T ~\ l / 2 \ / \ / \ r \ / - r \ \ f ^- i r a \

\ l ) 7/ I 1 T* 1 A I / / 7 /1 T n I / / 7 /1 T I 111 I 1* 1 I I I T > I I ' A A 1 J ^Z ikr 1

(5.3.65)has three solutions ^(^c); yi(%) and 2/2(2) given, respectively, by (5.3.62), (5.3.63)and (5.3.64) with a = 5/2, j3 = 3/2 and 8 = 1/2.

Remark 5.11 In Sections 5.2.1 and 5.3.1 we constructed the explicit solutionsto linear ordinary homogeneous differential equations with constant coefficientsof the same forms (5.2.1) and (5.3.1) involving Liouvill e and Caputo fractionalderivatives, respectively. Moreover, we gave conditions for when the obtainedsolutions are linearly independent and form fundamental systems of solutions.The results obtained show that these equations, generally speaking, have differentsolutions. This is due to the fact that Liouville and Caputo derivatives differ by aquasi-polynomial term (see (2.4.6)).

5.3.2 Nonhomogeneous Equations with Constant Coeffi-cients

In Section 5.3.1 we applied the Laplace transform method to derive explicit solu-tions to the homogeneous equations (5.3.1) involving Caputo fractional derivatives(2.4.1). Here we use this approach to find general solutions to the correspondingnonhomogeneous equations

MD%1y){x) + Aoy{x) = f(x) (x > 0; 0 < Ul < am) (5.3.66)fe=i

with real coefficients A G E. (k = 0, ,m) and a given function f(x) definedon M.+ . As in the case of the equations with Liouville fractional derivatives, thegeneral solution to the equation (5.3.66) is a sum of its particular solution andof the general solution to the corresponding homogeneous equation (5.3.1). It issufficient to construct a particular solution to the equation (5.3.66). In fact, ifIk — 1 < a/t < Ik and 0 = lo h = = lm, then applying the Laplace transformto (5.3.66), taking (5.3.3) into account and using the inverse Laplace transform,we derive the general solution to the equation (5.3.66) in the form

V(*) = Y, Cjyj{x) + / Gai,..,am{x- t)f(t)dtt (5.3.67)

Chapter 5 (§ 5.3) 323

where yj(x) (j = 1, , lm) are solutions to the corresponding homogeneous equa-tion (5.3.1)

Vj(x) =Y,A"( C~1\ IT- "T T {x)' (5-3-68)

k=q V Lî,-,«»1!JJ/

where lq S J = '?+i ~ 1 f° r Q = ,m — 1, PQl)...; Om(s) is given in (5.2.81),and Cj are arbitrary real constants.

The latter means that a particular solution to the nonhomogeneous equationwith Caputo fractional derivatives (5.3.66) coincides with a particular solutionto the nonhomogeneous equation with Liouvill e fractional derivatives (5.2.79).Therefore from Theorems 5.12-5.14 and Theorems 5.4-5.6 we derive the follow-ing statements.

Theo rem 5.15 Let l - l < a ^ l ( l e N), X £ M. and let f(x) be a given realfunction defined on M.+ . Then the equation

{cD%+y) (x) - Xy(x) = f(x) (x > 0) (5.3.69)

is solvable, and its general solution is given by

pX ' —1

y(x) = (x- tÊâ [X(x - t)a] f{i)dt + Y] CjXjEa j+1 (Xxa), (5.3.70)° jô

where Cj (j = 0, / — 1) are arbitrary real constants.In particular, the general solutions to the equation (5.3.69) with 0 < a 1 and

1 < a 2 have the forms

y(x) = [ {x - t)a-1Ea,a [X(x - t)a] f(t)dt + ClEa {Xxa) (5.3.71)

and

y(x) = f (x - trÊ [X{x - t)a] f(t)dt + cxEa (Xxa) + c2xEa<2 (Xxa),Jo

(5.3.72)respectively, where c\ and c2 are arbitrary real constants.

Remark 5.12 Gorenflo and Mainardi ([304], Section 3) obtained the explicit so-lution to the equation (5.3.69) with A = —1 in a form different from (5.3.70).Their arguments were based on applying the Riemann-Liouville fractional integra-tion operator Ift+ to both sides of (5.3.69) and then using the Laplace transform.They also discussed the solutions obtained in the cases 0 < a < 1 and 1 < a < 2(see also Gorenflo et al. [310]).

Theorem 5.16 Let I-I < a / (I £ N), 0 < 0 < a be such that a -1 +1 /?, letA , / J G I , and let f(x) be a given real function defined on M.+ . Then the equation

{cD»+y)(x)-X{cD^+y)(x)-fiy(x) = f(x) (x > 0; a > 0) (5.3.73)


is solvable, and its general solution has the form

y(x) = / (x - ty-iGaaxAx ~ t)f(t)dt + Y/cjyj(x), (5.3.74)

where Ga^-\^(z) and yj(x) (j = 0, , I — 1) are given by (5.2.97) and (5.3.24)-(5.3.25), respectively, and Cj (j = 0, , I — 1) are arbitrary real constants.

In particular, the general solution to the equation

(cD^+y)(x)-x{cD^+y)(x)^f(x) (x > 0; a > 0) (5.3.75)

is given by

y(x) = / {x - t)a 1Ea-8 aJo

(5.3.76)

where yj(x) (j = 0, - 1) are given by (5.3.27)-(5.3.28).

Theorem 5.17 Let m G N\{1,2} , / - 1 < a ^ I (I G N), and let /3 andcti i > ctm-2 be such that a > /3 > am_2 > > ai > ao = 0 with a — l + 1 /?,and let X,Ao, , Am~2 G K, and let f(x) be a given real function defined on E+.Then the equation

m-2

(cD%y)(x) - X(cD0y)(x) - J2 Ak(

cD^(cD%+y)(x) - X(cD%+y)(x) - ^ Ak(cD%.y)(x) = f(x) (x > 0) (5.3.77)

(m G N \ {1 ,2 } ; 0 = a0 < ax < < am _2 < /? < a; A, Ao, , Am_2 G R).

is solvable, and its general solution is given by

fx 1 —%y(x) = / (x - t)a~1Ga^...tam 2 ta-x{x -t)f(t)dt + 2_.Cjyj(x), (5.3.78)0 . . . . . ^

where Gau... tam_2^,a.tx(z) is given by (5.2.120), y,{x) (j = 0, ,1-1) by (5.3.58)-(5.3.60), and Cj (j = 0, , I — 1) are arbitrary real constants.


y'(x) - X (c£»o+2/) (x) - w(aO = f{x) (x > 0; 0 < /3 < 1; X,/J, G K) (5.3.79)

has its general solution given by

px ( oo „ [" (n +1,1)y(x) = / G^xAx - t)f{t)dt + c{Y] ^-xn 1*1

Jo {n=o n- [{n + 1,1-13)

Chapter 5 (§ 5.3) 325

n = 0 n!

(n + 1,1)

(n + 2 - /?, 1 - ,

Xxl~t (5.3.80)

where Gi!p-\jfJ,(z) is given by (5.2.105) and c is an arbitrary real constant.In particular,

y(x) = ltl/2.x^(x - t)f(t)dt + c

(5.3.81)is the general solution to the following equation:

y'{x) - X (°Dg+y) (*) = f(x) (x > 0; 0 < /3 < 1; A G R), (5.3.82)

while

(n + 1,1)

I , I /2;A, M (^-t)f{t)dtn =0 1,1/2)

Xx1'2

u = 0 n!

(n+1 ,1)Xx1'2 (5.3.83)

(5.3.84)

_ (n +3/2,1/2)

is the general solution to the equation

y'[x) - A (°bo+2/) (x) - ny[x) = f{x) (x > 0; A,


y"[x) - X (C^+J /) (x) - w{x) = f(x) (x>0; 0 < /? < 2; A,/i G R) (5.3.85)


y(x) = [ (x - t)G2,0-,\Ax ~ t)f(t)dt + clVl{x) + c2y2(x), (5.3.86)

where G2,f}-\,»{z) is given by (5.2.111), yx{x) by (5.3.52), j/2(a:) by (5.3.53) and(5.3.54) for the cases 1 < /3 < 2 and 0 < /? ^ 1 respectively, and ci and c2 arearbitrary real constants.


y"(x) - X (°Do+


fx

y{x) = {x- t)G2,i/2-,xAx

Jo

> 0; A, M G R) (5.3.87)

(5.3.88)

where G2)I/2;A,M( ;2 ;)' WIC1) an<l 2/2(2) are given by (5.2.111), (5.3.52) and (5.3.54)with /? = 1/2, and C\ and c2 are arbitrary real constants.



{ c ) = f(x) (x > 0; A . ^ e l ),

(5.3.89)with I - 1 < a ^ I (I 6 N) and 0 < 7 < /? < a, has its general solution given by

fx 1 4—i.f™\ — I /~, f\a~1/'"'' .-, . I T ^ f(i\ri+ -L- \ /> . . , . |T -I rfX ^ QHlU \*L \ I \JJ 01 \ST *y H CH'W," ^ / J \O fUiO ~\ 7 ^7 i / J V / ? \ * I

where G7^<a;\(x) and yj(x) (j = 0, , / - 1) are given by (5.2.129) and (5.3.62)-(5.3.64), respectively, and Cj (j = 0, , I — 1) are arbitrary real constants.

5.3.3 Cauchy Problems for Fractional Differential Equations

The results giving the explicit solutions to fractional differential equations, es-tablished in Sections 5.3.1-5.3.2, can be applied to obtain the explicit solutionsto initial and boundary problems for such equations. Here we illustrate such anapplication to Cauchy problems for fractional differential equations of order a(I - 1 < a ^ /; l e N) with the initial conditions of the form (4.1.59)

The first results for homogeneous equations (5.3.5), (5.3.22) and (5.3.56) followfrom Theorems 5.12-5.14 if we take relations (5.3.12), (5.3.14) and (5.3.36) intoaccount.

Proposition 5.7 Let I — 1 < a ^ I (I 6 N) and A G E. Then the Cauchy problem(5.3.91) for the equation (5.3.5) is solvable, and its solution is given by

1-1

y(x) = Y, bjXjEaJ+1 {\xa). (5.3.92)3=0

Proposition 5.8 Let a > 0 and (3 > 0 be such that I - 1 < a ^ / (I E N),0 < (3 < a and a - I + 1 /?, and let A , j u e i . Then the Cauchy problem (5.3.91)for the equation (5.3.22) is solvable, and its solution has the form

1-1

3=0

or1-2

+ (bl-1-Xb0)yl.1(x) {a-l-l=p), (5.3.94)

where yj(x) (j — 0, - 1) are given by (5.3.24)-(5.3.25).

Chapter 5 (§ 5.3) 327

Proposit ion 5.9 Let m G N\{1,2} , I - 1 < a ^ I (I G N), and Ze£ £ andoil, , am_2 6e sucft i/ioi a > ft > am_2 > > ai > ao = 0 and a — l + 1 ft,and Zei A,J4O)- - - J ^m-2 G M. TAera £/ie Cauchy type problem (5.3.91) for theequation (5.3.56) is solvable, and its solution has the forms (5.3.93) and (5.3.94)in the respective cases a — l — 1 > ft and a — l — 1 = ft, where yj(x) (j = 0, ,1 — 1)are given by (5.3.58)-(5.3.60).

Our next results for nonhomogeneous equations (5.3.69), (5.3.73) and (5.3.77)follow from Theorems 5.15-5.17.

Proposit ion 5.10 Let I - 1 < a g I (I G N), A G 1R and let f(x) be a givenreal function defined on M.+ . Then the Cauchy problem (5.3.91) for the equation(5.3.69) is solvable, and its solution is given by

y(x) = / (x - t)a-lEa,a [\[x - t)a] f(t)dt + Y" bjXjEatj+1 (Xxa). (5.3.95)

Jo j=o

Proposition 5.11 Let I - 1 < a I (I G N) and 0 < ft < a be such thata — I + 1 ^ ft, let X,fj, G E, and let f(x) be a given real function defined on K+.Then the Cauchy problem (5.3.91) for the equation (5.3.73) is solvable, and itssolution has the form

y{x)= / {x-t^Ga^xAx-VfiWt + Ysbjyjix) {a-l-l>ft), (5.3.96)

or

1-2

y(x) = (x- t)a-lGa^xAx ~ t)f(t)dt +

J° j=0(5.3.97)

with a — I — 1 = ft, where Ga^-xtll{z) and yj(x) (j = 0, , / — 1) are given by(5.2.97) and (5.3.24) -(5.3.25), respectively.

Proposit ion 5.12 Let m G N\{1,2} , I - 1 < a g I {I G N), and let ft anda\, , am-2 be such that a > ft > am-2 > > ot\ > Qo = 0 and a — l + 1 ^ ft,let X,Ao,--- , Am-2 G K, and let f(x) be a given real function defined on R+.Then the Cauchy problem (5.3.91) for the equation (5.3.77) is solvable, and itssolution has the form

y(x)= / {x-t)a-1Gai,..,am_2,0,a;x(x-J=0

(5.3.98)or

r>j I rtt 1 ^— I j ™ i ) (T Ft \ \ T '

Jo m


1-2

+ {bi-i - Xbo)yl-1(x) { a - I - 1 = 0 ) , (5.3.99)

j=o

where Gait... ,am~2,l3,a;^(z) an Vj{x) U = 0>'"" >' ~ 1) are given by (5.2.120) and(5.3.58)-(5.3.60), respectively.

Proof. Propositions 5.10-5.12 follow from Theorems 5.15-5.17 on the basis of therespective formulas (5.3.70), (5.3.74) and (5.3.78) and the directly verified relations

° (* = o, , J - 1 ) ,

°) = ° (fc = 0, , Z - 1),

if we also take the relations (5.3.12), (5.3.14) and (5.3.36) into account.

Example 5.24 The Cauchy problem

y'(x)-X (c£> +y) (x)-ny{x) = f(x), y{0) = b G E (x > 0; 0< 0 < 1; A,/i 6 R)(5.3.100)


(1) = (x xn

n=0

(n + 1,1)

(n+ 1,1-/3)Az1"^

n=0n!

(5.3.101)(n + 1,1)

(n + 2 - /3,1 - ,

where Giyp-\tli(z) is given by (5.2.105).


y"(x) - X (cD%+y) (x) - fiy(x) = f(x) (x > 0; 0 < /3 < 1; X,fi G R) (5.3.102)

j/(0) = 60, y'(0) = 6i ( 60 , 6 I G R) (5.3.103)


y(x) = f {x-Jo

n =0

(n + 1,1)

(2n +1,2- /3)

Chapter 5 (§5.4) 329

n = 0

n = 0

n!

(n+1 ,1)

(2n + 3 - /?, 2 - /?)

(n + 1,1)

(2n + 2,2 - /?)Aa;.2-/; (5.3.104)

where G2,i3-\^{z) is given by (5.2.111)

Remark 5.13 Gorenflo and Mainardi ([304], Section 4) used the Laplace trans-form to derive the explicit solutions to the Cauchy problems (5.3.100) and (5.3.102)-(5.3.103) with A < 0 in terms of the inverse Laplace transform C~l in (1.4.2),and discussed the solutions obtained by them. They also indicated that the ex-plicit solutions to Cauchy problem (5.3.102)-(5.3.103) will be different in the cases0 < / ? < l a n d l < £ <2 [see also Gorenflo et al. [310]].

Remark 5.14 Using Propositions 5.10 and 5.11, in Examples 5.24 and 5.25we have presented the explicit solutions to two Cauchy problems (5.3.100) and(5.3.102)-(5.3.103). Using Propositions 5.7-5.9 and 5.10-5.12, we can also derivethe explicit solutions to Cauchy problems for particular cases of the homogeneousequations (5.3.5), (5.3.22), (5.3.56) and of the nonhomogeneous equations (5.3.69),(5.3.73), (5.3.77), considered in Sections 5.3.1 and 5.3.2.

5.4 Mellin Transform Method for Solving Nonho-mogeneous Fractional Differential Equationswith Liouvill e Derivatives

In the following subsections, we present our investigations of the Mellin transformmethod involving the Liouvill e fractional derivatives.

5.4.1 General Approach to the Problems

In this section we apply the one-dimensional direct and inverse Mellin integraltransforms (1.4.23) and (1.4.24) to derive particular solutions to such linear non-homogeneous differential equations as

(x>0; a > 0)k=0

and

YjBkxa+k{Da_+ky){x)=f{x) (z>0; a > 0),

k=0

(5.4.1)

(5.4.2)

with constants Ak, Bk G K (k = 0, , m) and the left- and right- sided Liouvill efractional derivatives (D%+ky) (x) and (D°L+ky) (x) given by (2.2.3) and (2.2.4),respectively.


First we present a scheme for solving such equations by using the direct andinverse Mellin transforms M. and M~1 given by (1.4.23) and (1.4.24), respectively.The Mellin transform method for solving the equations (5.4.1) and (5.4.2) is basedon the following relations:

(Mxa+kD$ky) (s) = r{lT_{1

sZa_k

(Mxa+kD°L+ky) (s) = p(a) ( > < y ) ( s )' ( 5" 4 4)

These formulas, being valid for suitable functions y(x), follow from (2.2.43) and(2.2.46) if we take the property (1.4.28) into account.

Applying the Mellin transform to (5.4.1) and (5.4.2) and using (5.4.3) and

(5.4.4), we obtain

E Ak F( ! _s _ _ M (MV) (s) = (M^ ^ (5-4-5)

andm FCs + rv + A-ll

E B * - L T T ) (Aly)(s) = (A1/)(a)) (5A6)

lk=0 ' Jrespectively. Using the inverse Mellin transform in (5.4.5) and (5.4.6), we derivethe following solutions to the equations (5.4.1) and (5.4.2) in respective forms:

f^ AkT(sr-a-k) (5A7)

and

~ hT{St" + k). (5-4.8)fc=0

By analogy with (5.2.81), we now introduce the Mellin fractional analog of theGreen function:

Gl{x) = (M~x b a M ) {x)> p" {s) = t*/kn^^r (5A9)

/ ( < ^ + f c ) . (5.4.10)K—0

Applying the property (1.4.41) for the Mellin convolution (1.4.39):

M ^ ° ° k ( |) /(*) j ) ) (s) = (Mk)(s)(Mf)(s), (5.4.11)

Chapter 5 (§ 5.4) 331

we present the solution (5.4.7) in the form

/»OO

y(x) = / Gi(t)f(xt)dt, (5.4.12)Jo

and the solution (5.4.8) in the form of the Mellin convolution of G2a{x) and f(x)

Ga(f)/(*)j - (5-4.13)

5.4.2 Equations with Left-Sided Fractional Derivatives

We consider the equation (5.4.1) with m = 1:

fy) (x) + \xa (D%+y) (x) = f(x) (x > 0; a > 0; A £ R). (5.4.14)a+l

Particular solutions to this equation are different in the cases when A / ti + 1and A = n + 1 with n 6 No. In the first case a particular solution to (5.4.14) isexpressed in terms of the generalized Wright function (1.11.14) with p = 1 andq = 2 of the form

1 * 2

M ) 7(a)2Fi{a,l-b;c;-z) (a,b,c,z EC). (5.4.15)

7(6)7(0)

The following assertion can be derived from Theorem 1.5.

Lemma 5.5 The generalized Wright function (5.4.15) is defined for complexz eC such that \z\ < 1 and when \z\ = 1, provided that 91(6 + c — a) > 1.

T h e o r em 5 . 18 If a > 0 and A e l ( A / n + l ; n G N o ) , then the fractionaldifferential equation (5.4.14) is solvable, and its particular solution is given by

= f GitX(t)f(xt)dt,Jo

(5.4.16)

r ( i - A )T(a + 1-A)' — x

(5.4.17)

Proof. (5.4.14) is the same as the equation (5.4.1) with m = 1, A\ = 1 and AQ = A.And (5.4.5) takes the form

(A-a-a)r(l-s)(My) (s) = (Mf) (s). (5.4.18)

r(l-s-a)

Hence the Mellin fractional analog of the Green function in (5.4.9) is given by

T(s - a)(s - a - 1 + A)F(s)

(5-4.19)


and, in accordance with (5.4.12), a particular solution to (5.4.14) has the form

O

V(*) = / GitX(t)f(xt)dt, (5.4.20)Jo

First we show that GlaX{x) = 0 for x > 1. By (5.4.19), we have

(MGlx) (s) = (MGx) (s) (MG2) (s), (5.4.21)

where

(MG1)(s) = ^ - ^ - , (MG2)(s)= \ (5.4.22)

L (s) s — a — 1 + A

The direct application of the Mellin transform leads to the following relations:

{ x~a(l-x)a~1 rs . . -. ( _a_ i+A fl <- r <T 1

n ^ ' t i G 2^ ) = 0 ' x>!0, a; > 1; I u> a; > 1.

(5.4.23)Thus, according to (5.4.11), we have

H G l ( | )<?2( t )y , (5.4.24)and it follows from (5.4.23) that G^ x(x) - 0 for x > 1. Therefore, (5.4.20) yields(5.4.16).

Now we show that G^^x) is given by (5.4.17). By (5.4.19) and (1.4.24), wehave

G1aX(x) = / T i '——-x-sds. (5.4.25)

aM ' 2mJ ( s a l + A) r (a) V 'Since A ^ n + 1 (n £ No), then the pole s = a + 1 — A of the integrand in (5.4.25)does not coincide with any pole Sk = a — k (k G No) of T(s — a). If we choose7 > max[a + 1 — 9l(A), a], then evaluation of the residues at the above poles yields

r ( i - A ) + A - 1

( + l A )r ( a + l A ) tr k\ {X - k - l)T{a - k)K — 0

r ( i -A ) +x_, _ (-i)k r(fc + i-A ) +k

( + 1 A ) ^ A! r(fc 2 A ) r ( f c )

Hence, by (5.4.15) and (1.11.14), GxaX{x) is given by (5.4.17), and Theorem 5.18

is completely proved.

Chapter 5 (§ 5.4) 333

Example 5.26 Equation (5.4.14) with a = 1/2 and A n + 1 (n G No):

) (^) = f(x) (x>0; a > 0; A 6 E) (5.4.26)


y(x) = f G\/2,x{t)f{xt)dt, (5.4.27)Jo

where G\,2 x(x) is given by (5.4.17) with a = 1/2, while

(5-4-28)

(5.4.29)


x2y"(x)+Xxy'(x) = f{x) (x > 0; A # 1; n G No).

Note that, for A n +1 (n G No), (5.4.16) with a = 1 yields the solution (5.4.28),but it is directly verified that (5.4.28) satisfies the equation (5.4.29) with any A 1.

Now we give a particular solution to the equation (5.4.14) with A = n + 1(n G No):

zQ +1 (0S+1!/) (x) + (n + l)xa (D%+y) (x) = f(x) (x > 0; a > 0; n £ No),(5.4.30)

in terms of the psi function tp(z) defined in (1.5.17) and of the Euler constant 7[see Abramowitz and Stegun ([1], formula 6.3.2)]:

= - r ' ( l ) . (5.4.31)

The results will be different in the cases n eM and n = 0.

Theorem 5.19 (a) If a > 0 and n £ N are such that a j= 1, , n, then thefractional differential equation (5.4.30) is solvable, and its particular solution hasthe form

= fJo

(5.4.32)

where

I 1 \ n —\ /n\T(a - n )

0 0

log(z) +

(-1)

n - 1\ ^

k>k^k — a

1

— n+ tp(a --n) + 7

fc=0

(5.4.33)

(b) If n — 0, then the fractional differential equation


xa+1 (DZ+'y) (x) + xa (D%+y) (x) = f(x) (x > 0; a > 0) (5.4.34)

is solvable, and its particular solution is given by

y(x)= [ G^(i)/(xi)di ,Jo

(5.4.35)

\ h. U r

7 l - y , ' , ; M - (5-4-36)

Proof. By the proof of Theorem 5.18, the solution to the equation (5.4.30) has theform (5.4.16) with X - n + 1, which yields (5.4.32) and (5.4.35). To prove theexplicit representations for Gl

an+1(x), we use the relation (5.4.25) with A = n + 1:

The integrand in (5.4.37) has a pole of the second order at sn = a — n andsimple poles at su = a — k for k G No, (k n). If we choose 7 > a, then evaluationof the residues at the above poles, using the relation (1.5.4), yields

k=0, (fc^n)

I T(s-a + n+l)x V\( \ ( 4 ~\ \V( \ a - k)

r'(i ) - r(i)io g(2;) - r(i ) [ E ;

* = 0 ,

From here, in accordance with (1.5.6), (1.5.17) and (5.4.31), Gl,n+1(x) is givenby (5.4.33), which completes the proof of the assertion (a) of Theorem 5.19. Theassertion (b) of Theorem 5.19 is proved similarly.

Example 5.27 Equations (5.4.30) and (5.4.34) with a = 1/2:

x3'2 (Dl!?v) (X) + (n + l ) ^ / 2 [Dl'^y) (X) = f(x) (x>0; a > 0; n e N)(5.4.38)

Chapter 5 (§ 5.4) 335

andx3'2 (D3£y) (x) + x1'2 (ptfy) (x) = f(x) (x > 0)

have their respective particular solutions given by

f1y(x)= / G\/2tn+1{t)f(xt)dt

Joand

(5.4.39)

(5.4.40)

(5.4.41)y(x)= / G{/21(t)f(xt)dt,Jo

where G\ ,2 n+1(a;) and G\ ,2 x(x) are given by (5.4.33) and (5.4.36) with a = 1/2.

Example 5.28 The following ordinary differential equation of order n + m + 1:

= f{x) (x>0; a > 0; n, m G N)(5.4.42)


y{x) =

where

(5.4.43)

\n— 1,~ —m n - 1

n+m —1

E

+

\fc«fe-n-r

ip(m) + 7

fe=0,k\(n - k)(n + m - k - 1)!'

(5.4.44)

Example 5.29 The following ordinary differential equation of order m + 1:

xm+ly(m+l)^ + xmy(m)(x) = f (x > Q. m £ N) (5.4.45)

has a particular solution of the form

el

y(x)= (5.4.46)

where

In particular,

m - l ^k-m

Jois a particular solution to the equation:

x2y"(x)+xy'(x)=f(x) (x > 0).

(5.4.48)

(5.4.49)


Remark 5.15 In Theorem 5.19 we obtained a particular solution to the equation(5.4.30) for all a > 0 and n G No, except for the case when n £ N and a 1, , n.This is due to the fact that the function Gltn+1(x), given by (5.4.33), is not definedfor such a = k (k = 1, , n). This means that the Laplace transform method cannot be used to obtain a particular solution to the following ordinary differentialequation of order k (k = 1, n):

xk+1y(k+1\x) + (n + l)xkyW{x) = f(x) (x > 0; n G N; k = l , - - - , n ) . (5 .4 .50)

Remark 5.16 The solution to the equation (5.4.34) in the form (5.4.35) wasobtained by Podlubny ([682], Example 6.1), but his formula (6.10) for Gx

a t(x) isin error. In fact, log(a;) + ip(a) should be replaced by - log(a;) - ip(a), as it isgiven in the right-hand side of (5.4.36).

5.4.3 Equations with Right-Sided Fractional Derivatives

We consider the equation (5.4.2) with m = 1:

xa+i ( z r+ i y) (a;) + \x<* (Diy) {x) = f(x) {x > 0; a > 0; A G E). (5.4.51)

As in the case of the equation (5.4.14) with the left-sided Liouvill e derivatives, itsparticular solutions wil l be different for a + \ n and a + X = n (n G No). Inthe first case, a particular solution is expressed in terms of the generalized Wrightfunction (5.4.15).

Theorem 5.20 If a > 0 and A G C are such that a + A n (n G No), then thefractional differential equation (5.4.51) is solvable, and its particular solution hasthe form

J~l(^)f, (5.4.52)where

a+X— x (5.4.53)

Proof. (5.4.51) is the same as the equation (5.4.2) with m = 1, Ai = 1, Ao = A.The relation (5.4.6) takes the form

Hence the Mellin fractional analog of the Green function in (5.4.10) is given by

x) = : G U x h < 5 ' 4 5 5)

and, in accordance with (5.4.13), a particular solution to the equation (5.4.51) isgiven by

J~^)f- (5-4.56)

Chapter 5 (§ 5.4) 337

We show that G2aX{x) = 0 for x < 1. By (5.4.55),

(MG2atX) (s) = (MG3) (s) (MGA) (s), (5.4.57)

where

h s) = jrhrx- (5A58)

The direct application of the Mellin transform leads to the following relations:

Thus, according to (5.4.11), we have

O O { ^ ) d i , (5.4.60)

and it follows from (5.4.59) that G2a x(x) = 0 for x > 1. Therefore, (5.4.56) yields

(5.4.52).Now we show that G2

aX{x) is given by (5.4.53). By (5.4.55) and (1.4.24), weget

1 fj+ico V(

Since a + X n (n G No), then the pole s = —A — a of the integrand of(5.4.61) does not coincide with any pole Sk = —k (k G No) of F(s). If we choose7 > max[—a - £H(A), 0], then evaluation of the residues at the above poles yields

(-1)* T(-X-a + k) kr(" A ) ^ fc! r ( l - A - a + fc)r(a-,

Hence, by (5.4.15) and (1.11.14), G2aX{x) is given by (5.4.53), and thus Theorem

5.20 is completely proved.

Example 5.30 The equation (5.4.51) with a = 1/2 and A / n - l / 2 ( n £ No):

x3/2 (p3j2y) {x) + Xx1/2 (p^yj {x) = f(x) (x > 0; a > 0; A £ K) (5.4.62)

has a particular solution of the form

: , (5.4.63)


where G?,2 x(x) is given by (5.4.53) with a = 1/2, while

y(X) - _

~ 1 4-


x2y"{x)-\xy'(x) = f(x) (x > 0; A -1 ).

( 5 A 6 4 )

(5.4.65)

Note that, for A n - 1 (n £ No), (5.4.53) with a = 1 yields the solution (5.4.64),but it is directly verified that (5.4.64) satisfies the equation (5.4.65) with any

Now we give a particular solution to the equation (5.4.51) with X = n — a(n e No):

a+i (n _ a)x« > 0; « > 0; n£ No).(5.4.66)

As in the case of the equation (5.4.30) with the Liouvill e left-sided derivatives, thesolution will be expressed in terms of the psi function ij){z) and the Euler constant7, and the results will be different in the cases n € N and n = 0.

T h e o r em 5.21 (a) If a > 0 and n 6 N are such that a / l , - - , n, then thefractional differential equation (5.4.66) is solvable, and its particular solution hasthe form

2 ( ^ ) , (5.4.67)

where

_ n)log(a;) E

j=0 "

{-l) kxk

— n+ ip(a - n) + 7

(b) If n = 0, then the fractional differential equation

(5.4.68)

a+l 0; a > 0)

is solvable, and its particular solution is given by

where

(5.4.69)

(5.4.70)

Chapter 5 (§ 5.4) 339

Proof. By the proof of Theorem 5.20, the solution to the equation (5.4.66) has theform (5.4.52) with A = n - a, which yields (5.4.67) and (5.4.70). To prove theexplicit representations for G2

a n-a{x), we use the relation (5.4.61) with X = n — a:

X-sds. (5.4.72)7 T F 7 T X d s .7 o o (s + n)T(s + a)

The integrand in (5.4.72) has a pole of the second order at sn = —n and simplepoles at Sk = —k for k G No (k n). If we choose 7 > 0, then evaluation of theresidues at the above poles, using the relation (1.5.4), yields

>-« [s (s + n - l)T{s -a)\ k _ k$n - k)T(a - k)

n (_„)... (_i)r(a-n)

k\(n - k)T(a -k=0, v ' y

From here, in accordance with (1.5.6), (1.5.17) and (5.4.31), Ga,n-a(x) is givenby (5.4.68), which completes the proof of the assertion (a) of Theorem 5.21. Theassertion (b) of Theorem 5.21 is proved similarly.

Example 5.31 Equations (5.4.66) and (5.4.69) with a = 1/2:

(X) +(n~$ x1'2 (£>i/2y) (x) = f(x) {x>0; a > 0; n£N)

(5.4.73)

x3'2 (DTy) (X) - ^x1'2 (D^y) (X) = f{x) (x > 0) (5.4.74)

have their respective particular solutions in the forms

and

(5.4.75)

and

y(x) = I G21/2<_1/2 ( y ) f{t)j, (5.4.76)

where G\,2n__1,2{x) and G\,2 _1,2{x) are given by (5.4.68) and (5.4.71) witha = 1/2.


Example 5.32 The following ordinary differential equation of order n + m + 1:

xn+m+1y(n+m+l\x)+mxn+my(n+mHx) = f{x) {x > 0; n,m G N) (5.4.77)


dtO

2/(z) = (-1)"+™+! / G (5.4.78)

where

~ n ! (m- l ) !

n+m — l

n—1

ip(m) + 7

E (5.4.79)fc=0,

Example 5.33 The following ordinary differential equation of order m + 1:

x™+l y{m+l)( x) + mx™y(Tn)(x) = f(x) (x > Q; m G N) (5.4.80)


-, (5.4.81)

.-fc-1)!Specifically,

. (5.4.82)

(5.4.83)

is a particular solution to the equation (5.4.49). It is clear that this solutioncoincides with the solution given by (5.4.48).

Remark 5.17 In Theorem 5.21 we obtained a particular solution to the equation(5.4.66) for all a > 0 and n £ No, except for the case when n £ N and a 1, , n.This is due to the fact that the function G2

a n-a{x), given by (5.4.68), is not definedfor such a = k (k = 1, , n). This means that the Laplace transform method cannot be used to obtain a particular solution to the following ordinary differentialequation of order k (k = 1, , n):

(x > 0; n G N; Jb = ). (5.4.84)

Remark 5.18 Results presented in Sections 5.4.1-5.4.3 were proved by Kilbasand Trujillo [408]. In Sections 5.4.2 and 5.4.3 we have applied a general approach,developed in Section 5.4.1, to establish particular solutions to the equations (5.4.1)and (5.4.2) with m = 1. Such an approach can be also applied to the equations(5.4.1) and (5.4.2) with m € N\{1} .

Chapter 5 (§ 5.5) 341

5.5 Fourier Transform Method for SolvingNonhomogeneous Differential Equations withRiesz Fractional Derivatives

Our investigation of the Fourier transform method involving the Riesz fractionalderivatives will now be presented in the following subsections.

5.5.1 Multi-Dimensional Equations

In this section we apply the multi-dimensional Fourier transforms to derive partic-ular solutions to linear nonhomogeneous differential equations of the form (5.1.1):

J2 (x) + Aoy{x) = f(x) ( i£R»; n,m£N; 0 < cm < < am),k=l

(5.5.1)involving the Riesz fractional derivatives Daky (k = 1, , m), given by (2.10.23),and constants Aj £ I (fc = 0, , m)

First we present a scheme for solving such an equation by using the direct andinverse Fourier transforms T and T~l given by (1.3.22) and (1.3.23), respectively.The Fourier transform method for solving the equation (5.5.1) is based on therelation (2.10.27):

{JrDay) {x) = \x\a {Fy) [x) {x 6 1"; a > 0). (5.5.2)

Applying the Fourier transform T to both sides of (5.5.1) and using (5.5.2), wehave

{Ty){x) = I Aff)(x). (5.5.3)[z2k=iAk\x\ h +Ao]

Applying the inverse Fourier transform J7~1 in (5.5.3), we obtain a particularsolution to the equation (5.5.1) in the form

HK (55'4>By analogy with (5.2.81) and (5.4.9)-(5.4.10), we now introduce the Fourier

fractional analog of the Green function:

Applying the property (1.3.17) to the Fourier convolution (1.3.33):

/)) (x) := (V f k(x - t)f(t)dt) (x) = {Tk){x){Tf){x), (5.5.6)


we present the solution (5.5.4) in the form of the Fourier convolution of G^ ... Qm (a;)and f{x):

[ G%lt...taJx-t)f{t)dt. (5.5.7)

Next we find the explicit representation for G^ ... am(x)- For this we needthe following auxiliary assertion showing that the Fourier transform of a radialfunction is also a radial function and that it can be expressed in terms of theone-dimensional integral involving the Bessel function of the first kind Jv{z) givenin (1.7.1) [see Samko et al. ([729], Lemma 25.1)].

Lemma 5.6 There holds the relation

°° /2 (5.5.8)ffor any function ip(p) such that the integral in the right-hand side of (5.5.8) isconvergent.

T h e o r e m 5 . 2 2 L e t n , m € N , a i , , a m > 0 , and A o , , A m E l k s u chthat 0 < ai < < am, am > (n — l)/2; A® 0, Am ^ 0. Then the equation(5.5.1) is solvable, and its particular solution is given by (5.5.7), where

lx\(2-n)/2 /-oo ^

(5.5.9)

Proof. By (5.5.4) and (1.3.23), we have

pF iT\ - f 1 P-ixtHt

and (5.5.9) follows from (5.5.8) with <p(p) = [Y,T=i AkPa" + A)]" 1 a nd w i t h x

replaced by —a;. According to (1.7.7) and (1.7.8), for any fixed x G Kn, there existthe following asymptotic estimates of the integrand in (5.5.9) at zero and infinity:

|a; \("~ 2 ) /2

n\x\

(5.5.10)

(5.5.11)

By (5.5.10) and (5.5.11), and the hypotheses of Theorem 5.22, the integral in theright-hand side of (5.5.9) is convergent, and thus Theorem 5.22 is proved.

Chapter 5 (§ 5.5) 343

Corollar y 5.11 If n G N, a > (n - l ) /2, A £ I (A 0), then the differentialequation

(Day) (x) + Xy(x) = f{x) {x G K") (5.5.12)


y(x)= f GFa{x-t)f{t)dt, (5.5.13)

|T | ( 2 -n ) /2 />oo -I

G* {x) = i f c j ^ i wTxf'^n-M'Wp. (5.5.14)

Example 5.34 If n G N, then the equation

(pz/2y) (x) + \y(x) = f{x) (x 6 l " ; A e R) (5.5.15)


V(x) = f G /2{x - t)f(t)dt, (5.5.16)

where G /2(x) is given by (5.5.14) with a = 3/2.

When m £ N\{1} , 4o = 0 and Ai 0, the equation (5.5.1) takes the form

* = i

k(Daky)(x) = f(x) ( x £ l " ; n G N; m G N\{1} ; 0 < c*i < < am).

(5.5.17)

Theorem 5.23 Lein G N, m G N\{1} , a i , - -- , am > 0, andA-y,--- , Am G K 6eSMC/I i/iai 0 < ai < < am, ai < n, am > (n —1)/2; ^4i 0, Am 0. T/ieni/ie equation (5.5.17) is solvable, and its particular solution is given by (5.5.7),where

^ = ~îrI E ^ ^ I H - ] " ' 7 ' ^ 1 - ' * " * - (5' 518)

Proof. The proof of Theorem 5.23 is similar to that of Theorem 5.22, using thefollowing asymptotic estimates:

(5.5.19)

1 /*?

U ^ ^ ^ ^ l K 1 ) / 2 ] ^ (p->oo). (5.5.20)


Corollar y 5.12 If n G N, a > 0 and /3 > 0 are such that /? < a, a > (n - l ) /2,/3 < n and X G ffi (A 0), i/ien the differential equation

(Day) (x) + X (D0y) (x) = f(x) {x G Rn) (5.5.21)


V(x)= f Gl0(x-t)f(t)dt, (5.5.22)

U (2-") /P J

Remark 5.19 It follows from the condition am > (n — l)/2 that the explicitsolutions to the equations of the form (5.5.1) and (5.5.17) with the Riesz derivativeof order am = 1/2 can be obtained only in the one-dimensional case n = 1.

5.5.2 One-Dimensional Equations

Substituting n = 1 in the results of Section 5.5.1, we derive particular solutions tolinear nonhomogeneous differential equations of the forms (5.5.1) and (5.5.17):

m

Âk{T>aky)(x)+A0y{x) = f{x) ( i e l ; m £ N; 0 < e*i < < am),fc=i

(5.5.24)

AkCDa-y)(x) = f(x) ( i £ l ; m e N\{1} ; 0 < ax < < am), (5.5.25)fc=i

involving the one-dimensional Riesz fractional derivatives Daky given by (2.10.23)with n = 1:

( D Q * y ) (x) = , .; . I \ ' a ' ^dt (I > ak; k = l,---,m), ( 5 . 5 . 2 6)«i(',afc) JR \t\ ~f k

where di(l,ak) is given by (2.10.25) with n = 1 and a = ak- We note thatthe relations (5.5.9) and (5.5.18) for the Fourier analogs of the Green functionG^ ... Qm (a;) are simplified in view of the familiar relationship

/ 2 \ 1 / 2

J-i/2{z)= (—J cos*. (5.5.27)

The first result follows from Theorem 5.22 and (5.5.27).

Theorem 5.24 Let m G N, cei, , am > 0 and AQ, , Am G C be such that0 < ai < < am, Ao / 0, and Am ^ 0. Then the equation (5.5.24) is solvable,and its particular solution is given by

y{x ) = / GZlt...taJx-t)f(t)dt, (5.5.28)J -00

Chapter 5 (§ 5 . 5 ) 3 4 5

£,..., am (*) = I [ [E'-LiAkl lak+Ao] «»(P\*\)*P- (5-5.29)

where

<£,..., am (*) = I [Coro l la r y 5.13 / / a > 0, A g i ( A ^ O ) , then the differential equation

(T>ay)(x)+\y(x)=f(x) ( i£ l ) (5.5.30)


O

y(x)= GFa(x-t)f{t)dt, (5.5.31)

J — OO

(5.5.32)


(p1/2y) {x) + \y(x) = /(a;) ( i £ i ; A G E) (5.5.33)


y{x)= f GF/2{x-t)f{t)dt, (5.5.34)

J — OO

where G^,2(x) is given by (5.5.14) with a = 1/2.

Our next result follows from Theorem 5.23 and (5.5.27).

T h e o r em 5.25 Let m e N \ {1 } , Qi, , am > 0, and Ai,- , Am e l k suchthat 0 < ai < < am, cti < 1; A\ 0, Am 0. Then the equation (5.5.25)is solvable, and its particular solution is given by (5.5.28), where

<£,..., . » = \ [ {^JAklp]ak]cos(P\x\)dp. (5.5.35)

Corollar y 5.14 If a > 0 and (3 > 0 are such that (3 < a and /3 < 1 and A G E(A 0), i/ien t/ie differential equation

(Day) (x) + A (D"i/ ) ( i ) = f{x) (x G E) (5.5.36)


/.OO

V(x)= GZs(x-t)f(t)dt, (5.5.37)J — OOOO



(p3/2y) (x) + A (T>l/2y) (x) = f(x) (xGR, Ae l) (5.5.39)


O

y{x) = G%/2A/2{x-t)f(t)dt, (5.5.40)J — OO

where Gf/2 1/2{x) is given by (5.5.37) with a = 3/2 and j3 = 1/2.

Remark 5.20 The Fourier and Mellin transforms were used in [370] and [372]to present a scheme for solving linear one-dimensional nonhomogeneous equationswith constant coefficients of the form (5.1.1) and (5.4.1)-(5.4.2) involving the Liou-vill e fractional derivative D^_y and D°!_y given in (5.3.3)-(5.3.4), and the Hadamardfractional derivatives T>Q+y and Vty given in (2.7.9)-(2.7.10), respectively.

Chapter 6

PARTIAL FRACTIONALDIFFERENTIALEQUATIONS

The present chapter is devoted to the results for partial fractional differential equa-tions. We give a survey of results in this field and apply Laplace and Fourier inte-gral transforms to construct solutions in closed form of Cauchy type and Cauchyproblems for fractional diffusion-wave and evolution equations.

6.1 Overview of ResultsIn this section we consider methods and results for partial differential equationsand separately present the so-called fractional diffusion equations which arise inapplications. We also present investigations of more general abstract differentialequations of fractional order. More detailed information about some of theseresults can be found in a survey paper by Kilbas and Trujillo ([408], Sections 5-7).

6.1.1 Partial Differential Equations of Fractional Order

Gerasimov [279] derived and solved fractional-order partial differential equationsfor special applied problems. He studied two problems of viscoelasticity describingthe motion of a viscous fluid between moving surfaces, and reduced these problemsto the following two partial differential equations of fractional order:

and

347


with an unknown u = u(x,t), the given constants g and k, and a Caputo partialfractional derivative with respect to t of the form (2.4.53)

0 < a < 1 , ( , L 3 )

Zhemukhov ([927], [928]) studied the Darboux problem for the following second-order degenerate loaded hyperbolic equations

ymuxx-uyy = 2_âk(x,y)D%ltXf [u(x,O),ux(x,O),uxx(x,O)], (6.1.4)

f t ( \ T-'IrrW 1 — 0 (f\ 1 K \

k=l

d m , y u x x u y y + aux+buy + cu + Yj

involving the partial Riemann-Liouville fractional derivatives with respect to xdefined in (2.9.9)

( £ ) W + 1 ^ [ ^ (« > o; y > o; a > o),(6.1.6)

[a] and {a} being the integral and fractional parts of a. He reduced these prob-lems to equivalent nonlinear Volterra integral equations of the second kind andproved their solvability by the method of the contraction mapping principle, andthen solved Cauchy problems for these Volterra equations in order to obtain thesolutions u(x,y) to the Darboux problems considered.

Conlan [145] considered the following nonlinear partial differential equation

%+tXD$+iyu) (x,y) = [(6.1.7)

and used the Banach contraction mapping principle; he proved local theoremsfor the existence and the uniqueness of the solution u(x,y) of the correspondingintegral equation.

Fedosov and Yanenko [255] studied the following half-integer order partialdifferential equation

ak (Dk+ '?xD^k)'\) (x, y) = f(x, y), (6.1.8)

where ak {k = 1, ,ra) are constants and DkJ x and D™~ " 2 are the partialLiouvill e fractional differentiation operators of the form (2.2.3) defined by (2.9.11)with n = 2 and ai = —oo:

(6.1.9)

Chapter 6 (§ 6.1) 349

and similarly for V®+yu. They proved that the operator in the left-side of (6.1.8)can be represented as the composition of n invertible operators of the formT^+2

X + ^jT^+2y, where \j are the roots of a certain characteristic polynomial.

In particular, in the case n = 1, a0 = 0 and oi = 1, they gave the relation for thegeneral solution of the equation (6.1.8), which can contain an arbitrary function,and investigated the case n — 2 in detail [see Samko et al. ([729], Section 43.2,Note 42.13)]. Fedosov and Yanenko [255] also discussed a number of boundaryconditions for the uniqueness of the solution u(x,y) of the equation (6.1.1). Theystudied a similar problem in Fedosov and Yanenko [254] in connection with a mixedproblem for the wave equation in cylindrical polygonal domains.

Biacino and Miseredino [83] investigated the Dirichlet problem on a rectangle[a, b] x [a' x b'] for a two-dimensional differential operator

Lu = Eu + Pu, (6.1.10)

where E is a uniformly strongly elliptic operator of the fourth order, while Pis an operator that contains a special partial derivative Dau of fractional ordera = (ai, a2) (t*j > 0, j = 1,2), \a\ = a\ + a2, and proved an alternative theoremand an existence and uniqueness result for the Dirichlet problem for the operator(6.1.10). Biacino and Miseredino [84] obtained similar statements while study-ing the Dirichlet problem on an interval [a, b] for a one-dimensional differentialoperator (6.1.10).

A Cauchy type problem for the following partial fractional differential equation

{D^tu) (x,t) = ( ^+ ) X u) (x,t) (1 g a g 2; 1 (3 2) (6.1.11)

with

*M«) (* *) = ( | ) f i T W) SI JT^(6.1.12)

was considered by Fujita [267]. He proved the existence and the uniqueness ofa solution u(x,t) and a representation for such a solution. In the case when/3 = 2 and 1 ^ a ^ 2, he proved the positivity of the fundamental solution andstudied the asymptotic behavior of the solution u(x,t). The results obtained canbe interpreted as a phenomenon between the heat equation (a = 1; /3 = 2) andthe wave equation (a = 0 = 2).

In connection with problems in fractals, the fractional partial differential equa-tions of the forms

(D tu) (x,t) = -Ax-P^^- (o < a % i ; A > 0; /? ^ o) (6.1.13)

and

(D tu) (x,t) = -A p ^ £ ] ( ^(6.1.14)


with the Riemann-Liouville partial fractional derivative (6.1.12), were consideredby Giona and Roman [282] and Roman and Giona [715], respectively. Applying theLaplace transform (1.4.1) with respect to t, they reduced these equations to certainordinary differential equations for U(x,p) = (Ctu(x,t)) (p) = Jo°° u(x,t)e~ptdt,obtained explicit solutions U(x, t) of these equations under a certain normalizationcondition, and indicated that the solutions u(x,t) of (6.1.13) and (6.1.14) can beobtained by using the inverse Laplace transform C~l with respect to t.

Kolwankar and Gangal [436] considered the following partial fractional differ-ential equation

T>?u(x,t) = T{a^1)Xn(t)dUdi^t) (0 < a g 1; x G R; t > 0), (6.1.15)

where use is made of a special fractional derivative Dff(t) denned by

dx) JtT{n - a) \dx) Jt (y - t)a~n+ l

where n = [a] + 1 and xn(t) is the characteristic function of a set fi C R+:Xn{t) = 1 if t G Cl and xn(t) = 0 if t £ fi. They obtained the explicit solution of theCauchy problem for the equation (6.1.16) with the initial condition u(x, 0) = S(x),5(x) being the Dirac delta function. Kolwankar and Gangal [436] indicated thatthe partial fractional differential equation (6.1.15) is an example of more generalfractional differential equations of the form

T>?u{x, t) = L(x, t)u(x, t) (6.1.17)

with the special operator L(x,t), which are called the local fractional Fokker-Planck equations and which appear while studying a phenomenon taking place infractal space and time.

Atanackovic and Stankovic [38] and Stankovic [801] used the Laplace transformin a certain space of distributions to solve a system of partial differential equationswith fractional derivatives, and indicated that such a system may serve as a certainmodel for a visco-elastic rod.

Kilbas and Repin [388] studied the following equation of mixed type

^ > u) (,,») = 0 (, > 0), ( - , ) - ^ - ^ = 0 to < 0)(6.1.18)

with m > 0 and the partial Riemann-Liouville derivative (6.1.12) of order0 < a < 1 in a special domain of M2. They investigated a non-local problem forthis equation with boundary conditions involving generalized fractional integro-differential operators with the Gauss hypergeometric function (1.6.1) in the kerneland gave conditions for the uniqueness and existence of a solution of the consideredproblem. Earlier, Kilbas and Repin [387] established similar results for an analogof the Bitsadze-Samarskii problem for the equation of mixed type with the partialRiemann-Liouville fractional derivative.

Chapter 6 (§ 6.1) 351

Nakhushev [615] and Nakhushev and Borisov [619] studied certain problemsfor the following degenerate integro-differential equation:

n

uyv ~ {-y)muxx + aux +buy + Y, akD$.tiu(x, 0)+cu = d, (6.1.19)fe=i

and for the following so-called loaded parabolic equation:

n m

uv - kuxx + 'Y^'Yâl{x,y)D%'+Jkj{x,y)u{x:>,y)] = f(x,y), (6.1.20)

where D%+ û(x,0) are the partial Riemann-Liouville fractional derivatives or in-tegrals of order a* defined by (2.9.9) and (2.9.3), respectively, with n = 2.

To conclude this section, we note that the equations (6.1.13) and (6.1.14) fora = 1/2 were first obtained by Oldham and Spanier in [641] and [642], respectively,by reducing a boundary-value problem involving Fick s second law in electroan-alytical chemistry to a formulation based on the partial Riemann-Liouville frac-tional derivative Do+ x in (6.1.6). Oldham [640] and Oldham and Spanier ([643],Chapter 11) gave other applications of such equations for diffusion problems.

6.1.2 Fractional Partial Differential Diffusion Equations

It was indicated at the end of the previous section that Oldham and Spanier ([643],Chapter 11) first considered the partial fractional differential equations arising indiffusion problems. A series of papers was devoted to investigating such so-calledfractional diffusion equations, and in most of them formal explicit solutions tocertain boundary- and initial-value problems for the considered equations wereobtained. Wyss [898] studied the following fractional differential equation

^ l l ^ | M {x > o; 0 < a g 1; A > 0), (6.1.21)

where a;â~1/r(a) is understood as a distribution in the space V of generalizedfunctions (see Section 1.2). He investigated two problems for the equation (6.1.21)with

u{x, 0) = 0, u(0, T) = b (t > 0) (6.1.22)

When 6 = 0 and b = —1. He sought a solution u(x, t) to these problems in theform

u(x,t) = f(y), y = ra'2x, (6.1.23)

and reduced (6.1.21)-(6.1.22) to the following one-dimensional problem:

= (II?2/atlf)(y) (y > 0), /(0) = b, /(oo) = 1 + 6, (6.1.24)


where IZ%/a 1/ is the Erdelyi-Kober type fractional integral of the form (2.6.4).

Applying the Mellin transform (1.4.23) to the equation in (6.1.24) and takinginto account the initial conditions in (6.1.24), he arrived at the following relation:

Applying the inverse Mellin transform (1.4.24), Wyss [898] obtained the fol-lowing solutions to the above two problems in terms of the if-function (1.12.1) intheir respective forms:

!2,3

and

uyx, i) — n(l,l),(l,a/2)

(0,1), (1/2,1/2), (1,1/2)

Schneider and Wyss [746] considered the equation

u(x,t) = J2h{x)tk + - ! , / f . ^ - t (« - K a /; / = 1,2), (6.1.28)k=0 ' ° ^ '

where x 6 I " (n £ N), t > 0, fu(x) are the initial data, i.e.,

£ j u(x,t)\t=0 = fk(x) ( 0 ^ k ^ l - l ; 1 = 1,2), (6.1.29)

and A is the Laplacian (1.3.31). (6.1.28) represents the fractional diffusion andthe wave equation when I = 1, 0 < a ^ 1 and / = 2, 1 < a ^ 2, respectively.Applying the Laplace transform (1.4.1) with respect to t in (6.1.28), they reduced(6.1.28) to the form

j- i

AU(x,p)-paU(x,p) = -J2h(x)P~k~1 (' = 1,2), (6.1.30)k=0

where U(x,p) = (Ctu(x,t))(p). Applying the inverse Mellin transform (1.4.24) tothe general solution of (6.1.30), Schneider and Wyss [746] obtained the solutionto (6.1.28) in the form

i-i .

«(*,* ) = y ) / G%(\x-y\,t)fk(y)dy, (6.1.31)fc=o-'RB

1 /O

here y £ Rn, \x — y\ — [^2 =1(xk — 2/fc)2] , and the analogies of the Greenfunctions G^(r,t) (0 ^ k 5; ^ — 1; I = 1,2) are expressed via the H-function(1.12.1) by

Chapter 6 (§ 6.1) 353

(n/2-jfe/a,l/2),(l-Jfe/a,l/2) J(6.1.32)

Schneider and Wyss [746] also obtained an explicit solution to the fractionaldiffusion equation (6.1.28) with 0 < a ^ 1 in the half space D = E""1 x 1+ withthe boundary 3D = Wn~1 x {0} , supplemented by a special boundary condition.

For 0 < a ^ 1 and m = 1, Schneider [745] gave a more elegant solution of(6.1.28):

u(x,t) = f(x) + - L f f"^* ^ (0 < a 1), u(z,0) = f(x) (6.1.1 W ./o \l ~ s)

33)

for x e R" (n G N) and t > 0. Applying the Fourier transform (1.3.22) with respectto x and the Laplace transform (1-4.1) with respect to t, he reduced (6.1.33) tothe form

co-l

(TxCtu{x,t)){y,p) = ; + 2{Ff){y), y2 = yy. (6.1.34)

Taking the inverse Laplace and Fourier transforms (1.4.2) and (1.3.23) to(6.1.34), Schneider [745] obtained the solution of the problem (6.1.33) in the form(6.1.31)

u(x,t)= [ Ga(x-y,t)fk(y)dy, (6.1.35)

where

12(6.1.3

Wyss [898] and Schneider and Wyss [746] also investigated the properties of thefractional analogs of the Green function given by (6.1.32) and (6.1.36). Schneiderand Wyss [746] indicated a physical application of (6.1.28) to special types ofporous media as pointed out by Nigmatullin [623], and Schneider [745] constructeda stochastic process called grey Brownian motion, based on a probability measurecalled grey noise, and showed that such a process is related to fractional diffusionas ordinary Brownian motion is related to ordinary diffusion. We also mentionde Arrieta and Kalla [159] who treated some fractional diffusion equations of theforms (6.1.21) and (6.1.28) by using the Laplace transform (1.4.1) and the Mellintransform (1.4.23) of the .ff-function (1.12.1) and of special cases of the generalizedhypergeometric function pFq(ai, , ap; bi, , bq; z) given by (1.6.28).

Kochubei [427] considered the following Cauchy type problem:

(x,t) = Lu{x,t) (0 < a < 1; i e l " ; 0<t^T), (6.1.37)


u{x,O) = f{x) (xeM.n), (6.1.38)

where (cDla'u) (x,t) is the Caputo regularized partial fractional derivative withrespect to t:

(cD{a)u) (x,t) = — — / , ' S \ S - U^X: ' , (6.1.39)V * / v ' ; r(l - a) [dt Jo (t-s)a ta J ' v '

and L is the second-order elliptic differential operator in n variables:

Qk+j

where the summation is taken over repeated indices. Kochubei [427] proved thatthe Cauchy type problem (6.1.37)-(6.1.38) has a unique solution u(x,t) under theappropriate assumptions on f(x) and the coefficients ajy, a; and a of the operatorL, and also indicated the conditions when this problem with f(x) = 0 has anonzero solution. In the case when L = A is the Laplacian (1.3.31), he obtainedthe explicit form for the fundamental solution u(x,t) in terms of the /^-function(1.12.1), and showed its integralibility in t over M.n. The fundamental solution ofthe Cauchy problem for the equation (6.1.37), in which L is a uniformly ellipticoperator, was constructed in terms of the i?-function (1.12.1) by Eidelman andKochubei [219] .

Fujita [269] studied the following two-dimensional equation (6.1.33) with1< a < 2

u(x, t) = f(x) + - f ^ ^ l (1< a < 2; x G E; t > 0) (6.1.41)1 \a) Jo \l ~ ~)

in a certain subspace of the Schwartz space of rapidly decreasing functions (seeSection 1.2). Using the Fourier transform, he obtained its solution u(x,t) asfollows:

u(x,t) = - [ Pa(\x-t\,T)f(r)dT, (6.1.42)

Pa(X,t) = ~ f exp [-i|r|2/«e-7T«5n(r)i/2l g - t e r = 2 - l . (6.1.43)

Fujita [269] also showed that the fundamental solution of (6.1.41) takes itsmaximum value at x = at

a/2 for each t > 0, and hence these points propagatewith finite speed as in the wave equation

Chapter 6 (§ 6.1) 355

in the case a = 2, and that the support of u(-, i) is not compact for each t > 0 evenif / has compact support and thus, in that sense, there is an infinite propagationspeed as in the case a = 1 of the heat equation

at ox*

Fujita [268] obtained the solution to the equation

with K a ^ 2 , i £ l , and t > 0, in the form

fx+Ya(t) 1Ya(t))+f(x-Ya(t)) + g(t)dt\, (6.1.47)

x-Ya(t)

where Ya(t) is a continuous, non-decreasing and nonnegative stochastic processwith Mittag-Leffler distribution of order a/2, and E stands for the expectation.Using the Fourier transform and probability methods, Fujita [270] proved theenergy inequalities for the integro-differential equations of the form (6.1.41) and(6.1.46) which correspond to the energy inequality for the wave equation (6.1.44).

The equation (6.1.21) belongs to the so-called fractional diffusion equations[see Nigmatullin [622], [623] and Westerlund [874]]. In the simplest case of theone-dimensional diffusion, such an equation is given by

{D%+ttu){x,t) = \ 2 d 2 u } X ; l ) ( a > 0 ; A > 0 ) , (6.1.48)

with the partial Riemann-Liouville fractional derivative (D%+tu){x,i) denned in(6.1.12). When a = 1 and a = 2, (6.1.48) coincides with the heat (diffusion)equation (6.1.45) and the wave equation (6.1.44), respectively.

Mainardi [514], [517] [519] studied an equation of the form (6.1.48):

(cD tu)(x,t) = A 2 ^ | j M (Q > 0; A > 0), (6.1.49)

with i e l and t > 0 for 0 < a ^ 2. Here (CDQ+<t u)(x,t) is the Caputo partialfractional derivative with respect to t of order a > 0 (I — 1 < a < 1; I g N) defined,for any x £ M. and t > 0, by

a u)(xt) 1 [* {#u{x,T)/dT>)dTo+'tU}{X't}-T(n-a)Jo (t-r)°-'+i ' ( 6 5° }

(^0+tU){x,t) = ^ - . (6.1.51)

Mainardi [514] investigated the initial-value problem

c d U ^ l ) ( i £ l ; f > 0 ; 0 < a < l ) , (6.1.52)


u(x,O) = f(x) (xeM), lim u(x,t) = 0 (t > 0). (6.1.53)X—

Using the method suggested by Schneider [745] and applying the Laplace trans-form (1.4.1) with respect to t and the Fourier transform (1.3.1) with respect to x,he reduced the problem (6.1.52)-(6.1.53) to a form like (6.1.34). And, by applyingthe inverse Laplace and Fourier transforms, he found the solution u(x,t) of thisproblem in the form (6.1.35):

O

u{x,t)= Ga(x-T,t)f{r)dT. (6.1.54)J — OO

Here a fractional analog of the Green function Ga(x, t) is given by

G°{x,t) = l-jEa (-A2yV) cos(yx)dy = ' \ ( |, 1 - |; M*- "

(6.1.55)in terms of the integral of the Mittag-Leffler function (1.8.1) and the Wright func-tion (1.11.1).

Mainardi ([517], [519]) used the same approach to find the fundamental solu-tions of the Cauchy problem and of the so-called signaling problem for the equation(6.1.49) with 0 < a ^ 2 in terms of the Wright function (1.11.1) with z = \x\t~a/2,which reduces to the Gaussian function in the case a = 1 of the classical diffusionequation. For 0 < a £ 1, the initial conditions of the Cauchy problem are givenby (6.1.53) with f(x) = 5(x), while the initial conditions of the signaling problemhave the form

u(x,0) = S(x) (x > 0); u(0,t) = g(t), u(+oo,t) = 0 (t > 0). (6.1.56)

When 1 < a ^ 2, the condition

jû(z, t ) | t=o=0 (6.1.57)

must be added to the relations (6.1.53) and (6.1.56).

Podlubny [682] also applied such a technique to obtain the solution of theCauchy type problem for the equation (6.1.48) with the following initial conditions:

lim u(x,t)=0, (6.1.58)\x\—>oo

and for the equation of Schneider-Wyss type

with the initial condition (6.1.53), in the form (6.1.54), where

Chapter 6 (§ 6 . 1 ) 3 5 7

which, for a = 1, is reduced to the classical expression

G(x,t) = -L(,rt)-1/2exp ( - ^ ) . ( 6.L61)

Gorenflo and Mainardi [305] considered the fractional diffusion equation (6.1.49)with A = 1 in the quarter-plane R++ = {{x, t) e E2 : x > 0, t > 0}

^ * ^ ((M) £ E+ +; 0 < a 2), (6.1.62)

with the initial and boundary conditions of the form

u(x,0) = 0, u(0,t) =4>{t) (ftO), (6.1.63)

for 0 < a ^ 1, while, for 1 < a ^ 2,

«(*, 0) = û(a;, 0) = 0, JU(O, t) = ^(«) (* 0). (6.1.64)

They applied the Laplace transform to obtain the explicit solution to theseproblems in the form (6.1.54), where a fractional analog of the Green functionGa(x,t) is the inverse Laplace transform with respect to t of exp (-xta/2) and—£~a/2exp (—xta/2), respectively. In this regard we indicate the papers byMainardi [519, 520], Gorenflo and Mainardi [302], Gorenflo, Mainardi and Srivas-tava [310], Mainardi and Tomirotti [535] and Mainardi and Paradisi [530], wherevarious relations of such a function Ga(x,t) with special functions and extremalstable probability densities are worked out. We also mention Luchko and Gorenflo[506] who considered scale-invariant solutions for some diffusion partial diferentialequations of fractional order, and Buckwar and Luchko [109] who investigated aninvariance of such equations relative to a certain Lie group.

Equations (6.1.48) and (6.1.49) are the simplest examples of the approachto "fractalization" and unification of the classical diffusion and wave equationssuggested by Nonnenmacher and Nonnenmacher [636]. Another approach givenby Metzler et al. [581], in the simplest case, leads to the following partial fractionaldifferential equation:

(D%+ttu)(x,t) = Ax-P- Lp^^Pj (a > 0; A > 0; /? 0). (6.1.65)

Agrawal [7] applied the direct and inverse Laplace transforms to obtain the so-lution, in closed form, of the Cauchy problem (6.1.53) and of the signaling problem(6.1.56) for the following fourth order fractional differential equation

(cD°+tu){x,t) + fr2^"^ = 0 (x e 1; t > 0; 0 < a 2), (6.1.66)


with b £ M. and the Caputo partial fractional derivative (6.1.50).

Hilfer [344] obtained a fundamental solution of the following multi-dimensionalequation of the form (6.1.48) of order 0 < a < 1

(D%+ttu)(x, t) = X (Axu) {x, t) ( i e i » ; t > 0; A > 0) (6.1.67)

with the Laplacian (1.3.31) in terms of the .H^'"-function (1-12.1).

Chechkin et al. [135] studied the diffusion-like equation with the Caputo timepartial fractional derivative of distributed order, proved the positivity of its solu-tion, and established the relation of the considered equation with the continuous-time random walk theory.

Kilbas et al. [385] applied the direct and inverse Laplace and Fourier transformswith respect to x £ M. and t > 0 to find the explicit solution to the Cauchy problem

(CD%+ tu)(x, t) = \{D0_ xu)(x,t) ( i e l ; t > 0; 0 < a < 1; /? > 0; A £ R \ {0} )(6.1.68)

lim u{x,t)=0, u{x,0+)=g(x) (6.1.69)X—

with the Caputo and Liouvill e partial fractional derivatives denned for x £ K andt > 0 by

r^- / ' ,a dr ( 0 < a < l) (6.1.70)— OL) at Jo (t — Tja

and

OX J J— <x \ x '

respectively. Such a method was also used in Kilbas et al. [386] to construct theexplicit solution for the following Cauchy type problem

(£>£+ tu){x,t) = \{D0_ xu){x,t) (x G E; t > 0; 0 < a 1; /? > 0; A £ E\{0} ,(6.1.72)

lim u(x,t)=0, (D%7\u)(x,0+) = g(x) (6.1.73)x— T l

with the Riemann-Liouville partial fractional derivative (D$+tu)(x, t) with respectto t, given by (6.1.12). Applications were given in the above two articles to theanalysis of diffusion mechanisms with internal degrees of freedom while studyingthe square root of the standard linear diffusion equation in one space dimension,ut — uxx = 0 [see Vazquez [842] and Vazquez and Vilela Mendes [845]].

Voroshilov and Kilbas ([861], [862]) used the direct and inverse Laplace andFourier transforms to derive explicit solutions of the Cauchy type problems for the

Chapter 6 (§6.1) 359

one- and multi-dimensional homogeneous fractional diffusion equations (6.1.49)and (6.1.67) with 0 < a < 2, with the Cauchy type conditions (6.1.58) in the case0 < a ^ 1, while, for 1 < a < 2,

( £ > ? 4» (x, 0+) = fk(x) (xGRn;l<a<2;k = l,2). (6.1.74)

Here ( D Q ^ U) {x,t) is the partial Riemann-Liouville fractional derivative de-fined by (6.1.12), while (D%~2

tu) (x,t) = (l£+"u) (x,t) is the partial Riemann-Liouvill e fractional integral of the form (2.9.3)

^ L («>, , ). (6.1.75)

In particular, for 0 < a < 1, the results obtained coincide with those provedby Podlubny [682] for the Cauchy type problem (6.1.48)-(6.1.58).

Mainardi et al. [526] considered the fractional diffusion equation of the form(6.1.68) with the Caputo partial fractional derivative (6.1.50) of order 0 < a ^ 2

in which the Liouville partial derivative yD^_ XU J (X, t) is replaced by the so-called

Riesz-Feller partial fractional derivative [Dg.xu\ (x,t) defined, for 0 < /? 2 and

\0\ min[/3,2-/3] via the Fourier transform, by

(a, t) = |CT|/y(-g»(-)<V2 {TxU) (CTj t) ( , £ ! ; ( > o). (6.1.76)

They established the explicit solution of the form (6.1.64) for the equationconsidered with the initial condition (6.1.69), where g(x) = S(x), in the case0 < fi < 1 and with the additional condition û(x,0) — 0 when 1 < /? 51 2.Another representation for the fractional Green function Ga(x.t) in terms of thei?3'3-function was given by Mainardi et al. [529].

6.1.3 Abstract Differential Equations of Fractional Order

In this section we present investigations of certain abstract fractional differentialequations. First of all, we note that Berens and Westphal [79] first considered theabstract Cauchy type problem for the Riemann-Liouville fractional differentiationoperator Dfi+ (0 < a < 1) defined in (2.1.8), with a certain domain G(Da;p)and range in I/p(R+) (1 a{x) + D%+wa(x) = 0 (x > 0; 0 < a < 1), (6.1.77)(XX

Yvm+ \\ua{x) - fo\\Lp = 0 (1 <; p < 00). (6.1.78)

Berens and Westphal [79] proved that D%+ is a bounded linear operator whosedomain is dense in LP(E+) and that {a : a > 0} belongs to the resolvent setof DQ+ and showed, by using the Hille-Yoshida theorem, that the Cauchy type


problem (6.1.77)-(6.1.78) has a unique solution wa(x) for any given /0 in termsof a holomorphic contraction semigroup generated by D$+. They constructed the

solution of this problem in the form u>a(x) = Wa ' fo(x), where Wa is a semigroupof a class Co in LP(M+) for any x > 0.

Kochubei [424] studied the following Cauchy problem

(t) = Ay(t) (0 < t < T), y(0) = y0, (6.1.79)

where A is a closed linear operator on a Banach space X, j/o £ X, and Z)(")(0 < a < 1) is the regularized Caputo fractional derivative of the form (6.1.39):

He found conditions on the resolvent (A - AE)"1 of the operator A which yieldthe existence and uniqueness of the solution y(t) to the problem (6.1.78), and gavea formula for this solution. Kostin [443] considered the Cauchy problem for thefollowing fractional operator equation

(D%+y)(t) = Ay(t) (t > 0), (D^ky)(0+) = yk (k = 1,- ,n; n - l < a n),(6.1.81)

with the Riemann-Liouville fractional derivative (2.1.10) of any a > 0 and obtainedthe conditions for the uniform correctness on a compact set of this problem. When0 < a < 2, he applied the obtained result to the operator A = A2ra+1, A being theLaplacian (1.3.31), in the Banach space of bounded uniformly continuous functionso n l n .

Using ideas related to the theory of first- and second-order abstract differ-ential equations, Bazhlekova [75] gave necessary and sufficient conditions on anunbounded closed operator A in a Banach space X with dense domain D c Xsuch that the abstract Cauchy problem for the following differential equation offractional order

f°D%+y) (t) = Ay(t) (0 ^ t £ T; 0 < a 1), y(0) =x£X, (6.1.82)

with the Caputo derivative (2.4.17), can be solved. Bazhlekova [75] gave the condi-tions on A obtained for the solution y(t) to the problem (6.1.82) to be holomorphic,and relations between the solutions to this problem for different a (0 < a fi 1). Inparticular, these relations showed that (6.1.82) has a holomorphic solution when-ever A generates a Co-semigroup. Gorenflo, Luchko and Zabreiko [301] consideredthe Cauchy problem

(cD%+y) (t) = Ay(t) (O^KTôo; n-l<a^n; n 6 N), (6.1.83)

y ^ (fl) = X k e X { k = 0 , - - - , n - l ) , (6.1.84)

which is more general than (6.1.82), with an unbounded linear operator A in aBanach space X. They described those initial data of this problem for which the

Chapter 6 (§ 6.2) 361

solution may be represented via the Mittag-Leffler function (1.8.17) in the formn-l

y(t) = xktkEa,k+1 {Ata). (6.1.85)

fc=o

Perturbation properties of the solution y(t) to the problem (6.1.83)-(6.1.84) wereinvestigated by Bazhlekova [76], and the results obtained generalize the knownfacts about Co-semigroups and cosine operator functions.

El-Sayed in [225] and [229] studied abstract Cauchy problems for the differentialequation of fractional order (D +y)(t) = Ay(t) (0 ^ t ^ T; 0 < 7 ^ 2) with alinear closed operator A on a Banach space X. He reduced the problems consideredto certain integral equations containing the abstract operator A, and proved severalresults for such integral equations. But he did not prove the equivalence betweenthe initial-value problems and the corresponding integral equations. The sameconcerns his papers [227] and [228], where he studied some abstract equations withthe right-sided Riemann-Liouville fractional derivative D"_y and some nonlineardifferential-difference and functional-differential equations of fractional order.

Kempfle and Beyer [364] studied a formal linear operator of the form

(6.1.86)fc=0

with dk G 1 (k = 0, ,m) and 0 ^ a0 < ax < < am. They called (6.1.86) afractional differential operator and studied the following equation

Ax(t) = f{t) {t G 1; x(t) G L2(M)), (6.1.87)

called the related differential equation. They denned the associated symbol p(w)of A by

m

p(s) = Y,dk(is)ak (6.1.88)k=0

with the special choice of the branch of the power function (is)ak (k = 0, , m).Kempfle and Beyer [364] considered the representation of A as follows

A = F-^-iD)?, D = 4~, (6.1.89)ax

in terms of the direct and inverse Fourier transforms (1.3.1) and (1.3.2). Using(6.1.89), they obtained the explicit solution x(t) e L2(M) of (6.1.87) in the form

x(t) = (A-'mt) = (2TT)-1/2 J" | ^ - I ^ _ L^ j {T)f{t - T)dr. (6.1.90)

Physical applications were considered, and a special case of the operator (6.1.86)in the form DM + aD" + b was studied. See also Beyer and Kempfle [82], Kempfleand Gaul [365] and Kempfle [363] in this regard. Kempfle and Schaefer [366]compared the above approach for certain fractional differential equations with theusual approach based on the Riemann-Lioville fractional derivative (2.1.10), andthey concluded that the results from both approaches coincide in the case of linearequations with constant coefficients.


6.2 Solution of Cauchy Type Problems forFractional Diffusion-Wave Equations

In this section we consider a fractional differential equation of the form

(Z>S+,tu)(s:,t) = A2(Axu)(x,i) ( i £ l " ; * > 0; 0 < a < 2; A > 0), (6.2.1)

involving the partial Riemann-Liouville fractional derivative (Dfu)(x,t) of ordera > 0 with respect to t > 0 defined by (6.1.12), and the Laplacian (Axu)(x,t)with respect to x 6 E":

^ ^ f ^ (n€N). (6.2.2)

In particular, when n = 1, (6.2.1) takes the form of the equation (6.1.32) knownas the fractional diffusion-wave equation (see Section 6.1). Therefore, we call theequation (6.2.1) for n ^ 2 the multi-dimensional fractional diffusion-wave equation.We apply the Fourier and Laplace transforms to obtain an explicit solution of theequation (6.2.1) with the Cauchy type initial conditions (6.1.56):

= fk(x), (6.2.3)

where x £ En, k = 1 for 0 < a ^ l,and k = 2 for 1 < a < 2.

First we consider the two-dimensional case n = 1.

6.2.1 Cauchy Type Problems for Two-DimensionalEquations

We consider the partial differential equation (6.1.31) of order 0 < a < 2

(D%+ttu)(x,t) = A 2 ^ ^ ' * ) ( i 6 l ; t > O ; A > 0), (6.2.4)

with the Cauchy type initial conditions of the form (6.2.3) for n = 1

= fk(x), (6.2.5)

where x G R, k = 1 for 0 < a ^ l,and k = 2 for 1 < a < 2.

To solve this problem, we apply the Laplace transform with respect to t:

/»OO

(£tu)(x,s)= / u(x,t)e~stdt ( i £ l ; s > 0) (6.2.6)Jo

and the Fourier transform with respect to a; £ I :

(?xu){a,t) = / u(z,t)e'XCTda; (cr £ E; i > 0). (6.2.7)J-<x

Applying the Laplace transform (6.2.6) to (6.2.4), and taking into account theformula of the form (5.2.3)

Chapter 6 (§ 6.2) 363

Boa+,f«) (x, s) = sa (Cu) (x, s)-Y,SJ"' ( ^ > ) (*>

with i f l , Z — 1 < a ^ Z and 1 6 N. If Z = 1 and I = 2 in the respective cases0 < a ^ 1 and 1 < a < 2, and the initial conditions in (6.2.5), we have

) (x,s) = J^£tuS) (x,s) (I = 1,2).

Applying the Fourier transform (6.2.7) and using the formula (1.3.11) with

d2u{x, t)dx2

we arrive at the following relation:

= -\<j\ 2{Txu)(a,t), (6.2.9)

{ T x C t u ) {a, s ) =A |(J|

{Tx}k) (a) (a e 1; t> 0; I = 1,2). (6.2.10)

Now we obtain the explicit solution u(x,t) by using the inverse Fourier trans-form with respect to a:

1 f°°(T'û) {x,t) = — / u{a,t)e~laxda (a e E; <> 0)

and the inverse Laplace transform with respect to s:

(6.2.11)

p+iooi p+ioo

{C-1u)(x,t) = — I estu{x,s)ds ( i £ l ; 7 = (6.2.12)

From readily-accessible tables of the Fourier and Laplace transforms, we have

2c(c>0; a (6.2.13)

2Asf

Hence the relation (6.2.10) takes the form

{TxCtu) (CT, S)=\TXK 1 2 = 1,2),


or, in accordance with the convolution property (1.3.17),

{TxCtu){o,s) = hx p ^ - ' - t e - ^ '1 *, fk(x)X\ (a) (I = 1,2).

From here, applying the inverse Fourier transform (6.2.11), we derive the fol-lowing relation:

1^ k ~ 1 ' ^ ^ s i *X fk(x) (xGR; s>0; 1 = 1,2). (6.2.14)

Applying the inverse Laplace transform in (6.2.14), we can obtain the explicitsolution to the Cauchy type problem (6.2.4)-(6.2.5). For this we need to know

I ic I a

the inverse Laplace transforms of the functions sft~1~f e~~8 (k = 1,2). Thesefunctions are expressed via the Laplace transform of the Wright function (1.11.1)of the form </>(-a/2, b; -z). If 0 < a < 2, then c/>(-a/2, b; -z) is an entire functionof z (See section 1.11). It is easily proved by using (6.2.6) and (1.11.1) that

("f , § - * + 1; J-f r*) ] ) (s) = , * - i - *e-^ . (6.2.15)

for k = 1,2.Applying the inverse Laplace transform in (6.2.14) and taking (6.2.15) and

(1.3.13) into account, we obtain the following result.

Theorem 6.1 // 0 < a < 2 and A > 0, then the Cauchy type problem (6.2.4)-(6.2.5) is solvable, and its solution u(x, t) is given by

u(x,t) = } I G%(x-T,t)fk(r)dT {l = lforO<a<:l; l = 2forl<a<2),

(6.2.16)

(A; = 1,2), (6.2.17)

provided that the integrals in the right-hand side of (6.2.16) are convergent.

Corollar y 6.1 7/0 < a ^ 1 and A > 0, then the Cauchy type problem

(Dg+>t u)(x,t) = A 2 ^ ' * ) , (D tu)(x,0+)=f(x) ( i e i ; t > 0 ) (6.2.18)

is solvable, and its solution has the form

Chapter 6 (§ 6.2) 365

u(x,t) = G?(x-T,t)f{T)dT, (6.2.19)J — oo

( 6 - 2 - 2 0 )


Corollar y 6.2 If 1 < a < 2 and A > 0, then the Cauchy type problem

(D tu) (x, t) = A2 ( x £ i ; f > 0 ) (6.2.21)

j 2 = f2(x) ( i £ l ) (6.2.22)


O *OO

U{x,t)= G?{x-T,t)f1{T)dT+ G%(X-T,t)f2{T)dT, (6.2.23)J — oo J— oo

where Gi(x,t) is given by (6.2.20), that is, by

(6-2.24)


Example 6.1 The following Cauchy type problem (6.2.18) with a = |

= f{x) (xeR;t>0) (6.2.25)


O

u(x,t)= G\/2{x-T,t)f(T)dT, (6.2.26)J — OO

where G\/2(x,t) is given by (6.2.20) with a = 1/2.

Example 6.2 The following Cauchy type problem (6.2.21)-(6.2.22) with a = §

(Do%>) (x, t) = A2 ( i £ l ; i > 0 ) (6.2.27)

( > ) /2(a;) (* e R) (6.2.28)



u{x,t) = G31/2{x-T,t)f1(T)dT+ G3

2/2(x-T,t)f2{T)dT, (6.2.29)

J — oo J — oo

where Gi/2(a;,i) and G32/2(x,t) are given by (6.2.20) and (6.2.24) with a = §.

Example 6.3 The following Cauchy problem for the heat equation (6.1.45)

du(^t)_ = A 2 ^ j M , u ( M ) = /(* ) ( i e l ; l > 0 ) (6.2.30)


O -j 2

u[x,t)= I G(x-T,t)f{T)dr, G(x,t) = —^t-1/2e-^l. (6.2.31)y 2AV7T

This well-known result follows from Corollary 6.1, if we take into account thefollowing relation for the Wright function

^ F C " * ' (6-2-32)

which is readily verified by using (1.11.1).

Remark 6.1 The explicit solution of the Cauchy type problem (6.2.18) in theform (6.2.19) with Gf(x,t) given by (6.2.20) was obtained by Podlubny ([682],Example 4.4), who used a different notation.

6.2.2 Cauchy Type Problems for Multi-DimensionalEquations

In this section we extend the results of Section 6.2.1 to the multi-dimensionalfractional diffusion-wave equation (6.2.1) of order 0 < a < 2 with the Cauchy typeconditions (6.2.3). To solve this problem, we apply the Laplace transform withrespect to t:

/»OO

(Ctu)(x,s)= u{x,t)e~stdt ( i £ l " ; s > 0) (6.2.33)./o

and the multi-dimensional Fourier transform with respect to x G W1:

(Fxu){a,t) = [ u(x,t)eix-adx (a £ W1; t> 0; 1 = 1,2). (6.2.34)

Applying the Laplace transform (6.2.33) to (6.2.1), taking into account theformula (6.2.8) with x € M" and the initial conditions in (6.2.3), we have

Chapter 6 (§ 6.2) 367

i

sQ(Ctu)(x,s) = ^ y - 1 / f c ( z ) + A2(Ax£(M)( ;r,s) (/ = 1,2).

Applying the Fourier transform (6.2.34) and using a formula of the form (1.3.32)given by

{FxAxu) {a,t) = -\a\2 (Jû) (a,t), (6.2.35)

we arrive at the following relation of the form (6.2.10):

i

\J-xLtU) (a, s) = y — ryj—jj (J~xJk) (&) (,cr t JK ; f > U; i = 1,1). yo.A.6o)

Now we obtain the explicit solution u{x,t) by using the inverse Fourier trans-form with respect to a:

[Tû) (x,t) = 1 /2 [ u{o-,t)e-iaxd<7 (a E l " ; t > 0) (6.2.37)

and the inverse Laplace transform with respect to s:

(Cû) (x,t) = — / estu{x,s)ds (x eR"; 7 = Re(s) > ov). (6.2.38)

To apply the inverse Fourier transform (6.2.37) to (6.2.36), we need an auxiliaryassertion involving the Fourier transform of the Macdonald function -K^^j-id^l )defined by (1.7.25).

Lemma 6.1 For n £ N and c > 0, there holds the following relation:

[Tx [|a;|(2-")/ 22ir {T1_2)/2(c|a:|)]) (a) = { ^ J ^ ^ (a € R"; n e N).(6.2.39)

Proof. Using (1.3.32) and (5.5.8), we have

I TpK(n-2)/2{cP)J(n-2)/2{\G\p)dp. (6.2.40)

Using a known formula in Prudnikov et al. ([689], Vol. 2, formula 2.16.21.1)and taking (1.6.8) into account, we have

pKin_2)/2{cp)j{n_2)/2{\a\P)dp = ac(n/2)+1 2F1 f | » i ; | ; — ^ -


an(-2)/2 !

2 'c ) c* + \a

Substituting this relation into (6.2.40) yields the result in (6.2.39).

By (6.2.39) with c = -, we have

jH a/2 (a) = 7. (6.2.41)

Hence (6.2.36) takes the form

{TxCtu){a,s) =' sfc-l+a(n-2)/4

^ A(2ATT)" / 2

with / = 1,2, or, in accordance with the convolution property (5.5.6),

- A(2ATT)" / 2

Thus, by applying the inverse Fourier transform (6.2.37), we derive that

' ,A-l+a(n-2)/4( 2) (X, 8) = Y, A(2A7r)n/2

with i e M.n, s > 0, and 1 = 1,2.Applying the inverse Laplace transform in (6.2.38), we can obtain the explicit

solution of the Cauchy type problem (6.2.1)-(6.2.3). For this we need to know theinverse Laplace transform of the functions

sk-1+a{n-2)/AK(n_2)/2 ( ^ sQ / 2 ) (* = 1,2). (6.2.43)

We show that these functions are expressed in terms of the Laplace transform ofsome special J?-functions (1.12.1) of the form

^ U(a, 1/2), (6, a/2) ] _

3.2.44)

where a,b,c,d S R and an infinite contour C separates all poles of the gammafunctions T(c + r) and T (d + §) to the left. By (1.11.16), A = (2 - a)/2. ThusTheorem 1.4 yields the following assertion.

Chapter 6 (§ 6.2) 369

Lemma 6.2 / / 0 < a < 2 and a,b,c,d G R, then the H-function (6.2.44) isdefined for C = Ci7OO and |arg(^)| < [(2 - a)n/4\, z 0.

The functions (6.2.43) can now be expressed in terms of the Laplace transformof the .ff-function (6.2.44).

Lemma 6.3 7/0 < a ^ 1 (k = 1) or 1 < a < 2 (k = 1,2), then there exist therelations

1 W 2' 0

2

(n 1\ /-i i. a(n-2) a\

\2 ' y ' V2 4 ' 2/

(6.2.45)

Proof. Use the representation (6.2.44) for the .ff2,'2~function m the integrand of(6.2.45) and choose C such that <R(r) > f + . T h e n - f c - ^^ + f <R(r) > - 1for 0 < a ^ 1 (k = 1) and 1 < a < 2 (& = 2), which ensure the convergence of theintegrals below. By (6.2.33) and (6.2.44), we have

;2,02,2

(R l\ (\ h a(n~2) a\V 4 > 2 / ' ^ " 4 ' 2)

4 '

M 2 L ^ ' ) L \2 2 ^ 2 ^

\4 ^ 2/

g(n-2)

21 1 /" r (R — 1 4. T-1) r (k _ 2. J. Z\ / |T |

2^ ir r(^ + ) vT

1,2L s2

4 ' 2)

Vj ' ?.f a _ 1 1) (k-R k)\ 2 x > x / ' V 2 4 ' 2 / J

(6.2.46)

By (1.12.57), we get

(6.2.47), 2 " ~ X ' V ' V2 ~ T ' ~ " "

Hence (6.2.46) yields (6.2.45), which completes the proof of Lemma 6.3.

Using (6.2.45), rewriting (6.2.42) in the form

(Ctu)(x,s) =2-n|a.|(2-n)/2

t-k-'- j-2,0'2, 2

(R l\ (A h °("~2) a\U ' 2) ' l 1 K 4 ' 2 )

(n.-\\\ (I _ R k)V2 ' A / ' V2 4 ' 2)


and applying the inverse Laplace transform (6.2.38) we get the following explicitsolution u(x,t):

u(x,t) =k=l

rj-2, 0 I ~io o | -—

Aj2,2

In k\ f-\ h a("~2) aV4' 2) ' ^ * 4 ' 2 fk(x), (6.2.48)

^ 2 *-> L ) > \ 2 4 1 2 )

with I = 1 and / = 2 in the cases 0 < a ^ 1 and 1 < a < 2, respectively.

Thus we obtain the following result.

Theorem 6.2 If 0 < a < 2 and A > 0, then t/ie Cauchy type problem (6.2.1)-(6.2.3) is solvable, and its solution u(x,t) is given by

(6.2.49)

with I = 1 for 0 < a ^ 1, I = 2 for 1 < a < 2, and

r/» n h _ i. _ a("~ 2) q 'V 4 ' 2 ' V ft 4 ' 2 ,

1,2 Li L) ' V2 4' 2/(6.2.50)

(fc = 1,2), provided that the integrals in the right-hand side of (6.2.49) are con-vergent.

Corollar y 6.3 If 0 < Q ^ 1 , n £N and A > 0, then the Cauchy type problem

with x £ E™, t > 0, is solvable, and its solution has the form

u(x,t)= G%(x-T,t)f(r)dT, (6.2.52)J Rn

G?(x,t) = -TT2'0-"2,2

(H. I) f »("-U ' 2) ' I 4

V2" ~ ' / ' V2 ~ 4" ' 2

(6.2.53)provided that the integral in the right-hand side of (6.2.52) is convergent.

Corollar y 6.4 If 1 < a < 2 and A > 0, then the Cauchy type problem

{D%+ttu)(x,t) = X2(Axu){x,t) (x 6 E"; t> 0) (6.2.54)

( £ > £» (*,0+) = h(x), {D%~2tu) (x,0+) = f2(x) (x 6 E") (6.2.55)

Chapter 6 (§ 6.2) 371


u(x,t)= I G^(x-T,t)f1{T)dr +

where Gf (x,t) is given by (6.2.53) and

9-n|™|(2-n)/2 , ,. fG?(*,t) = lXl .-2-^11^2,0 1

(6.2.56)

™ I" \ ^_ 1 _ a ( " - 2 ) a \4> 2/ > \ 1 4 »2J

' V2 4' 2)(6.2.57)


Example 6.4 The following Cauchy type problem (6.2.51) with a = 1/2,and t > 0:

(D1o/+

2ttu)(x,t) = X2(Axu)(x,t); (l£tu) (x,0+) = f(x) ()

has solution, and it is given by

u{x,t)= I G\/2{x-T,t)f{T)dT,

where G/ (x, t) is given by (6.2.3) with a = 1/2.

(6.2.58)

(6.2.59)

Example 6.5 The following Cauchy type problem (6.2.54)-(6.2.55) with a = 3/2,x 6 W1 and t > 0:

(D30£tu) (x,t)= A2 (Axu) (x, t) (6.2.60)

(l?5i>) (x, 0+) = /!(!) , ( / ^ t u) (x, 0+) = /2(a;) (6.2.61)

has solution, and it is given by

u(x,t)=f Gl/2(x-r,t)f(T)dT+ f G32/2(x-T,t)f2(T)dT, (6.2.62)

where G\/2{x,t) and G2/2(:M) are given by (6.2.53) and (6.2.57) with a = 3/2.

The result of Corollary 6.3 is simplified for a = 1. This is based on the followingassertion.

Lemma 6.4 The function G"(x,t) in (6.2.53) for a = 1 has the form

G(x,t) := G\(x,t) = _L_i- i e-££. (6.2.63)

Proof. In accordance with the property (1.12.43) of the .ff-function, we have


2-n X\ . 1V2' 2) ,

(o,i ) J-(6.2.64)

If 0 < a < 2, then the direct evaluation of the integrand in

Jf ))

where a > 0, 6 G E and an infinite contour C = £j7OO separates all poles s = - j(j £ No) of the gamma function T(s) to the left, yields the following relation:

o l *-Then (6.2.64) takes the form

G{x,t) :=G\(x,t) = -—7—T1 1

A"7r(r

Thus the result in (6.2.63) follows from (6.2.32).

Example 6.6 The following Cauchy problem

du(x,t) _ 2

(6.2.66)

(6.2.67)

dt= \2(Axii)(x,t), u{x,0) =

has its well-known solution given by

u(x,t)= f G(x-T,t)f(r)dT, G(x,t)=

(6.2.68)

(6.2.69)

Remark 6.2 If n = 1, then (6.2.39) coincides with (6.2.13), because (see (1.7.27))

)'2e-\ (6.2.70)

while the relations (6.2.45) coincide with (6.2.15). Indeed, by the property (1.12.43)of the i?-function, we have

and, in accordance with (6.2.70), the formula (6.2.45) takes the form

T> l £ '!£!,-§A

By the property (1.12.45) of the i?-function with a — 1/2, we have

[\ — k 4- 2L SL

f(0,1)

(6.2.71)

(6.2.72)

(6.2.73)

Then, in accordance with (6.2.66) and (6.2.70), (6.2.45) with n = 1 yields (6.2.15).

Chapter 6 (§ 6.3) 373

Remark 6.3 When n = 1, then, in accordance with (6.2.71),(6.2.73) and (6.2.66),the functions G%(x,t) (k = 1,2) in (6.2.50) coincide with those in (6.2.17). ThusTheorem 6.1 and Corollaries 6.1-6.2 follow from Theorem 6.3 and Corollaries 6.3-6.4, respectively. This also means that Examples 6.1-6.3 are particular cases ofExamples 6.4-6.6 for n = 1.

Remark 6.4 By analogy with Green functions for partial differential equations,functions Gf (x, t) and G%(x,i), given in (6.2.20), (6.2.53) and (6.2.24), (6.2.57) canbe called fractional Green functions (see Section 6.1). In particular, the functionsG\(x,t) yield the usual Green functions G(x,t) in (6.2.31) and (6.2.69).

Remark 6.5 The fundamental solution of the Cauchy type problem (6.2.51) with0 < a < 1 was obtained by Hilfer [345] in terms of the E.\'"-function (1.12.1).

Remark 6.6 Results presented in this section were given in Voroshilov and Kilbas[861], [862] and Kilbas at al. [413].

6.3 Solution of Cauchy Problems for FractionalDiffusion-Wave Equations

In Section 6.2 we established explicit solutions of Cauchy type problems for frac-tional diffusion-wave partial differential equations involving the Riemann-Liouvillepartial fractional derivatives (Dg+tu)(x,t) of order a > 0 . In this section we con-sider the fractional differential equation of the form (6.2.1)

(cD?fi+u)(x,t) = X2(Axu)(x,t) ( x e f ; t > 0; 0 < a < 2; A > 0), (6.3.1)

involving the partial Caputo fractional derivative (cD"o+u)(x,t) with respect tot > 0, and the Laplacian (Axu)(x,t) with respect to a; 6 I " given in (6.2.2). Justas in (2.4.1), the above Caputo derivative of order a > 0 and I — 1 < a < I (! £ N)is defined in terms of the Riemann-Liouville partial fractional derivative (6.1.12)by

, t) = (D^ \U(X, T) - £ ^ y ^ l \ (*, t). (6.3.2)

When, for any fixed x 6 K, u(x,t) 6 Cl(M.+ ) as a function of t > 0, then(CD%+ jii)(x,t) has the representation (6.1.50). Following an approach developedin Section 6.2, we apply the Fourier and Laplace transforms to obtain an explicitsolution to the equation (6.2.1) with the Cauchy initial conditions

d u{x,0) = j (a- e M«. * = 0 for 0 < a g 1; k = 1 for l < a < 2). (6.3.3)

a°u(x,o):=u{x,0). (6.3.4)

dt°As in Section 6.2, we begin from the two-dimensional (case n = 1).


6.3.1 Cauchy Problems for Two-Dimensional Equations

We consider the following partial differential equation (6.2.1) of order 0 < a < 2

- ( i e l ; f > 0 ; A > 0 ) , (6.3.5)

with the Cauchy conditions (6.3.3).Applying the Laplace transform (6.2.6) with respect to t and the Fourier trans-

form (6.2.7) with respect to x £ E and using the following relation of the form(5.3.3)

(Ct cD^ tu) (x, s) = s" (Cu) (x, s) - £ S«-& -1

N),

(6.3.6)

where i E l , l - l < a ^ ! , I e N , and with I = 1 and I = 2 in the respective cases0 < a ^ 1 and 1 < a < 2, the initial conditions in (6.3.3) and the formula (6.2.9),we arrive at the relation of the form (6.2.10)

a, s) = 6 R; * > 0; / = 1,2). (6.3.7)fc=0 ' '

By (6.2.13), we have

1 Ixl

2XS2 6

and hence (6.3.7) takes the following form:

or, by the convolution property (1.3.17),

H-i "| \

2A \ I

Applying the inverse Fourier transform (6.2.11) to this formula, we obtain therelation

fk(x) (1 = 1,2). (6.3.8)k=0

When 0 < a < 2, then the functions s? k le s2 (k = 0,1) are expressedvia the Laplace transform of the Wright function (f>(—a/2, b; z) as follows:

Chapter 6 (§ 6.3) 375

**"** (-f,fc +1 - f; -Mr* ) ] ) (,) = . t - ^ e - ^ * (* = 0,i).(6.3.9)

Applying the inverse Laplace transform to (6.3.8) and taking (6.3.9) and (1.3.13)into account, we obtain the following result.

Theorem 6.3 7/0 < a < 2 and A > 0, then the Cauchy problem (6.3.5), (6.3.3)is solvable, and its solution u(x, t) is given by

l-l ,oou{x,t) = J2 G%{x-T,t)fk{T)dT (I = 0 for 0 < a I, I = 1 for 1 < a < 2),

(6.3.10)

(fc = 0, l), (6.3.11)


Corollar y 6.5 7/0 < a < 1 and A > 0, then the Cauchy problem

^ ' 0 , u(x,0) = f(x) ( x e I ; t > 0 ) (6.3.12)


u{x,t)= f G?(x-T,t)f{r)dT, (6.3.13)J — oo

(6.3.14)


Corollar y 6.6 If 1 < a < 2 and A > 0, then the Cauchy problem

(6.3.15)

u(a;,0)=/o(aO, £ ' = / i ( ») (* G R) (6.3.16)


/>OO /-OO

u{x,t)= G?(x-T,t)fo{T)dT+ G%(x-T,t)fi{T)dT, (6.3.17)J— oo J— oo

where Gf(x,t) is given by (6.3.14) and

| ' 2 - f ; - x r * ) ' (6-3-18)provided that the integrals in the right-hand side of (6.3.17) are convergent.


Example 6.7 The following Cauchy problem (6.3.12) with a = |

l^TY'l ti\(^r i\ — X^ *__ !_ ! . IIIT Ci\ — f tT\ IT- d H$* i ^ fi\ {f\ 1 1 Q~\\ ^Q-i- t ) \ 1 1 — Q 2 ' LL\XIV) — J \X) \X c JfiL, I ^> \J) yv.O.Lv)

has its solution given byO

u(x,t) = G\/2(x-T,t)f(T)dT, (6.3.20)

J—oo

where Gx (x,t) is given by (6.3.14) with a = \.

Example 6.8 The following Cauchy problem (6.3.15)-(6.3.16) with a = |

" " ( i £ l ; l > 0 ) (6.3.21)

' — f, (v\ (a Q oo\

has its solution given byO /-OO

u{x,t)= Gl/2(x-T,t)f0(T)dT+ G32/2{x-T,t)h{T)dT, (6.3.23)

o / —o o

where Gi/2(a;,i) and GÔM) are given by (6.3.14) and (6.3.18) with a = §.

Remark 6.7 The explicit solution of the Cauchy problem (6.3.12) in the form(6.3.13) with G%(x,t) given by (6.3.14) was constructed by Mainardi [515] [seealso Gorenflo et al.[310] and Podlubny ([682], Example 4.5)].

6.3.2 Cauchy Problems for Multi-Dimensional Equations

In this section we extend the results of Section 6.3.1 to the multi-dimensionalfractional diffusion-wave equation (6.3.1) of order 0 < a < 2 with the Cauchyconditions (6.3.3). Applying the Laplace transform (6.2.33) with respect to t > 0and the Fourier transform (6.2.34) with respect to a; € M.n, and using the relation(6.3.6), the initial conditions (6.3.3) and the formula (6.2.35), we arrive at thefollowing relation of the form (6.3.7):

{F tCtu) (a, s)=y2 / \ 2 I ,2 {F*fk) (<T) (creM.n;t>0;l = 1,2). (6.3.24)fc=o

s + A |CT|

SL

According to (6.2.39) with c = j-, this formula takes the form

{TxCtu){a,s)

Chapter 6 (§ 6.3) 377

with Z = 1,2, or, in accordance with the convolution property (5.5.6),

{FxCtu){a,s)

(a).

Now, by applying the inverse Fourier transform (6.2.37), we derive that

) (x,s) =fc=O

(6.3.25)

(s; £ 1"; s > 0; I = 1,2).

The functionsa(n+2) - f c - 1 (^-saÂ (Jb = O,l) (6.3.26)

are expressed via the Laplace transform of the H2 '2-function (6.2.44). There existsthe following assertion, which can be proved as Lemma 6.3.

Lemma 6.5 // 0 < a ^ 1 (k = 0,1) or 1 < a < 2 (k = 1), i/ien there exist thefollowing relations:

r Z2,2U ' 21 > + K 4 ' 2

f n _ i i \ f l _ n 1\V2 ' / ' V2 4 ' 2/

(6.3.27)

Using (6.3.27), rewriting (6.3.25) in the form

, _ 2-"|;r|(2-")/2

fc=0 TU ' 2 ' fc- a(n+2)

" _ 1 I1! (I _ n 1\2 -1' -L-'' V2 4 ' 2/

{*),

and applying the inverse Laplace transform (6.2.38), we are led to the explicitsolution u(x,t):

2,02,2 A

U ' 2-< ' *x fk(x),(R _ 1 11 (I _ R 1\V2 ' X7 ' V2 4 ' 2 /

with / = 1 and Z = 2 in the respective cases 0 < a ^ 1 and 1 < a < 2.Thus we obtain the following result.

(6.3.28)


Theorem 6.4 7/0 < a < 2 and A > 0, then the Cauchy problem (6.3.1), (6.3.3)is solvable, and its solution u(x, i) is given by

l-l

u{x,t) = 2 / G%{x-T,t)fk{T)dT {I = 0 for 0 < a 1; I = 1 for 1< a < 2),

(6.3.29)G%(x,t)

k=o

-n|_|(2-n)/22,0

——I

(.2 ' / ' \2 4' 2)\l+%7r(n-l)/2t

L \ 2 ~> ~) ' \2 4 ' 1)

(6.3.30)where k = 0,1, provided that the integrals in the right-hand side of (6.3.29) areconvergent.

Coro l lar y 6.7 7 / 0 < a < l , n £ N and A > 0, then the Cauchy problem

^ -t-'n_i_ + " ^ y ^ j ^/ — » \^-*x^/ V » /? t*^j / 5 vjj — j yjyj yju t ir^ , fc ** \)) \\J.O.Oi.j


„.(„ 4\ _ / (la(T — T A f(T\rlT (fi ^9\

* ^(V2 4 ' (6.3.33)


Corollar y 6.8 If 1 < a < 2 and A > 0, then the Cauchy problem

(cD$+tu){x,t) = \2{Axu){x,t) ( i € l " ; f > 0 ) (6.3.34)

— =/ i (x) (a; G E") (6.3.35)LL\iL, Ul — /Qlti/Ij


u(x,t)= f Gf(x-T,t)fo(T)dT + (6.3.36)

where G"(x,t) is given by (6.3.33) and

2-n|a.|(2-n)/2 O(T I+2) f l ^ ^ ("n 1\ /Q "(n+2) a\

2 ' A1+;8'7r(n-1)/2 2'2 A f ! » _ i i \ ( I _ " 1L V2 x ' / ' V2 4 ' 2 /

(6.3.37)provided that the integrals in the right-hand side of (6.3.36) are convergent.

Chapter 6 (§ 6.4) 379

Example 6.9 The following Cauchy problem (6.3.31) with a = 1/2

(cDl£tu)(x,t) = \2{Axu)(x,t), u(x,0)=f(x) ( i e i " , t > 0 ) (6.3.38)


u(x,t)= f Gl/2(x-T,t)f(T)dT, (6.3.39)

where G\/2{x,t) is given by (6.3.33) with a = 1/2.

Example 6.10 The following Cauchy problem (6.3.34)-(6.3.35) with a = 3/2

(°Do£itu)(x,t) = \2{Axu)(x,t) ( i6 l " ; (>0) (6.3.40)

u(x,0) = /„(*) , ^ ^ - = h(x) ( i £ l " ) (6.3.41)


u{x,t)=f Gl/2(x-r,t)f(T)dr+ [ G32/2(x - r,t)f2(T)dT, (6.3.42)

where G\/2{x,i) and G32/2(x,t) are given by (6.3.33) and (6.3.37) with a = 3/2.

Remark 6.8 It can be verified, by using the properties (1.12.43) and (1.12.45)of the ^-function (1.12.1) and the relation (6.2.73), that the formula (6.3.27) forn = 1 coincides with (6.3.9).

Remark 6.9 When n = 1, then it is directly verified, by using the properties(1.12.43) and (1.12.45) of the -function and the formula (6.2.73), that the func-tions G%{x,t) {k = 0,1) in (6.3.30) coincide with those in (6.3.11). Thus Theorem6.5 and Corollaries 6.5-6.6 follow from Theorem 6.7 and Corollaries 6.7-6.8, respec-tively. This also means that Examples 6.7-6.8 are particular cases of Examples6.9-6.10 for n = 1.

Remark 6.10 The functions G%(x, t) and G%{x, t), given in (6.3.14), (6.3.33) and(6.3.18), (6.3.37) provide two- and multi-dimensional fractional Green functions.

Remark 6.11 In Sections 6.2 and 6.3 we constructed explicit solutions to theCauchy type and Cauchy problems for the fractional diffusion-wave equations(6.2.1) and (6.3.1) of the same form involving the Riemann-Liouville and Ca-puto partial fractional derivatives, respectively. The results obtained show thatthese problems have different solutions in the case of non-integer a 6 I (a f! N).This is caused by the fact that the Riemann-Liouville and Caputo partial frac-tional derivatives differ by a quasi-polynomial term (see definitions in (6.1.12) and(6.3.2)).


6.4 Solution of Cauchy Problems for FractionalEvolution Equations

In this section we consider the fractional differential equation (6.1.68):

(cD tu)(x, t) = X(D0_tXu)(x, t), (6.4.1)

with z e E, t > 0, a > 0, /? > 0, A G M \ {0} , and involving the Caputopartial fractional derivative (cDft+tu)(x,t) with respect to t > 0 of order a > 0(I — 1 < a I; l e N) and the Liouvill e partial fractional derivative (D/txu)(x, t)with respect to x 6 K of order 0 > 0 denned by (6.3.2) and (6.1.71), respectively.We apply the Fourier and Laplace transforms to establish the explicit solution ofthe equation (6.4.1) with the following Cauchy conditions of the form (6.3.3):

u(x,O) = fo(x), U a ^ > =fk(x), (k = 1 , . . . , /- 1; xeR). (6.4.2)

As in Section 6.3, we begin from the simplest case of the equation (6.4.1) with

(cD%+tu)(x,t) = A "{f ' ' ( z £ K ; i > 0 ; a > 0). (6.4.3)OX

6.4.1 Solution of the Simplest Problem

In this section we consider the Cauchy type problem (6.4.3)-(6.4.2). Applyingthe Laplace transform (6.2.6) with respect to t > 0 and taking into account theformula (6.3.6) and the initial conditions in (6.4.2), we have

l - l r>

sa(jCtu)(x,s) - y V - * - 7 f c ( z ) = X—(Ctu)(x,s). (6.4.4)*=o dx

Applying the Fourier transform (6.2.7) with respect to x € K and taking intoaccount the following formula for the Fourier transform of derivatives

(TxD™u) (a, s) = (-ik)m(Txu)(a, s) (Dx = £ ; m £ N) (6.4.5)

with m = 1, we arrive at the relation for {J-xCtu) (a, s):

l-l a-k-l

(^Ctu) (a, s) = ^2 -^ —- (Fxfk) (<J)- (6.4.6)fc=o s % a

Applying the inverse Fourier transform (6.2.11) with respect to a e M. andthe inverse Laplace transform (6.2.12) with respect to s (£H(s) > <T ) and taking

Chapter 6 (§ 6.4) 381

(6.4.11) and (6.4.12) into account, we obtain the solution of the problem (6.4.2)-(6.4.3) in the form

(6.4.7)If [Tjk){a) (k = I,--- , Z) satisfy some additional conditions, the inner integralsin (6.4.7) can be evaluated by using the residue theory [see, for example, Ditkinand Prudnikov ([195], Chapter 2)].

On the basis of (6.4.6), we can give another representation for the solutionu(x, t) in terms of the

Mittag-Lefner function Ea^{z) denned in (1.8.17). According to (1.10.9) withP = k + 1, we thus have

$a-k-i

(C [tkEa,k+1 {-i\ata)}) = (\Xas~a | < 1; k = 1, , I). (6.4.8)S ~T~ %ACJ

Applying the inverse Laplace transform (6.2.12) in (6.4.6) and using (6.4.8),we obtain

{Txu){o, t)=Y, tkEa,k+i (-i\<rt a) {Txfk) (a). (6.4.9)ife=0

Then the inverse Fourier transform (6.2.11) yields the following solution u(x,t):

u(z,i) = 5Z o~ / £<*,*+! (-«A<rfQ) (Jx/fc) (o-)e"itrxda. (6.4.10)fc=o 00

Thus we have established the following result.

Theorem 6.5 Let a > 0 be such that I - 1 < a ^ I (I 6 N). TTien £/ie Cauchyproblem (6.4.3), (6.4.2) is solvable, and its explicit solution u(x,t) is given by(6.4.7) or (6.4.10), provided that the integrals on the right-hand sides of (6.4.7)and (6.4.10) exist.

Corollar y 6.9 / / 0 < a ^ 1, then the Cauchy problem

(cD$+ttu)(x,t) = A ^ g ^ , u(x,0) = f{x) ( i e l ; f > 0 ) (6.4.11)


u(x,t) = / sa~1estds— / e~luxda (7 € R) (6.4.12)2-ni 77_i oo 2?r J^CC sa + iXa

or, equivalently,


Ea (-i\ata) {Txf) ( a )e -~AT, (6.4.13)

provided that the integrals in the right-hand sides of (6.4.12) and (6.4.13) exist.

Corollar y 6.10 If 1 < a ^ 2, £/ien the Cauchy problem

(cD tu)(x,t) = A ^ ^ (xeR; t>0) (6.4.14)

U(s,0) = fo(x), ^ | ^ = / i ( . ) (6.4.15)


=J-2TTI 77

r m2TT J sa + i\a

' - oo s " + «A(T

or, equivalently,

u(x t) — —\ 1 / rt

0 (<r)e—da, (6.4.17)



( cD^>) (x, t) = A ^ ^ , u(x,0) = f{x) (x € E; * > 0) (6.4.18)


+ OO1'2) e - ^ ^ - (6-4.19)


9™ ( i £ » ; « > 0) (6.4.20)

u(x,0) = fo(x), ^ ° l = /1(a:) (6.4.21)

Chapter 6 (§ 6.4) 383


OO

+OO

f

u(x, t) = -^J E3/2 (-i

+ 2^ J

(6-4.22)

The solution u(x,t) of the Cauchy problem (6.4.2)-(6.4.3) can be representedin a form different from that in (6.4.7) and (6.4.10), if the given functions fk(x)(k = 1, ) are analytic.

Theorem 6.6 Let a > 0 be such that 1 -1 < a ^ I (I e N), and let fk{x)(k = 1, , I) be a n a l y t ic f u n c t i o ns of t he real v a r i a b le i £ l s u ch t h at

l i m f(kj) { x ) = 0 {k = 1, ; j e N o ) . ( 6 . 4 . 2 3)

Then the solution of the Cauchy problem (6.4.3), (6.4.2) is given by

(6.4.24)

fc=O j=0

provided that the series in the right-hand side of (6.4.24) converges for any x € Rand any t > 0.

Using (6.4.10) and interchanging the order of integration and summation (whichis permissible by the uniform convergence of the series represented by the entireMittag-Leffler function), we have

+ OO

fc=l

I oo

(6-4.25)fe=l j = 0

By means of the formula (1.3.11), we have, for the inverse Fourier transform,

-I /- + O

— / (k = I,--- ,/; j e No).

(6.4.26)Using (6.4.26), from (6.4.25) we derive the representation for the solution u(x,t)of the problem (6.4.2)-(6.4.3) in the form (6.4.24).


du{x,t){cD%+tu){x, t) = A K' ' ( i £ l , ; t > 0 ; I - l < a g i; / £ N), (6.4.27)


(bk e l ; f t > 0; k = O,---,l-l) (6.4.28)


J-I

u(x, t) = Y, bktke-Êa,k+1 (sign(x)X kt

a). (6.4.29)fc=0

In particular, if 0 < a ^ 1, then

u(x, t) = be~Êa (sign(x)XiJ,ta) (6.4.30)

is the solution to the following Cauchy problem:

{cD%+tu)(x,t) =\du(^t\ u{x,0) = be- ( i £ l ; f > 0 ) . (6.4.31)

6.4.2 Solution to the General Problem

In this section we generalize the results of Section 6.4.1 for the fractional differentialequation (6.4.1) with the Liouvill e fractional derivative (D_ XU\ (x,t) with respectto x G M. of any order /3 > 0. For this we need the following formula for the Fouriertransform of such a derivative [see ([729], formula (7.4))]:

{Tx{Dt tXu)){a,t) = {-iaf{Tu){a,t) (/3 > 0), (6.4.32)

where the function (—ia)@ is understood as

{-ia)p := lafe'^1^9"^ (a G E; /3 > 0). (6.4.33)

Following Section 6.4.1, we apply the Laplace transform (6.2.6) with respect tot > 0 and the Fourier transform (6.2.7) with respect to x G M. to the equation(6.4.1). Using the relation (6.2.8) and (6.4.32), we arrive at the following relationof the form (6.4.6):

{TxLtu) (a, s) = g 8° * 1 (?xfk) (a). (6.4.34)

k=o s -*\-lcr r

Using the inverse Laplace and Fourier transforms and applying the same argumentsas in Section 6.4.1, we obtain the solution u(x,t) to the problem (6.4.1)-(6.4.2) inthe form (6.4.7) and (6.4.10) as follows:

u(x,t) = Y^ / sa k 1estds— I x,, \ae %axda (7 £ R)

K ' ' ^ 2m 77_i o o 2TT . / .^ sa - \(-ia)P K' '(6.4.35)

and1 ik

~* ' " ' % a ct\ ( T t ^ 1 \ —i<jx j la A oe\I \J~xJk) ^(Jje OCT. (D.4 .ODJ

Thus we have proved the following result.

Chapter 6 (§ 6.4) 385

Theorem 6.7 Let a > 0 be such that I - 1 < a ^ I (I £ N) and let /3 > 0. TTien£/ie Cauchy problem (6.4.1)-(6.4.2) is solvable, and its explicit solution u(x,t) isgiven by (6.4.35) or (6.4.36), provided that the integrals on the right-hand sides of(6.4.35) and (6.4.36) exist.

Corollar y 6.11 7/0 < a ^ 1 and /3 > 0, then the Cauchy problem

(cDa ii)(r t) — \ I Dp 11 I (r /"I n(r (Y\ — f(r) (r (= 1R- t > (Tl Cfi 4 7"!\ \J —]— ts / \ ' / \ *tj § \ ' / ' \ ' / V \ / \ I / \ /


u(x,t) = -—l sa-1estds— \J,.\fle~iaxd(7 ( 7 6 I) (6.4.38)7T« J-y-ico 2 7 r ^ - 00 S ° ~ A( —1(T)P

or, equivalently,

e--â, (6.4.39)


Corollar y 6.12 // 1 < a ^ 2 and f3 > 0, i/ien the Cauchy problem

l ; t > 0 ) , (6.4.40)

u(x,0) = fo(x), ^ l ^ - = f1(x) (xeR) (6.4.41)


7 _ i 0 O 2TT

c

+ : "ioo 7r J-00 S " ~ A(—l(T)p

or, equivalently,

«M) = - r 0 0 ^ (A(-^)^ a) (^/o) (o-)e--

Ea>2 (\(-iafta) {Txh) (a)e-^da, (6.4.43)



Corol lary 6.13 If a > 0 (I - 1 < a I, l e N ), then the Cauchy problem

{cD$+ju)(x, t) = A " ^ ' *> ( i e l ; t> 0) (6.4.44)

u(x,0) = fo(x), ^ d(*' O) = fk(x) (k = 1, ,1 - 1; x 6 R) (6.4.45)


u(x t) - V — r+l°° sa-k-1estds—fr 0 7 i C o J_x s*

(6.4.46)or, equivalently,

( ^ e " i < T X ^ , (6-4.47)



( c D l £ t u ) ( x , t) = A ( D i , x u ) (x,t) u(x,0) = f(x) ( x £ R ; t > 0 ; f 3 > 0 ) (6.4.48)


u(x, t) = —J E1/2 (Xi-io-ft1/2) (Fxf) (a)e-^da. (6.4.49)


(cD3o£tu){x,t) = A ( l £ ,au) {x,t) {x G E; t > 0; P > 0) (6.4.50)

u(x,0) = fo(x), ^p-=f1{x) (6.4.51)


u(x,t) = E33 / 2

+ OO

^3/2,2 (A(-ia)^t3/2) {Txh) (a)e-^da. (6.4.52)

Chapter 6 (§ 6.4) 387


— ( i £ l ; (>0; |,m£N), (6.4.53)

u(x,0) = fo(x), ^ U ^ O) = h(x) (k = 1, , I - 1; x 6 R) (6.4.54)


E ^ (X(-i(>)mtl) (-^/*) W^da. (6.4.55)fc=0


R; t > 0; /, m e N), (6.4.56)

(fc = 1, , I - 1; x 6 1) (6.4.57)


j.k />+oo

/_ . / _, J.\ k / IT1 [ \ / "1 \ 771 _.277l J.1 I / *£" -f \ //T^vj — MTX J — (/? A CQ\

In particular,

u(a;,f) = — / E2 (-A<72^2) (J^/o) (<r)e"itrxd(T^ n J — oo

i /»-|-OO

t I / O O\

2 7 r J-oo

is the solution to the following Cauchy problem:

'- ( i £ l ; f > 0), (6.4.60)

u(x,0) = fo(x), ^ S . = f1(x) ( i e l ) . (6.4.61)

Remark 6.12 If A > 0 and A is replaced by A2, then the relations (6.4.46),(6.4.47) with / = 1 and / = 2 yield, in different form than (6.3.13) and (6.3.17),explicit solutions of the Cauchy problems, (6.3.12) and (6.3.15)-(6.3.16). It is easilyverified that these solutions coincide for "suitable" functions fo(x) and fi(x).


Remark 6.13 The results of Corollaries 6.9 and 6.11 with 0 < a < 1 were givenby Kilbas et al., [385]. Similar results were established in Kilbas et al. [386] forthe Cauchy type problem for equations of the form (6.4.11) and (6.4.37), withthe Caputo derivative replaced by the Riemann-Liouville derivative, and with theinitial condition (6.1.58).

6.4.3 Solutions of Cauchy Problems via the il-Functions

We show that, in the case I 2m (I, m G N), the solution (6.4.58) can be expressedin terms of the iJ-functions (1.12.1) of the form

k — Is)(6.4.62)

where a,b,c,d G M. and an infinite contour £ separates all poles of the gammafunctions F(s) to the left and poles of gamma functions F(l - s) and F(l — 2ms)to the right. By (1.11.16), A = I - 2m. Thus Theorem 1.6 yields the followingassertion.

L e m m a 6.6 / / l,m £ N (I ^ 2m) and k = ,1 — 1, then the H-functions(6.4.62) are defined in the following cases:

l>2m, £ = £_oo, (6.4.63)

/ < 2m, C = £+ 0 o. (6.4.64)

Now we prove that the functions

Ei,k+1 {X(-l)ma2mtl) (k = 0,---,l-l) (6.4.65)

can be expressed in terms of the Fourier transform of the H-functions (6.4.62).

Lemma 6.7 / / l,m G N are such that I 2m, then, for k = 0, ,/ — 1, therehold the following formulas:

= Eltk+1 [(-l) m\a2mtl] . (6.4.66)

Chapter 6 (§ 6.4) 389

Proof. By (6.2.7) and (6.4.62), we have

1 „ ! r>. \ , , >™ At,|2m

r(s)r(i - s)r(i - 2ms)L v ' J -dx.

2ni Jc T(ms)T(l - ms)T(l + k - Is)(6.4.67)

Using (5.5.8) and (5.5.27), we find for the inner integral in (6.4.67) that

We can choose s G C in (6.4.67) such that 0 < 9t(s) < 5^. Then, usingthe following known formula [see, for example, Brychkov and Prudnikov ([108],formula 7.12)]

= sin Pf) (0 (6.4.69)o P

and the functional equation (1.5.8) for the gamma function (1.5.1), we evaluatethe integral in (6.4.68) as follows:

00 Axu

\l-2msdx = 2-rr-

T(2ms)(I - ms) T (| + ms)'

- 2m (6.4.70)

Substituting (6.4.70) into (6.4.67), we have

rl,2\~\2m


1 f T(s)T(l - s)= —f 2TIT(1 - 2ms)T(2ms)cT{l + k- Is) [r(ms)r(l - ms)T (§ - ms) T (| + ms)

mtl\a\2m]~sds.

In accordance with the Legendre duplication formula (1.5.9), we find that

2TIT(1 - 2ms)Y{2ms)r(ms)r(l - ms)T [\ - ms) V (| + ms) = 1

and, therefore, that

2m [a)

2-wi Jcds. (6.4.71)

By (1.8.32) with a = I, /? = k + 1 and z = A(-l) ra</|<7|2m, (6.4.71) yields theresult in (6.4.66), and thus Lemma 6.7 is proved.

By the convolution property (1.3.17) and (6.4.66), we obtain following resultanalogous to (6.4.58).

Theorem 6.8 Ifl,m £ N are such that I 2m, then the Cauchy problem (6.4.56)-(6.4.57) has its solution given by

' />OO

U(x,t) = 22 Glmk(X ~ T,t)fk(T)dT,u—i J —ooK—1

(6.4.72)

where

(6.4.73)

Corollar y 6.14 If I GN (I 2), then the solution to the Cauchy problem

d'u(x,t) _ 82u(x,t) , _ „ „ , . _ „ . ,(6.4.74)

u(x,0) = fo(x), d"UÔ) = fk(x) (k = 1, ,I - 1; x € R) (6.4.75)

Chapter 6 (§ 6.4) 391

u[x,t) =

/.ooG2k(x - T, t)f(r)dr, (6.4.76)

( 0 ' 2 )


du(x.t) , d2mu(x,t)0f

has its solution given by/»OO

u(a;, i) = / Gm{x - r,«/ —CO

™|2m(0,2m), (0,m)(0,1), (0,m)

In particular, the solution to the Cauchy problem

du(x,t) d2u(x,t)dt = X- 8x2 -, u(x,0) = f(x) (a; e l ; t > 0),

is given by

o

= /J -

(6.4.77)

(6.4.78)

(6.4.79)

(6.4.80)

(6.4.81)

G2(x,t) = j - ^ M | (-1) ^ (Oj l ) j(0,1)

(6.4.82)

Remark 6.14 The functions Glmk(x, t) ( / ,m£N; k = 0, - 1) and Gm(x,t)

(m £ N), given in (6.4.73) and (6.4.80), yield the Green functions for the ordinarypartial equations (6.4.56) and (6.4.78).

Remark 6.15 The explicit solution (6.4.72) of the Cauchy problem (6.4.56)-(6.4.57), presented in Theorem 6.8 via the Green functions (6.4.73), is valid forl,m S N such that I / 2m. In particular, such Green functions cannot be con-structed for the classical Cauchy problem for the wave (diffusion) equation:

d2u(x,t) d2u(x,t) du(x,0)dt2 8x2 dt

= /i(x), u(x,0) = f2(x). (6.4.83)

Therefore, the problem of constructing Green functions for the problem (6.4.56)-(6.4.57) for the case / = 2m (in particular, for the problem (6.4.83)) is still open.


Chapter 7

SEQUENTIAL LINEA RDIFFERENTIA LEQUATION S OFFRACTIONA L ORDER

As mentioned in Chapter 3, the problem of studying linear differential equationsand systems of linear differential equations of fractional order has been dealt withby numerous authors throughout history, particularly in recent years. We alsonoted that many of the questions related to this topic have only been partiallyresolved, and in a less than systematic fashion.

As for the problem of obtaining explicit solutions to linear differential equationsand to systems of linear differential equations of fractional order with constant co-efficients, Miller, Ross, Gorenflo, Mainardi, and Podlubny, mainly, have providedin recent years some methods based primarily on the use of Laplace transformsfor certain equations associated with the Riemann-Liouville DQ+ or the CaputoCDO+ derivatives. This leaves certain problems with initial values involving thefractional derivatives D"+ or CD"+ (a G IR), or those associated with boundaryconditions, for example, beyond their scope.

This chapter presents a general theory for sequential linear fractional differen-tial equations, and systems of these equations with Riemann-Liouville and Caputoderivatives which provides a transparent and systematic approach for obtaining ageneral solution for the corresponding cases with constant coefficients. The con-tents of this chapter are based essentially on the works of Bonilla et al. [97, 98].See also Dattoli et al. [155] and Vazquez [842], and Vazquez and Vilela MendezVazquez [845].

Specifically, we will give the explicit solution, in the case of constant coefficients,for both the homogeneous and the non-homogeneous problem, using a fractionalGreen function which generalizes the corresponding ordinary function.

393


Also in this chapter, we apply series of fractional order to investigate the so-lutions of certain linear fractional differential equations with variable coefficients,including a generalization of the well-known Frobenius theory.

In general, in this chapter it will be understood that 0 < a 1, a G M.,A G Mnxn(M) and B G Mnxi(E), MmXn(M) being the set of real matrices oforder m x n. A real matrix function of order n x n with domain in M. will berepresented by A(x).

We will use frequently the Mittag-Lemer type functions e*' and e^*m, whichwere introduced in Chapter 1; see (1.10.11) and (1.10.29) in Section 1.10.

7.1 Sequential Linear Differential Equations ofFractional Order

This section introduces a general theory for the fractional case, analogous to thatof the normal case, applicable to Riemann-Liouville (R-L) sequential linear frac-tional differential equations (LFDE). General methods, independent of the inte-gral transforms, are developed for obtaining the general solution to a LFDE withconstant coefficients, by using the roots of the characteristic polynomial of thecorresponding homogeneous equation, or for finding a certain Green function inthe non-homogeneous case.

It must be pointed out here that a theory, similar to that developed in thischapter for the R-L fractional differential operator D"+, is feasible for the case ofLFDE with the differential operator of the Caputo CD"+ type, keeping in mindproperties of the Mittag-Leffler functions presented in Chapter 1. In any case, it isalso possible to connect this theory for the Riemann-Liouville derivative with thecorresponding theory for the Caputo derivative operator if we use the followingclose connection between D"+y and cD"+y (see (2.4.4) in Section 2.4):

{cDaa+y){x) = {Da

a+W) ~ Via)]) (*) (0 < <*(<*) < !) (7.1.1)

We begin this section with the following definitions.

Definition 7.1 Let n G N. We shall call linear sequential fractional differentialequation of order na the equations of the form

n

J2h(x)y{ka(x)=f(x) (a<x<b), (7.1.2)fc=o

where bk(x) are given real functions, y^°(x) := y(x) , and y^ka{x) :— (T>^_y) (x)(k = l , . . .,n) represents a fractional sequential derivative. For example, for theR-L derivative, T>"+y is

y := D%+y

=V"V^-1)ay (* = 2 3 ) [ '

Chapter 7 (§ 7.1) 395

For the case k = 2 and 0 < a < 1/2, the relationship between Vîs given by

and

(7.1.4)

by applying Property 2.4.

If Va: G [a,6], 6n(a;) ^ 0, equation (7.1.2) may be expressed in its normal formfor the R-L derivative as follows:

n - l n - l

hi I A, J I ty * (/ I I A, I ^*^ (I I A, I™ \ / \ Q,~\-& / \ / if \ /

fc=O fc=O

Note that the equation (7.1.5) reduces to the system

(DZ+Y)(x)=A(x)Y(x)+B(x),

with

A(a:) =

/ 0 10 0

\ —0,0

01

(7.1.5)

(7.1.6)

(7.1.7)

B(x) =

/

V

00

Y(x) =

yi(x) \

\ Vn\

(7.1.8)

just by changing the variables

y1{x)=y(x); (D%+yj) (x) = yj+1(x) (j = l ,...,n - 1). (7.1.9)

Definitio n 7.2 For a 6 R, we call a-Wronskian of n functions Uj(x)(j = l,...,n), having fractional sequential derivatives up to order (n — l)a inthe interval V C (a, 6], the following determinant:

[au[a(x)u2(x(

u(2a{x)

. . un(x)

. . u{na(x)

To simplify the notation, this will be represented by |Wa(a;)|.(7.1.10)


We will use WQ(a;) for the corresponding Wronskian matrix.

Definitio n 7.3 We shall call fundamental system of solutions of the equationI>na(y) = 0 in V C [a,b] a family of n functions linearly independent in V, whichare solutions of this equation.

Definitio n 7.4 The term general solution of the fractional differential equationI jna(l') = f{x) refers to any solution to this equation, which depends on n inde-pendent constants.

Next we wil l prove two theorems on the existence and uniqueness of globalsolutions to the equation L n a(y) = f(x) with certain initial conditions.

Theorem 7.1 Let x0 > a , let y* G R (k = 0,1, . . . , n- 1) be given real numbers,and let a,j(x) G C[a,b] (j = 0, l , . . ,n - 1) and f(x) G C[a,b] be given continuousfunctions on [a,b]. There exists a unique continuous solution y(x) G C(a,b] of theCauchy problem

[Wv)](aO = fix) (7.1.11)( % y * (fc = 0 , l , . . . , n - l ), (7.1.12)

In addition, this solution y(x) is a-singular of order a, that is,

lim (a; - aY~ay{x) < oo, (7.1.13)x—>a-\-

and satisfies the inequality (ll+ ay) (x) < oo.

Proof. Theorem 7.1 follows immediately upon application of the correspondingtheory developed in Chapter 3 and the solution prolongation method until themaximal solution is found.

Theorem 7.2 Let dj(x) G C([a,b}) (j = 0, l , . . ,n - 1) and f(x) G d-a([a,b]) begiven functions on [a, b]. Then there exists a unique continuous solutiony(x) G C(a,b] for the equation

= f(x), (7.1.14)

such that, for k = 0,1,.. ., n — 1,

\\m{x-a)l-a{Vka%y){x)=bk (bk G R) (7.1.15)

x—ya-\-

or{ll+ aVk

a%y) (a+) = bk (bk G E). (7.1.16)

Proof. The proof of Theorem 7.2 is analogous to that of Theorem 7.1.

Proposit ion 7.1 Any linear combination of solutions of the homogeneous equa-tion

[Wy)](* ) = o (7.1.17)

is also a solution to this equation.

Chapter 7 (§ 7.1) 397

Proof. This property is immediately evident keeping in mind the linearity of LnQ.

Proposition 7.2 Let XQ G (a,b] (or xo — a) and let a,j(x) £ C((a,b])(j = 0,1, ..,n — 1) be such that (x — a)1~aaj(x)\x=a < oo (j = l , ...,n). Thenthe homogeneous differential equation (7.1.17) with the initial conditions

y(ja(xo) = 0 (or [(x - a)1-ayâ(x)}x=a+= 0) (j = 0,1, ...,n - 1). (7.1.18)

has only the trivial continuous solution y(x) = 0.

Proof. Proposition 7.2 follows clearly from Theorems 7.1 and 7.2.

Proposition 7.3 Let {wj(a;)}" =1 be a family of functions which admit fractionalsequential derivatives up to order (n — l)a in (a,b], satisfying for j = 1,2, ...,nand k = 0, ...,n — 1, the condition

lim (7.1.19)

i/iai is, £/ie functions {u\ a} j = i,j „ are a-singular of order a.^ fc = 0 , 1 , , . . , n - 1

If functions (x — a)x~aUj(x) (j = 1, -n) are linearly dependent in [a, b], thenfor all x £ [a,b],

(a ; -a )n- n a|WQ(a j ) |=0, (7.1.20)

where |WQ(a;)| is defined by (7.1.10).

Proof. Since {(x - a)l~aUj(x)}J=1 are linearly dependent in [a, b], there exist nconstants {CJ}" = 1, not all zero, such that, Va; G [a,b],

ix~a)1 aUj(x) = 0, (7.1.21)

and therefore, Va; £ (a, b],

= 0. (7.1.22)A= l

Successive application of the sequential derivatives P*" (k = 1,..., n — 1) to(7.1.22) lead to the following relation:

- 0 (7.1.23)

withCl

C = (7.1.24)

\ Cn


and therefore, |Wa:(a;)| = 0 Vcc G (a, 6].

In addition, by the hypothesis (7.1.19), we can also conclude that

lim (x - a)1" QWa(a;)C = 0, (7.1.25)x~>a-f-

and hencelim (a; - a)n-na\Wa(x)\ = 0, (7.1.26)

x—»a+

which proves Proposition 7.3.

Proposit ion 7.4 // the conditions of Proposition 7.3 are valid and there ex-ists an x0 G [a,b] such that [(x — a)n~na\Wa(x)\]x=Xo ^ 0, then the functions{(x - a)1~aUj(x)}1j==1 are linearly independent.

Proof. By Proposition 7.3, Proposition 7.4 is proved by reductio ad absurdum.

Proposit ion 7.5 Let solutions Uj(x) (j = 1,- , n) of equation [L nQ(y)](:r) = 0in (a, b] be a-singular of order a. Then the functions

{(x - a)1-auJ(x)}?=1 (7.1.27)

are linearly dependent in [a, b] if, and only if, there exists an x0 G [a, b] such that

(x - a)n-na\Wa(x)\)x=Xo = 0. (7.1.28)

Proof. If {(x - a)1~aUj(x)}" =1 are linearly dependent, (7.1.28) follows as a conse-quence of Proposition 7.3.

To demonstrate the converse, we will first consider the case XQ a and considerthe system of equations

W a(zo )C = 0, (7.1.29)

where C is given by (7.1.24) and {c°}™ =1 are independent constants, which yieldthe solution Co 0.

For the functionn

]Uj(x), (7.1.30)

defined in (a, 6], which is a solution to L n a(y) = 0 in (a,b] (because it is a linearcombination of solutions) it holds that

, (kal \ r> (vjr, r i - i \ f7 1 o i ly \%0} — U ^v/C — U, 1, . . . , 71 — 1^ yi .L.oL)

Therefore, by Proposition 7.2, y(x) = 0 (Vz G (a, b]), or, equivalently, the func-tions {uj(a;)}™ =1 are linearly dependent in (a, b] and consequently{(x — a)1~aUj(x)}" =1 are linearly dependent in [a,b\.

For the case x0 = a, the result is similarly obtained by considering the system[(x-a)1-aWa(x)]x=X0C = 0.

Chapter 7 (§ 7.2) 399

Proposition 7.6 // the conditions of Proposition 7.5 are satisfied, thenVx 6 [a, b],

(x-a)n-na\Wa(x)\=0 or (x - a)n-na\Wa{x)\ 0, (7.1.32)

where Wa(x) =Wa(ui,...,un)(x).

Proof. Proposition 7.6 follows from Propositions 7.3 and 7.5.

Proposition 7.7 If {UJ(X)}'J'=1 is a fundamental system of solutions to the equa-tion [Lna(y)](x) = 0 in a certain interval V C (a,b], then the general solution tothis differential equation in V is given by

i, (7.1.33)fc=i

where {cfc}^ =1 are arbitrary constants.

Proof. Since yg(x) is a solution to (7.1.17), we need only show that any othersolution is a special case of yg{x). Let u(x) be a solution to the equation (7.1.17)which satisfies, for a given XQ G (a, 6], the following conditions:

u(ka{x0) =ug e 1 (fc = 0 , l , . . . , n - l ). (7.1.34)

Then, applying Theorem 7.1, there exist certain particular constants {CJ}™=1 suchthat

u(x) = }_^ckuk{x), (7.1.35)

k=i

which proves the proposition.

Proposition 7.8 The set of all solutions of the differential equation[L aa(y)](x) = 0, in a certain interval V C (a, b], is a vector space ofn dimensions.

Proof. Proposition 7.8 follows directly from Proposition 7.7.

Proposition 7.9 There exist a fundamental system of solutions of the differentialequation of fractional order [Li na(y)](x) = 0 in an interval V C (a, b], such that

[(x - a ) " -n a|Wa( i ) | ] I = o = — ^ . (7.1.36)

Proof. Proposition 7.9 follows directly from Theorem 7.2.

Proposition 7.10 // yp(x) is a particular solution to the equation[Lna(y)](z) = f(x)> then the general solution to this equation is given by

yg(x)=yh(x)+yp(x), (7.1.37)

where yh(x) is the general solution to the associated homogeneous equation[L na(y)}(x)=0.

Proof. Proposition 7.10 is evident.


7.2 Solution of Linear Differential Equations withConstant Coefficients

In this section we introduce a method independent of the Laplace transform, anal-ogous to that for the ordinary case, for obtaining a fundamental system of solutionsto the equation LnQ,(y) = 0, which also yields an explicit expression for the generalsolution to the non-homogeneous equation Lna(j/ ) = f(x).

7.2.1 General Solution in the Homogeneous Case

Letn - l

[L na(y)](x) := (P™) (x) + £ ak (V%) y(x) = 0, (7.2.1)k=0

where the coefficients {aj}™Ii are real constants.

As in the usual case, we shall seek the solution of (7.2.1) in the formy{x) = ea{x~a), where ea( l " o) is defined by (1.10.11). It follows from (7.2.1)and (1.10.54) that

[ ( ) ] (x) = Pn(\)exJ*- a\ (7.2.2)

wheren - l

Pn(\)=\n + Y,ak\k (7.2.3)

is the characteristic polynomial associated with the equation [Laa(y)](x) = 0.

The following assertion is true for complex A G C.

Lemma 7.1 If X G C is a root of the characteristic polynomial (7.2.3), then

Proof. Lemma 7.1 follows from the linearity of the operators ^ and L na and from(1.10.28).

Proposition 7.11 // Ai is a root of multiplicity /ii of the characteristic polyno-mial (7.2.3), then the functions {yi,i(x)}£t.Ql:

.. ,(T\ — (T _ n\la Ai(a-o) /y 2 fi\

where ea / ^s defined by (1.10.31), are solutions of the equation

[L na(y)](x)'=0.

Chapter 7 (§ 7.2) 401

Proof. Taking (7.2.2), (7.2.4) and (1.10.28) into account and using the classicalLeibniz rule, we have

Since

f ^ | = 0 (j = 0 , l , . . . ,M l - l ) , (7.2.9)

by the hypothesis, then [Lna(yiti)](x) = 0, and the proposition is proved.

Corollar y 7.1 Let {Xj}j =1 be k distinct roots of multiplicity {A*J}J= I °f the char-acteristic polynomial (7.2.3). Then the functions

:-a)lae:l{x~a)Y"1 (7.2.10)

m=l "" ~

are linearly independent solutions of the equation (7.2.1).

Proof. Corollary 7.1 follows from Propositions 7.11 and 7.4.

Proposit ion 7.12 7/Ai and Ai (Ai = b + ic, c 0) are two complex solutions ofmultiplicity O\ of the characteristic polynomial (7.2.3), then the functions

l=o

and

1=0

form 2o\ linearly independent real solutions of the equation [Lna(y)](x) = 0.

Proof. Since Xi = b + ic is & root of multiplicity order o\ of the polynomial (7.2.3),i t follows from Proposition 7.11 that VI = 0,1,. . ., o\ - 1,

,. ( r \ — (r _ n\la Ai(x-a) /_, „ i o \

is a complex solution to (7.2.1). Using similar arguments for Ai , and taking intoaccount (1.10.33), (1.10.34), and Proposition 7.4, we complete the proof of Propo-sition 7.12.


Corollar y 7.2 Let {Xm,Xm}m_1, Xm = bm + icm (cm ¥" 0), be all distinct pairsof complex conjugate solutions of multiplicity {<7m}m=i of the characteristic poly-nomial (7.2.3) for the fractional differential equation (7.2.1). Then the functions

c r m - l

1=0

and(7m—1

m=l {j=0

determine a linearly independent set of solutions to the equation (7.2.1).

Proof. To derive Corollary 7.2, it is sufficient to apply Propositions 7.4 and 7.12.

Theorem 7.3 Let Pn(X), given by (7.2.3), be the characteristic polynomial for thelinear fractional differential equation (7.2.1). Let {-\ j} =1 be all real and distinctroots of (7.2.3) of the multiplicity {fij}j =1, and let {rj,fj}^ =1 (rj = bj + icj) bethe set of all distinct pairs of complex conjugate roots of (7.2.3) of the multiplicity{ajVj=i su°h that Ylij=i A*j + 2 Y^j=i °~j — n- Then the functions

(7.2.16)m=l ^ "~'

Om-1

_ (7.2.17)m=l [j=0 W>'- ' J l=1

andP I oo 9.44-1

)abm(x-a)

1=1

form the fundamental system of solutions of the differential equation (7.2.1).

Proof. Theorem 7.3 follows from 7.1-7.2 and Proposition 7.4.

Note that in the case a = 0, operational methods such as the Laplace transformmay be also applied to solve the problem involving constant coefficients.

Example 7.1

1. Consider the fractional differential equation y(-2a(x) + /j,2y(x) = 0 (/x > 0), withy(2a{x) = (Vl+y) (x). The characteristic polynomial (7.2.3) for this equation has

Chapter 7 (§ 7.2) 403

the form -P2(A) A2 +^t2 = 0, and Ai = fii and A2 = —fii are its roots. Thereforefrom Corollary 7.2 we obtain the following fundamental system of solutions:

{cosQ[^(x - a)], sinQ[ M (x-a)]} , (7.2.19)

where

s O* - a)} = g(-l)y^) <* 2fêt2-l (7-2.20)

{X - a)U) (7 2 21)

and

s i n ^s - a)} = f>l)> ( 2J>( a: ( ° y ) % ^ a ) (7.2.22)

f 7 2 23)r[(2j + l)a] ( 7-2 > 2 3)

2. The fractional differential equation y(2a(x) — y(x) = 0 has the following funda-mental system of solutions:

{eix~a\e-^}. (7.2.24)

This result follows from Corollary 7.1 if we take into account that Ai = 1 andA2 — —1 are the roots of the characteristic polynomial -P2(A) := A2 — 1 = 0.

3. The fractional differential equation y(2a(x)-2y(a(x)+y(x) = 0 has the followingfundamental system of solutions:

{e£-a\(x-a)ae%:a)}. (7.2.25)

Since Ai = 1 is the root of multiplicity two of the characteristic polynomialP2(A) := A2 - 2A - 1 = 0, Corollary 7.1 yields the above result.

7.2.2 General Solution in the Non-Homogeneous Case. Frac-tional Green Function

Consider the non-homogeneous linear differential equation of fractional order

= f{x) (7.2.26)

denned by (7.1.5). As in the case of ordinary differential equations, the generalsolution to this equation is a sum of a particular solution to the equation (7.2.26)and a general solution to the corresponding homogeneous equation (7.2.1).

In this section we apply the operational method and the Laplace transformmethod to derive a particular solution yp(x) of (7.2.26). Note that the Laplacetransform method is valid only when a = 0, since methods similar to that of


the variation of parameters are not applicable due to a noticeable lack of basicproperties in the R-L derivative. Note that a particular solution to the equation(7.2.26) may be also obtained by reducing this equation to a non-homogeneouslinear system.

A. Operational method.

Proposition 7.13 Let {Aj}^ =1 be k different roots of the multiplicity {/Xj}jf =1

( S I /j,j = n) of the characteristic polynomial -Pn(A) = 0 associated with theequation [Lna(y)](x) = 0, where y(ka(x) = (V^_y)(x). Then the particular solu-tion to the equation (7.2.26) with a given function f(x) on (a,b] is given by

kn Or) (7.2.27)3 = 1 \n=0 /

provided that the series in the right-hand side of (7.2.27) are uniformly convergent.

Proof. We can consider the inverse operator to the left for LnQ, in the form

- T (7-2-28)

( - ^ ) n 4 ; + 1 ) a >) , (7-2.29)j=l 3 = 1 \n=0 /

which yields the formal solution to the equation (7.2.26) in (7.2.27). Applying theoperator

Lna = n(^+-A, r (7-2-30)3=1

to the function yp in (7.2.27) and using the uniform convergence of the seriesconsidered, we derive that yp(x) is the solution to the equation [Lna(y)](a:) = f(x).

B. Laplace Transform Method

The Laplace transform method is only useful for the case a = 0, and has been usedby various authors as already discussed. We will apply this method to explicitlyderive a particular solution for the non-homogeneous equation

[Lna(y)](*) = /(*) (7-2-31)

By (2.2.36), the Laplace transform C of DQ+f(x), for a suitable f{x), is givenby

{£D%+f) (s) = sa(Cf)(s) (7.2.32)

and, therefore,(s)} = sna(Cf)(s). (7.2.33)

Chapter 7 (§ 7.2) 405

Proposition 7.14 A particular solution to the non-homogeneous LFDE (7.2.31)with y(ka(x) — (T>Q"y)(x) is given by

k Mj

Vp(*) = E E cJ^(9j,m * f)(x), (7.2.34)j=0 m=l

where

(

(7.2.35)Cj>m (j = 1,2, ...,&; m = 1,2,...,Hj) are constants defined by the following decom-position into simple fractions:

1 l ~'"m (7.2.36)

and (g * /)(#) represents the Laplace convolution of g(x) and f(x) defined by(1.4.10).

Proof. Applying the Laplace transform (1.4.1) to the equation (7.2.31), using(7.2.36) and taking the inverse Laplace transform (1.4.2), we find the particularsolution to (7.2.31) in the form

yp(x) = C - l

j = l m =l

(x). (7.2.37)

This yields the result in (7.2.34), if we take into account (7.2.35) and the Laplaceconvolution formula (1.4.12).

C. Operational decomposition method

In this section we extend the result obtained in Proposition 7.14 for the equation(7.2.31) with y(ka{x) = (V$$y)(x) to the case y(ka{x) = {Vk

a%y){x) Va € E. Firstwe consider the simplest equation.

Proposition 7.15 Let f(x) E Lx(a,b) DC[{a,b)]. Then the LFDE

y(a(x)-\y{x)=f(x) (x>a), (7.2.38)

where y(a(x) = (D"+y) (x), has the general solution

yg(x)=ce^-a^+yp(x), (7.2.39)

whereyp(x) = (exj *° f){x) (7.2.40)


is a particular solution to (7.2.38), *a being the following convolution:

(9*af)(x)= f'g(x-t)f(t)dt. (7.2.41)J a

In addition, if f(x) is continuous in [a,b], then yp(a+) = 0, while iff{x) G Ci_a([a,6]), then &+ ayp) (a+) = 0.

Proof. By Proposition 7.7, y(x) = cea ' is the general solution to the cor-responding homogeneous equation (7.2.38) yâ(x) - Xy(x) = 0. Therefore it issufficient to verify that yp(x) is a particular solution to (7.2.38). Applying Lemma2.10 and taking (2.1.56) into account, we have

(DZ+yp) (x) = (D?+ J* e^'-r)/(r)dr) (x) (7.2.42)

= f {DZ+4lx-a)}(t)f{* - * + a)* + /(«) t^+{ll+ aeXJx-a)}(t) (7.2.43)

= X [X e^-^fix + a- t)dt + f{x) (7.2.44)J a

= X f e^*-«/(f )de + f(x) = Xyix) + fix). (7.2.45)J a

This concludes the proof of Proposition 7.15,

Theorem 7.4 Let {Aj}* =1 be the k distinct complex roots of the multiplicity{o~j}j =1 of the characteristic polynomial (7.2.3) for the following non-homogeneousLFDE:

x) = f f[{D2+ - Xj^y) (x) = f(x) {x > a). (7.2.46)

Then the particular solution to (7.2.1) is given by

yP = iGa*af)ix), (7.2.47)

where Gaix) is given by

I!* " (fl'Â J (x). (7.2.48)

Furthermore, if fix) 6 Ci_a([a,6]), then (la+ayP) (a+) = 0, wMe 2/p(o+) = 0when fix) is continuous in [a,b].

Additionally, ( J âGa) (o+) = 0.

Chapter 7 (§ 7.3) 407

Proof. It is sufficient to apply successively the result of Proposition 7.15 whiletaking into account the weak singularity of the function e*x.

The function Ga {x—£) plays the role of the fractional Green function associatedwith the non-homogeneous LFDE [LnQ. (y)] (:r) = f(x), analogous to the case ofordinary differential equations with constant coefficients.

The result (7.2.47) is given, in general, by the function of a complex variable.Its real part will be a particular solution to (7.2.46) and its imaginary part will bea solution to the corresponding homogeneous equation [Lna(y)](a;) = 0.

7.3 Non-Sequential Linear Differential Equationswith Constant Coefficients

The results derived in Sections 7.1 and 7.2 may be applied to certain cases ofnon-sequential fractional differential equations of the Riemann-Liouville type.Without going into specifics, we consider the following case of order 2a:

(D2a%y) (x) + ai(x) {DZ+y) (x) + ao(x)y(x) = f(x) (0 < a < \). (7.3.1)

Recall that the relationship between (D\°^y) (x) and (V^y) (x) for 0 < a < \is given by (7.1.4):

( O ) (*) = {DI% L ) - ( f k y + ) (t ~ a)""1] ) (x) (7.3.2)

and that{D:+(t-a)a-1)(x)=0. (7.3.3)

Corollar y 7.3 LetO < a < \, ak(x) G C{[a,b]) (k = 0,1) andf(x) € d_a([a,6]).Then the Cauehy type problem for the non-homogeneous LFDE (7.3.1)

D2a%y + ai(x)DZ+y + ao(x)y = f(x) (7.3.4)

limxâ+(x - a^^yix) = b0 , - „ , ,l im B_a+(i - a)1" " (DZ+y) (x) = h ^ ^ ^

has a unique continuous solution y(x) G C(a,b].

Proof. The above result follows from Theorem 7.2 and Property 2.4, because theCauehy type problem in (7.3.4)-(7.3.5) is equivalent to

(V2a%y) (x) + ai(x) {V«a+y) (x) + ao(x)y(x) = f(x) (7.3.6)

Jim+(a; - a)1'0 (Vka%y) (x) = bk (k = 0,1). (7.3.7)


Corollar y 7.4 Let 0 < a < | and / G Ci_a([a,6]). T/ien the LFDE (7.3.4) wirea/ constant coefficients ao a r^ ai has the solution given by

y{x) = ClZl{x) + C2z2{x) + zp{x) - -—{x - a)""1. (7.3.8)

Here Zj (j = 1,2) are two linearly independent solutions of the homogeneous se-quential LFDE

{V2aa+z) (x) + ai {Va

a+z) (x) + aoz(x) = 0, (7.3.9)

andzP(x) = (zi * a z2 * a [f(t) + a0C{t - a)"-1}) (x) (7.3.10)

is a particular solution to the non-homogeneous equation

(V2a%z) (x) + ai (T%+z) (x) + aoz(x) = f(x) + a0C{x - a)a~\ (7.3.11)

where C, C\ and G2 are real constants satisfying the following relationships:

a) C\ + C2 — C for the case when the roots of the characteristic polynomial asso-

ciated with (7.3.9) are distinct;

b) C\ = C for any other case.

Proof. The result in the corollary follows from Theorems 7.3 and 7.4.

Example 7.2 Let 0 < a < § and / e Ci_Q([a,6]). The equation

{D2aa+y)(x)-y(x)=f(x) (x > a) (7.3.12)

has the general solution

yg(x) = d e + (C - Ci)e-(x-a) + h{x) - ^—{x - a)""1, (7.3.13)

where

n\x) = I ea * ea * \I\ ' rr ) ^ a> I

C and C\ being two arbitrary real constants.

Example 7.3 Let 0 < a < \ and / e Ci-a([a,b]). The equation

{Dl a+y) (x) - 2 (D2+y) (x) + y(x) = f(x) (x > a) (7.3.15)

has the general solution

yg(x) = Ce<Ta) + C2e(:~a) +u(x) - ^ ( x - a)""1, (7.3.16)

where

u(x) = (e^~a) *a e^7a) * a \f(t) + ppr(* - a)a~M ) (x), (7.3.17)

C2 and C being two arbitrary real constants.

Chapter 7 (§ 7.4) 409

7.4 Systems of Equations Associated withRiemann-Liouville and Caputo Derivatives

This section deals with the study of the following system of linear fractional dif-ferential equations (SLFDE) associated with R-L and Caputo derivatives

where

(DaY) (x) :=

/ on (a;)

A(x) =

(x)=A(x)Y(x)+B(x),

/ h(x) \\

B(x) =

)

(7.4.1)

(7.4.2)

\ anl(x) ann{x) J

are matrices of known real functions and Da represents the R-L or Caputo frac-tional derivative.

We give explicit solutions for the homogeneous equation with variable coeffi-cients and for the non-homogeneous equation with constant coefficients. In the lastcase we use a fractional Green function generalizing the corresponding ordinaryfunction.

7.4.1 General TheoryFirst we prove two theorems which establish the existence and uniqueness of globalsolutions for the system (7.4.1), where Da = D"+, with certain initial conditions.An analogous theory could be developed for the Caputo case.

Theorem 7.5 Let 0 < a < 1, let x0 and Yo be fixed real numbers, and letA(x) and B(x) be continuous matrices on [a,b]. Then the SLFDE (7.4.1) whereDa = D"+, has a unique continuous solution Y (x), defined in (a, b], which satisfies

Y(x0) = Yo. (7.4.3)

This solution is a-singular of order a, that is,

lim [(a; - a)l-aY{x)] < oo (7.4.4)x—>a+

and, in addition,(ll+ aY) {a+) < oo. (7.4.5)

Proof. Theorem 7.5 follows by applying the theory developed in Chapter 3.

Theorem 7.6 Let 0 < a < 1, let A(x) be continuous in [a,b], and letB(x) 6 Ci-a([a, b]). Then the SLFDE (7.4.1) has a unique continuous solution on(a, b] such that

lim [(a; - o)1-Qr(ar)l = b (or (/]+"?) (o+) = b) , (7.4.6)

where be W1.


Proof. The proof of Proposition 7.6 is analogous to that of Theorem 7.5.

Proposit ion 7.16 Any linear combination of solutions of the homogeneous sys-tem

Yâ(x) =A{x)Y{x) (7.4.7)

is also a solution to the system (7.4.7).

Proof. The proof of Proposition 7.16 is evident.

Definitio n 7.5 We shall call the fundamental system of solutions of the homoge-neous sequential LFDE (7.4.7) inVC (a, b] a family of linearly independent realvectorial functions {Ui}" =1, which are solutions to (7.4.7) in V.

Definitio n 7.6 We shall call the fundamental solution-matrix of the system (7.4.7)a square matrix X(x) of order n, the columns of which form a fundamental systemin (7.4.7), and which thus satisfies the matrix equation

X (Q(z) = A(a;)X(2:). (7.4.8)

Definitio n 7.7 IfX is a fundamental matrix solution for the system (7.4.7), thenwe shall call the general solution to (7.4.7) the following relation:

Yh(x) = X(x)C, (7.4.9)

where

C =

\

(7.4.10)

is a vector whose components are arbitrary constants.

Theorem 7.7 The general solution to the non-homogeneous sequential LFDE(7.4.1) has the form

Yg(x)=X(x)C + Yp(x), (7.4.11)

where X(a;) is a fundamental matrix of (7.4.7) and Yp(x) is a particular solutionto (7.4.1).

Proof. The proof of Theorem 7.7 is evident.

Theorem 7.8 The matrix

X(z) = e£(*-a> (A£M n x n( l ) ) (7.4.12)

is a fundamental solution-matrix for the system

Yâ(x) = AY(x). (7.4.13)

Chapter 7 (§ 7.4) 411

Proof. It is sufficient to recall that

(Daa+et{t-a)) (x) = Ae£(*-<0 (7.4.14)

and to note thatlim \(x-a)n-na\e^x-a^\\ ^ 0. (7.4.15)

x—)-a+ L J

Therefore, the general solution to (7.4.7) is given by

Yh(x) = e^x~a)C (C G W1). (7.4.16)

Corollar y 7.5 27ie following initial-value problem

Yâ(x) = A{x)Y(x) (7.4.17)

Y(x0) = 0(xo> a) (7.4.18)

/ms only the trivial solution Y(x) = 0.

Proof. Corollary 7.5 follow from Theorem 7.5.

Theorem 7.9 The following initial-value problemYîx) = AY(x) (7.4.19)

Y(x0) = % (x0 > a), (7.4.20)

where A G Mn(R) anrf Z?Q = .D"+) /jas a unique continuous global solution Y(x)on (a, oo) C M, which is a-singular of order a, that is,

lim [(* - a)l-aY{x)} < oo. (7.4.21)

T/iz's solution is given by

Y{x) = e^x- (eMxo-^y1 fo_ (7.4.22)

Proof. Theorem 7.9 is proved by an application of Theorems 7.5 and 7.7.

Theorem 7.10 The following initial-value problem

Yâ(x)=AY(x) (7.4.23)

lim [(a; - af^Yix)] = Yo (Yo 6 Rn), (7.4.24)x—»a+

where A £ Mn(E) and Da = I?"+, /ias /is unique continuous global solutiondefined in (a, oo) C M., given by

Y(x) = e^x-a^Y0. (7.4.25)

Proof. Theorem 7.10 is deduced by an application of Theorems 7.6 and 7.7.


Example 7.4 Consider the following equation:

{Vl%y) (x) + y(x) = 0. (7.4.26)

Making the substitutions

y(x) = Vi(x)(a, s t , 7.4.27

we reduce equation (7.4.26) to the following system of equations with constantcoefficients:

Yâ{x)=A(x)Y(x) (7.4.28)

with

( i * ) ( £ ) (7429)According to (7.2.19), its general solution has the form

Yg(x) = e*xC = ( C°SaX siUaX ) C. (7.4.30)9K ' a \ -sinax cosaa; J v '

Then the general solution to (7.4.26) is given by

y(x) = yi{x) = cicosQa; + c2sinQa;. (7.4.31)

7.4.2 General Solution for the Case of Constant Coefficients.Fractional Green Vectorial Function

We analyze the general solution to the following non-homogeneous sequentialLFDE

(DaY) (x) = AY + B(x). (7.4.32)

with constant matrix A(a:): A S Mnxn(R):

Theorem 7.11 The system

(Daa+Y) (x) = AY(x) + B(x), (7.4.33)

where A G M nxn(E),B G Ci_Q((a,6]) , has the general solution given by

Y(x) = e^x-a^C + [ e^x-^B{0dt, (7.4.34)J a

whereGa(x-0=e^x-V (7.4.35)

is the Green function associated with (7.4.33).In addition, ifB(x) is continuous in [a,b], , then the particular solution

Yp(x) = {Ga *a B) (x) = f Ga(x - OB(O« (7-4-36)

J a

satisfies the relation Yp(a+) = 0, while (ll+ aYp) (a+) = 0 ifB(x) G Ci_Q([a,6]).

Chapter 7 (§ 7.4) 413

Proof. First we show that Yp(x) is a particular solution to (7.4.33). For that weuse the following operational method:

{DZ+[1 - A7«+]y ) (x) = B(x) (7.4.37)

from whereYp(x) = ({i?2+[l - A/^+]} * B) (X) (7.4.38)

= ([1 - A / ^ ] - 1 (J«+B) (t) + K[\ - A l ^ U t - a)""1} ) (x). (7.4.39)

For the case K = 0, it follows that yp(a:) is equal to

j ^ "A'B) (,) = f ; _ _ L ^ / % _ 0 ( ^ ) - i A*B(Ode (7.4.40)

= fJa

i=0K1 + l)a\

(7.4.41)

= (Ga *a B) (a). (7.4.42)

The above arguments are formal. Now we show that, in fact, Yp(x) is a par-ticular solution to (7.4.33). For this we will use Lemma 2.10, and the relations

( ) " , (7.4.43)

andlim (ilê^^) (x) = E. (7.4.44)

Then we have

(Daa+Yp) (x) = (D%+ I eA(*-«B(0de) (x) (7.4.45)

= I" (Daa+z*{x~a)) (0B(x + a - £)d£ + B(x) lim fc^^'-")) (x) (7.4.46)

Ja \ / x—¥a+ \ /

= A / e^-a)B{x + a-Z)d£ + B(x) (7.4.47)Ja

= A f e^x-VB(Od£ + B(x). (7.4.48)J a

This concludes the proof of Theorem 7.11.

Example 7.5 The general solution to the following equation

?{a(x) = ( _°! I ) ) + ( 0 ) (7A49)

is given byYg(x) = Yh(x)+Yp(x), (7.4.50)


wherey. lT\ _ PA(x)n _ / >»a-<< auiax \ „ , ,

w Q y -sinQa; cosQa; J v 7

and

Yp(x) = fX ea{x~° ( o ) dZ = [X ( 1MZ^X~JI) ) d (7-4-52)

For the system of equations with constant coefficients involving the Caputoderivatives, the following statement holds.

Theorem 7.12 The initial-value problem

(cDaa+?) (x) = AY(x), (7.4.53)

Y(a) = b (be I T ) , (7.4.54)

where A G Mnxn(R), has a unique continuous global solution Y(x) in [a, oo) C M.given by

Y(x)= I eA{x-Âbdi + b. (7.4.55)J a

Proof. Since (CD°+Y) (X) = (D%+[Y(t) - Y(a)]) (x), the system (7.4.53)-(7.4.54)is equivalent to

(DZ+Z) (x) = AZ(x) + Ab, (7.4.56)

Z(a) = 0, (7.4.57)

where Z(x) = Y(x) - b.

Theorem 7.13 The differential equation

(cDaa+Y) (x) = AY(x) + B(x), (7.4.58)

where A G Mn(M.) and B G Ci-a([a, b]), has its general solution given by

Y(x) = f e^*-V [B(0 + AY(a)]d£ + Y(a) (7.4.59)J a

Proof. In order to establish Theorem 7.13 it is sufficient to use the definition of aCaputo derivative (1.4.1), the inequality y(a) < oo, and Theorem 7.7.

Theorem 7.14 The following initial-value problem

(x) = AY{x) + B(x), (7.4.60)

Y{a) = b [be Rn), (7.4.61)

where A G Mn(M) and B G Ci_a([a,b]), has its unique solution given by

Y{x) = r e£(*-«)[B(£) + Ab\di + b. (7.4.62)J a

Proof. Theorem 7.14 is an immediate consequence of Theorem 7.13.

Chapter 7 (§ 7.5) 415

7.5 Solution of Fractional Differential Equationswith Variable Coefficients. Generalized Me-thod of Frobenius

In this section we will consider solutions around a point XQ G [a, b] for differentialequations of the form

[L na(y)}(x) = g(x), (7.5.1)

where

[W*)](s ) = Vina(x) + E ik(x)y{ka(x), (7.5.2)k=0

where a G (0,1]; n G N, real functions g(x) and a,k(x) (k = 0,1, ,n — 1) aredefined on the interval [a, b], and y(ka(x) represent sequential fractional derivativesof order ka of the function y(x) (see (7.1.3)).

We note that the series method, based on the expansion of the unknown solu-tion y(x) in the fractional power series to obtain solutions of fractional differentialequations, was first suggested by Al-Bassam [17]. Using that idea, he derived theformal solution to the Cauchy type problem (4.1.34) with a = 0. Al-Bassam in[18], [19] [see also [21], [22] and [23]] considered more general LFDE of the form(7.5.16) and (7.5.34) with series coefficients p(x) and q(x) given by (7.5.17) and(7.5.36), respectively. Such a method for obtaining solutions for certain particularcases of the equation (7.5.16) was also used formally by Oldham and Spanier ([643],p. 103) and by Hadid and Grzaslewicz [322]. In this regard see also Miller andRoss ([602], Chapter VI , Section 3), Podlubny ([682], Example 3.2 and Sections6.2.1-6.2.3.) and Kilbas and Trujill o ([408], Section 4 and [377]).

On the other hand, we must indicate that, so far we know, the study of so-lutions around an environment of a singular point of linear fractional differentialequations has been no development til l now, see [709]. Here, we point out thatthe generalization of the power fractional series and the Frobenius methods couldbe very interesting as a natural tool to apply the separation method to obtain thesolution to many fractional models connected with Complex Systems.

7.5.1 Introduction

In this section we analyze solutions of the homogeneous equation [Lna(y)](a;) = 0,which are expressed as a series of powers of xa. For this we introduce the concept ofa-analyticity which generalizes the concept of an analytic function in the ordinarycase; [in this regard see Bonilla et al.[94]].

Definitio n 7.8 Let a G (0,1], f(x) be a real function defined on the interval [a, b],and xo G [a,b]. Then f(x) is said to be a-analytic at xo if there exists an intervalN(xo) such that, for all x G N(xo), f(x) can be expressed as a series of natural

oo

powers of (x-xo)a. That is, f(x) can be expressed as ^ cn(x-x0)

na (cn£R),n=0


this series being absolutely convergent for \x — xo\ 0). The radius ofconvergence of the series is p.

Example 7.6 a) The Mittag-Leffler function (1.8.1)

feriyis an a-analytic function around the point xo = a with radius of convergencep = oo.

It follows from Example 4.9, that y(x) = CEa(xa) is the general solution to

the following FDE of order a e (0,1):

{cDZ+y) (x) = y(x) (x>a; a £ R) (7.5.4)

b) The function

8=0 V '

is a-analytic around the point XQ = 0 with radius of convergence p = oo (see(1.10.12)).

By Corollary 4.6, y(x) = C exa is the general solution to the following FDE:

(D%+y) (x) = y{x); (x > 0; 0 < a < 1). (7.5.6)

Definitio n 7.9 A point Xo G [a, b] is said to be an a-ordinary point of the equation\^>na{y)\(x) — 0, if the functions ak{x) (k = 0,1, ,n — 1) are a-analytic in XQ.A point Xo G [a, 6] which is not a-ordinary will be called a-singular.

By analogy with the ordinary case, we classify the a-singular points as regulara-singular and essential a-singular as follows:

Definitio n 7.10 Let xo 6 [a, b] be an a-singular point of the equation[Ijna(y)]{x) = 0. Then xo is said to be a regular a-singular point of this equa-tion if the functions (x - xo)

(-n~k">aak(a) are a-analytic in x0 {k = 0,1, ,n- 1).Otherwise, XQ is said to be an essential a-singular point.

If xo is a regular a-singular point, then the equation L n a(y) = 0 can be ex-pressed as follows:

n-l

(x - xo)nay{na{x) + £ ( * - xo)

kabk{x)y(ka{x) = 0, (7.5.7)fc=0

where the functions bk{x) (k = 0,1, ,n - 1) are a-analytic around x0.

Chapter 7 (§ 7.5) 417

Example 7.7 a) Any point x = XQ > 0 is an ordinary point for the followingequations:

yâ(x)-xay(x)=0 (7.5.8)

y(2a{x)-xay{x) =0 (7.5.9)

x2a y{2a(x) - xay(a{x) + x2a y{x) = 0. (7.5.10)

b) Any point x = XQ > 1 is an ordinary point for the following equations:

(x-l)ay(-a(x)-y(x)=0 (7.5.11)

(x - l)2a y(2a(x) + (x- l)a y(a(x) + (x - 1)2« y(x) = 0 (7.5.12)

c) The point x = 0 is a regular a-singular point for the equation (7.5.10), whilethe point x — 1 is a regular a-singular point for the equations (7.5.11) and (7.5.12).

In the previous examples, y(a represents the R-L fractional derivative £>"+ aswell as the Caputo fractional derivative cD"+.

Next for n = 1,2, we develop a generalization of the usual method used forobtaining the power series solutions for a linear equation with variable coefficientsaround ordinary points, as in the equation (7.5.6), when the fractional derivativesare of the Riemann-Liouville or Caputo type. We also extend the well-knownFrobenius method so as to obtain solutions to the equation (7.5.7) around regulara-singular points.

Remark 7.1 We should point out that we consider the case n = 1, since for lineardifferential equations with variable coefficients of order a, a G (0,1), there is nodirect method for obtaining a general solution as there is for the case when a = 1.

We shall make use of the following property, whose general proof is analogousto that in the case a = 1.

Property 7.1 Let a £ (0,1]. If f(x) is a-analytic at XQ, with convergence radiusp (p > 0), then

( OO \ OO

K+ ( E c»(* - xo)na) (*) = E c« (^ ( * - *or ) (*)n=0 / n=0

(7.5.13)(a; G [xo,xo +p)).

Next, on the basis of Property 2.1, we give the following definition:


Definition 7.11 Let a e (0,1], a 6 E and 9t(/3) £ E\Z~. Then the derivative oforder a of (x — a)'3 is defined as follows:

Z+(t - a)**) (x) := f T)(x - af~a (x > a). (7.5.14)

In accordance with this definition, we have the following property:

Property 7.2 Let a e (0,1], a 6 1 and x0 Z a- If 91(13) £ Z, then

(D2+(t - xof) (x) = r ( ^ ^ ( s - *of~a (x > x0). (7.5.15)

Proof. For x0 = a, (7.5.14) holds. If xQ > a, notice that

n=0

this series being uniformly convergent for \x\ < 1. Therefore,

(l%+{t - xo)*) (x) = fa [(* - af (l - j ^

n=0

Since *H(/3 —n) 6 E\Z~ (n 6 No), applying (7.5.14) and taking the same argumentsas above we obtain (7.5.15).

Remark 7.2 Properties 7.1 and 7.2 stay valid if the derivative D"+ is replacedby the Caputo derivative cD"+.

7.5.2 Solutions Around an Ordinary Point of a FractionalDifferential Equation of Order a

In this section we analyze the existence of solutions for the equation

\La{y)](x) := y(a{x) +p(x)y(x) = 0 (7.5.16)

around an a-ordinary point zo € [a, b] , with p(x) defined in the interval [a, b] anda G (0,1). We consider separately the cases where y(a represents the Riemann-Liouvill e and Caputo fractional derivatives of order a of the function y(x). SinceXQ is an a-ordinary point, p(x) can be expressed as follows:

Chapter 7 (§ 7.5) 419

n(x-x0)na, (7.5.17)

n=0

this series being convergent for x G [XQ, XQ + p], with p > 0.

Theorem 7.15 Let a G (0,1] and a0 G R, and let x0 G [a, 6] 6e an a-ordinarypoint for the equation

[L a(y)](x) := (DZ+y) (x) + p(x)y(x) = 0. (7.5.18)

Then there exists a unique function

n=0

which is the solution to the equation (7.5.18) for x G (xo,xo+p) and which satisfiesthe initial condition a0 = lim (x - xo)

1~ay{x).X—>Xo

Proof. We shall seek a solutions of (7.5.18) as follows:

y(x) = (x- xo)"-1 J an(x - xo)na = J2 a^x ~ ^ o ) ^1 ^ " 1 . (7.5.19)

n=0 n=0

Substituting (7.5.19) in (7.5.18) and using (7.5.15), we obtain the followingrecurrence formula which allows us to express an (n > 0) in terms of ao'.

n-kak (* = 0,1,2,- . .) (7.5.20)1 ^ n + i)ai k=o

We show that, for x G (XQ,XQ + p), the series in (7.5.19) converges.

Let r < p. Since (7.5.17) converges, there exists a constant M > 0 such that

Mrka

and, therefore,r[( n + 2)q] M ^T[(n + 2)a], . , M "

Denoting b0 = \ao\, by recurrence we define bn (n G N) as follows:

" + " l ) n i i = — ^2 bkrna (n G No).

I t is clear that 0 ^ \an\ ^ bn for n G No- The series

oo oo

n=0 n=0


is convergent for \x — xo\ < r, because in accordance with the asymptotic repre-sentation (1.5.15), there holds the following estimate:

limbn+1(x -

bn{x - xo)na

\x - x0 < 1.

This proves the convergence of (7.5.19) for x G (xo, xo +/£>) The uniqueness of theabove solution follows from the corresponding existence and uniqueness theoremin Chapter 3.

Theorem 7.16 Let a G (0,1] and ao G K. Let xo G [a, b] be an a-ordinary pointfor the equation

M)](x) := {CDaa+y) (x) +p(x)y(x) = 0 (7.5.21)

with p(x) defined in [a,b]. Then there exists a unique a-analytic function y(x) inxo as a solution to (7.5.21) for x G [XO,XQ + p], such that the initial conditiony(xo) = ao is satisfied.

Proof. The proof is analogous to that of Theorem 7.15, by seeking the series rep-resentation for the solution in the form y(x) = X^ Lo an{% — Xo)na and using thefollowing recurrence formula for coefficients an:

TIY i -n i i i "

1 [{n + l)a + 1J _ Y ^ f7Koo\r(na + l) ^—'

Example 7.8 Consider the following FDE of order a (0 < a < 1):

The point x0 = — 1 is an a-ordinary point for this equation. Let us find the generalsolution to (7.5.23) around the point xo = —1. We seek solutions of the form

oo

y{x) = (x + I ) 0" 1 Y, <*n(x + l) nQ. (7.5.24)n=0

Substituting (7.5.24) in (7.5.23), we have

a2k+i =0; (k G N), (7.5.25)

a2k = (-l)k Q(k) a0, (7.5.26)

with

Chapter 7 (§ 7.5) 421

Therefore, the general solution to (7.5.23) has the form

y{x) =ao{x \a-l (7.5.28)k=l

The particular solution to (7.5.23) which satisfies y(0) = 1 is given by (7.5.28),with ao given by:

a0 =1

(7.5.29)

k=i

Example 7.9 The general solution to the following FDE of order a (0 < a < 1)

(°DZy) (x) + (x + l)ay(x) = 0 (a < -1) (7.5.30)

is given by

with

y{x) = a0

oo\2koc

* = 1

*

TTl=l r(2ja

(7.5.31)

(7.5.32)

The particular solution to (7.5.30), which satisfies y(0) = 1, is given by (7.5.31)with

«o = 5 ^ (7-5-33)

7.5.3 Solutions Around an Ordinary Point of a FractionalDifferential Equation of Order 2a

In this section we shall consider the solutions around an a-ordinary pointx0 G [a, b] to the equation

:= y<-2a{x) +p(x)y^(x)+q(x)y(x) = 0 (7.5.34)

where a e (0,1], p{x) and q(x) are defined on an interval [a,b], and y2a and ya

represent the Riemann-Liouville or Caputo sequential derivatives of order 2a anda, respectively, of the function y{x).

Since XQ is an a-ordinary point of (7.5.34), then by definitions 7.8 and 7.9,

0' x° (7.5.35)n =0


andoo

Yln^x ~ x^na fa e bo. o + P2]; Pi > 0). (7.5.36)n =0

Theorem 7.17 Let a G (0,1], let oo,ai G K, and Ze£ #0 G [a, 6] 6e an a-ordinarypoint of the equation

[W»)](sO == (â+2/) (*) +P{x) {Daa+y) (x) + q(x)y(x) = 0. (7.5.37)

Then there exists a unique solution to the equation (7.5.37) given by

n=0

for x G (XQ,XQ + p) with p = min{gi,p2}, which satisfies the following initialconditions:

li m [(a; - a;0)1~Q:2/(a;)] = a0

Proof. We seek the solution to the equation (7.5.37) in the form

oo oo

y(x) = (x- z o ) ^1 Y, a»fa - xo)na = J2 a«fa - ^ o ) ^1 ' " " 1 . (7.5.38)n=O n=0

Calculating (D%+y) (x) and (D^y) (x), taking (7.5.15) into account and sub-stituting these values in equation (7.5.37), we arrive at the following recurrenceformula for the coefficients an:

T[(n + 3)a] ^ T[(k + 2)a]a2 = g P

Therefore the coefficients an (n ^ 2) are expressed in terms of ao and a\.

We show that the series in (7.5.38) converges for x e {xo,xo + p) withp = min{pi,p2} - Let us fix r (0 < r < p). Since the series (7.5.35) and (7.5.36)are convergent for x G [a o, x$ + r], there exists M > 0 such that

Mrka

\Pn-k\ S—^r (n E No; 0 S k S n) (7.5.40)

and

\qn-k\ ~^r ( n G N0 ; O^k^n) (7.5.41)

According to (7.5.39), (7.5.40) and (7.5.41), there holds the following estimate:

Chapter 7 (§ 7.5) 423

Let 60 = |oo|, b\ = |oi|, and let bk {k > 1) be defined by the followingrecurrence formula:

T[(n + 3)a] u _ M |T[(n + 2)a], , 1 r ,Q

T[(n + l)a]n r»

Since 0 ^ \an\ ^bn (n £ No), the series (7.5.38) converges for x £ (a;o,a;o + p) ifthe series

00 00

n=0 n=0

converges. This series is convergent because the estimate

limn—>oo

bn+1(x -

bn(x - xo)na

IX - X0\ \

follows from the recurrence relation for bn, using the asymptotic relation (1.5.15).The uniqueness of the obtained solution follows from the corresponding exis-

tence and uniqueness theorems in Chapter 3.

Theorem 7.18 Let a € (0,1], and a0, a\ £ M., and let x0 € [a, b] be an a-ordinarypoint of the equation

[L 2ay](x) := {°D2aa+y) (x) +p(x) (cDa

a+y) (x) + q(x)y(x) = 0. (7.5.42)

Then there exists a unique solution y(x) of (7.5.42), for x £ (zoi o + p) with p =min{pi,p2}- This solution is a-analytic function in xo and satisfies the followinginitial conditions:

lim y(x) = a0X-+XQ

andlimx-+0 {CD%+y) (x) = ax.

Proof. Seeking a solution to equation (7.5.42) in the form

n=O

we arrive at the following recurrence formula for the coefficients an:

A T[(k + l)a +1kQn+2 Y{na + 1) =

fc^ iPn-k T(ka + 1)

The rest of the proof is analogous to that in Theorem 7.17.

Example 7.10 Find the general solution to the following FDE of order 2a(0 < a < 1), which could be called the fractional Ayre equation:


(V2ay) (x) - xay(x) = 0 (7.5.44)

with V(2ay = DaDay, for Da := D%+ (case (a)), as well as for Day := cD%+y(case (b)).

a) As shown earlier, we seek a solution in the form (7.5.38)

(7.5.45)

(7.5.46)

n=0

Substituting (7.5.45) in (7.5.44), we have

It follows from here that O2k-i = 0 (fc G N). Supposing that ao and arearbitrary constants, we obtain the following general solution to (7.5.44):

y{x) =

where

+k=i fc=l

=n r(3j - l)a) T(3ja)l =lr((3i

(7.5.47)

(7.5.48)

b) The general solution to (7.5.44) with Day := cD^+y is obtained similarly to theprevious case, and this solution is given by

y(x) = a0 1 + V akx3ka

fe=l

withr(3j - 2)q r((3j -

=

(7.5.49)

(7.5.50)

7.5.4 Solution Around an a-Singular Point of a FractionalDifferential Equation of Order a

In this section we obtain the fractional power series solutions of a homogeneousLFDE of order a (0 < a ^ 1), around a regular a-singular point x0 of the equation.It will be more convenient, for our purposes, to consider the differential equationwritten as

\La(y)](x) := (* - xo)ay{a(x) + q(x)y(x) = 0, (7.5.51)

where q(x), due to the regular a-singular character of xo, is an a-analytic functionaround XQ.

Chapter 7 (§ 7.5) 425

Theorem 7.19 Let x0 ^ a be a regular a-singular point of the differential equa-tion (7.5.51) of order a, and let

ln{x-xo)na (7.5.52)

n=0

be the power series expansion of the a-analytic function q(x). Then there existsthe solution

oo

y(x; a, 8l) = (x - zo)Sl ^ an(x - xo)na (7.5.53)

ra=0

of equation (7.5.51) on a certain interval to the right of XQ. Here OLQ is a non-zeroarbitrary constant, si > —1 is the real solution to the equation

F(s - a + 1) ' w ~ "

and the coefficients an (n ^ 1) are given by the following recurrence formula:

T(na + s - a + 1) ô-n = ^ 7 ; — n— > J 0,1 qn..[.

Moreover, if the series (7.5.52) converges for all x in a semi-interval 0 < x — Xo <R (R> 0), then the series solution (7.5.53) of equation (7.5.51) is also convergentin the same interval.

Proof. Seeking a solution to equation (7.5.51) in the form (7.5.53), taking fractionaldifferentiation of y(x;a,si) with using (7.5.14) and substituting the result and(7.5.52) into (7.5.51), we get

My(*;«,«))](*) - E [^rgfrrî) ] +E^--'](»-«or+'

ao [ qo\{x - XoY = °' (7-5-54)

If we now put fo(s) = — —- + qo, then, from (7.5.54), we obtain1 (s — a + 1)

aofo{s) = 0 (7.5.55)

andn- l

fo(na + s)


Suppose that a0 7 0, then /o(s) = 0. Thus, if s\ is the only real root of the equa-tion fo(s) = 0, then the expression (7.5.56) provides, by recurrence, the coefficientsan of (7.5.53) in terms of a®.

Now we prove the convergence of the series. Let 0 < r < R. Since the series(7.5.52) is convergent, there exists a constant M > 0, such that

Mr la

Applying now (7.5.57) to the relation (7.5.53), we have

M «A \an-k\

(7.5.57)

\fo(na £k=\

Now we define c0 = |ao| and cn =M

rka

0-n-k for n > 1. Then

Cn+1 M fo{na-\fo((n +l)a + s)\ /o((n + l)a

and using the asymptotic representation (1.5.15), we get

cn+i(x — xo)(n+1^

cn{x-x0)na

/ |a -a ;o |\

when n -> oo. Therefore the series Y^ cn(x - xo)na converges for all x such that

n=0

0 < x — x0 < r.From this we conclude that (7.5.53) converges for 0 < x — XQ < R.

Example 7.11 Consider the FDE

(x-l)ay(a(x)-y(x)=0, (7.5.58)

where yâ(x) represents either the Caputo or the Riemann-Liouville fractionalderivative.

Since the point x = 1 is a regular a-singular point of (7.5.58), we shall seek asolution to this equation around the point x = 1 in the form

(7.5.59)n =0

As we have seen earlier, s must be a real solution (which always exists) of the frac-tional index equation associated with (7.5.58); namely, of the following equation:

T{s-a= 1 (7.5.60)

Chapter 7 (§ 7.5) 427

Suppose that the above-mentioned solution is s = ft, (—1 < j3 < 0).I t is directly verified that an = 0 (n e N). Then the general solution to

equation (7.5.58) has the following form:

y{x) = ao{x-lf (aoÔ). (7.5.61)

Naturally, the result would be the same if we directly seek a solution as

y{x) = c(x - l ) s. (7.5.62)

Lastly, let us see the general solution to equation (7.5.58) for a = 0.01, 0.1, 0.3,0.5, 0.6, 0.9, 0.95, 0.99. The solution is given by

y(x) = C(x - if, (7.5.63)

where the /? with the corresponding a are given in the following table:

a P

0.01 0.466640.1 0.512010.3 0.615060.5 0.72118 (7.5.64)0.6 0.775390.9 0.942670.95 0.971240.99 0.99423

7.5.5 Solution Around an a-Singular Point of a FractionalDifferential Equation of Order 2a

We now focus our attention on the following homogeneous sequential linear frac-tional differential equation of order 2a (0 < a ^ 1):

[Ua(y)](x) := (x - z o ) 2 V 2 a ( z ) + (X- xo)ap(x)y(-a(x) + q(x)y(x) = 0. (7.5.65)

If x0 ^ a is a regular a-singular point of equation (7.5.65), then the functions p(x)and q(x) are a-analytic around x0 and, therefore, they have the following seriesexpansions:

oo

p(x) = J2Pn(x-xo)na (7.5.66)

71=0

and

q(x) = Y/qn(x-x0)na, (7.5.67)

n=0

valid on a semi-interval 0 < x — xo < R for some R > 0. Our goal is to find asolution to (7.5.65) in the form

oo

y{x; a, s) = {x - xo)s an(x - xo)

na (7.5.68)n=0


with a0 j^ 0, s being a number to be determined.Differentiating (7.5.68) we get for 9t(s) > a - 1 and s £ Z~,

n =0 nr(na + s-a + l) {x x°)na+'~a

and

„ r\ \ ' — » i - f

Substitution of these expressions into the equation (7.5.65) yields

[l<2a(y(t; a, s))]{x) = aofo(s)(x - xo)s

n-l

[ ] = 0, (7.5.69)n=l (=0

where

and

fk(s)=Pkr, \ \ +gfc- (7.5.71)1 [s — a + 1)

Thereforeaofo(s) = 0 (7.5.72)

andn - l

/o(na + s)

Since we assumed that ao ¥" 0, then from (7.5.72) we get

Equation (7.5.74) is the so-called "fractional indicial equation" of the differ-ential equation (7.5.65). We will assume that the equation (7.5.74), for values swith fH(s) > —1, has two roots which, if complex, must be complex conjugates(because T(z) = T(z) for all z eC).

If si is the larger real root of /o(s), then we could get a solution to (7.5.65) inthe form (7.5.68), which we denote by y{x;a,s\). The coefficients an, determinedby the recurrence formula (7.5.73), for any n ^ 1 have the following explicit

Chapter 7 (§ 7.5) 429

representations:

(-!)»

/ i

h

fn-fn

(«i) fo{si+a)(si) /i(«i + oi)

i(si)/n_2(si +a)

(*i) /«-i(si+a)

/o(-

/n-3

/n-2

0Si + 2a)

h 2a)h2a) ...

fo(si

00

+ ( n -+ ( n -

l )a)l )a)

/o(si + a) /0(si +2a) / 0 ( s i + n a ) a0

(7.5.75)

If s2 is the other root of the indicial equation fo(s) = 0, then the expression(7.5.75) with si replaced by s2 yield a second solution to equation (7.5.65), exceptfor the case si = s2 + not for all n 0.

The above arguments can be summarized as follows:

Theorem 7.20 Let XQ a be a regular a-singular point of the equation (7.5.65)and let the series (7.5.66) and (7.5.67) be convergent on a semi-interval0 < x — XQ < R with R > 0. Let si, s2 > a — 1 be two real roots of the fractionalindicial equation (7.5.74) with si > s2 and si — s2 na with n £ No- Then, inthe interval 0 < x — XQ < R, the equation (7.5.65) has two linearly independentsolutions:

0 0

= (x- xo)Sl ^2 an(si){x - xo)

na, ao(si) 0 (7.5.76)n=0

and

y2(s; a, s2) = {x - xo)S2 an(s2)(a; - xo)

na, ao(s2) 0. (7.5.77)n=0

Here the coefficients an(si) and an(s2) are given in terms of ao(si) and ao(s2) bythe recurrence formula (7.5.73) with s replaced by si and s2, respectively. More-over, the series

n=0

Jân(Sl)(x-x0)na (7.5.78)

-x0)na (7.5.79)

n=0

converge for 0 < x — XQ < R.

Proof. By the above arguments, to complete the proof of this theorem we wouldshow the convergence of the series (7.5.77) and (7.5.78). The details of this proofare analogous to those used to prove the convergence of the Frobenius series so-lution near a regular singular point of an ordinary differential equation of secondorder. In addition, we must take into account the above-mentioned asymptoticrelation (1.5.15).


There still remains the problem of how to find a second solution to equation(7.5.65) in the case when si - s2 = na (n £ No). The following theorem gives ananswer to these special cases.

Theorem 7.21 Let x0 ^ a be a regular a-singular point of the equation (7.5.65),and let the series (7.5.66) and (7.5.67) be convergent on a semi-interval0 < x — xo < R with R > 0. Let s\ and s2 be two real roots of the fractionalindicial equation (7.5.74) with si,s2 > a — 1 and si ^ s2. Then, in the interval0 < x — x0 < R, the equation (7.5.65) has one solution of the form (7.5.68):

-xQ)na (ao(si) ^ 0), (7.5.80)n=0

where the coefficients an{si) are given in terms of ao(si) by the formula (7.5.73)with s replaced by The following assertions also hold:

a) // si = S2, then a second solution to (7.5.65) has the following form:

where

y2(x; a,Sl)= Vl(a, 8l) lg(x - x0) + £ M * - zo)" Q+Sl, (7.5.81)n=0

bn = | - KOS

(7.5.82)

and an(s) is given by (7.5.73).

b) If s\ — S2 = na with n £ N, then a second solution to the equation (7.5.65)is given by

y2(x; a, s2) = Vl{a, 8l) A{x) lg{x - x0) + ^ cn(s2){x - xo)na+S2, (7.5.83)

n=0

where A(x) is a function obtained by evaluating the derivative

A(x) = §;[(*- *2) y{x;a,«)] [ = (7.5.84)

Moreover, the series (7.5.80),), (7.5.81), and (7.5.82) are convergent for thevalues of x on the semi-interval 0 < x — x0 < R.

Proof. First consider the case a). It is clear that yi(x;a,s\) in (7.5.66) is thesolution to the equation (7.5.65). To get the second solution we make argumentsanalogous to the case of ordinary differential equations.

Consider s as a continuous variable and determine the coefficients an(s) ofy(x; a, s) as functions of s, using formulas (7.5.73). Then

[L 2a{y{t; a, s))}(x) = a0fo(s){x - xo)s. (7.5.85)

Since /o(s) = (s — si)2g(s) (with analytic g{s)), then

[L 2 a[ — V(*;«>*) | ]](* ) =0, (7-5.86)

Chapter 7 (§ 7.5) 431

and so the second solution y2{x; a, si) is obtained from specifying the derivative

^-y(x;a,s) (7.5.87)as s=si

In the case b) the proof is also analogous to that of the ordinary case. Consider

y*(x; a, s) = (s - s2) y(x; a, s), (7.5.88)

where we again consider the coefficients an(s) of y(x; a, s) as functions of s givenby (7.5.73). Then there exists a function A(x) such that

y*{x; a,s2) = A(x) y(s;a,Si). (7.5.89)

On the other hand, by using (7.5.85), there also holds that

o) = 0, (7.5.90)

and, therefore,

^-Us-s2)y(x;a,S)]\ (7.5.91)OS L J \s=s 2

will be the second solution. Finally, evaluating the derivative (7.5.91) and taking(7.5.88) into account, we obtain the desired result (7.5.83).

Remark 7.3 Let us emphasize the fact that, for the case when the Riemann-Liouvill e fractional derivative is replaced by the Caputo derivative, the resultsobtained for solutions around regular a-singular points coincide exactly with thosein the Riemann-Liouville case.

Example 7.12 Consider the following sequential FDE of order 2a (0 < a < 1):

(x - xo)2ay{2a(x) + P0(x - xo)

ay{a(x) + Qoy(x) = 0, (7.5.92)

where yâ{x) is the Riemann-Liouville fractional derivative (D"+y) (x) or theCaputo derivative {^D"+y) {x), and Po and Qo are constants. We shall seek atleast one solution to the equation (7.5.92) around the regular a-singular pointx = xo > a (a € M.). Let us look for a solution in the form:

y = (x - xoy, (7.5.93)

This expression with s — 0i, is the solution to the equation (7.5.92), if /3i is asolution to the fractional indicial equation associated with (7.5.92):

T V . , I 1 \ T V - I 1 N

- + Qo = 0. (7.5.94)r ( a - 2 a + l) u T ( s -a + l) ' ^ u

Depending on the values of Po, Qo, and a, the equation (7.5.94) will have onereal solution, two distinct real solutions, two double real solutions, or two complexconjugate solutions. If there are two real solutions, we take the greater one.


Next we present a table where, for given values of Po, Qo and a, the corre-sponding roots 0\ and 02 of (7.5.94) are given.

a Po Qo 0i 02

(7.5.95)

0.9 50.6 50.30.1io- 3

0. 70.6

111111

1.9

- 1- 1- 1- 1- 10. 7

1.252 9

-0.8239 9-.09798 1-0.1470 3-0.6895 3-0.9969 9

-0.6965 1 + 0.2314 7 i0.75708 3

0.9347 90.5059 4

———

-0.6965 1 - 0.2314 70.7578 3

So, for example:a) If a = 0.95, Po = 1 and Qo = —1, then the following linearly independent

solutions of (7.5.92) around x = x0 exist:

t/i(z) = (z -z0) - 0- 8 2 3 99 (7.5.96)

2/2(z) = (z-z0) 0- 9 3 4 79 (7.5.97)

b) If a = 0.3, Po = 1 a nd Qo = ~lj then only one solution exists of (7.5.92)around x = xo, given by:

2/i(z) = (x - so)"0 1 4 7 03 (7.5.98)

c) If a = 0.7, Po = 1 and Qo = 0.7, then two linearly independent solutions of(7.5.92) around z = zo exist:

yi = Re[{x - xo)c+di] (7.5.99)

2/2 = Im[(x - xo)c+di] (7.5.100)

where c = -0.69651 and d = 0.23147.d) If a = 0.6, Po = 1.9 and Qo = 1.2529, then the equation (7.5.92) has the

following linearly independent solutions around x = xo-

2/i = (z - xo)0-757083 (7.5.101)

2/2(z) = £-\(x- xo)s] =(x- zo)°-757O83L(z - zo) (7.5.102)

OS L J s=0.757083

Example 7.13 Consider the following generalized Bessel equation of order v = 0:

x2ay{2a(x) + xay{a{x) + x2ay(x) = 0 (0 < a < 1) (7.5.103)

Let us find two linearly independent solutions of (7.5.103) around the regulara-singular point x = 0.

The relations (7.5.66) and (7.5.67) with the corresponding constants pn and qn

(n ^ 0) take the following forms:

p{x) = 1 and po = 1; pn = 0 (n ^ 1),

Chapter 7 (§ 7.6) 433

q(x) = x2a and q0 = 0; qi = 0; q2 = 1; qn = 0 (n ^ 3).

Therefore, the indicial equation associated with (7.5.103) is given by

r(a-2o+l)

This yields the following solutions for different values of a:

a 0i 02

0.990.80.650.628lO"4

-0.459-.6784-0.9629-0.1312-0.9999

0.081940.0535

-0.1051—

(7.5.105)

This table lets us obtain a solution around x = 0 of (7.5.103) for a = 0.628 and fora = 10~4, and two linearly independent solutions in the cases a = 0.99, a = 0.8,and a = 0.65.

Remark 7.4 As in the case of LFDE, the theory presented above can be appliedto cases including non-sequential derivatives of Riemman-Liouville by the use ofthe relation (7.1.4).

7.6 Some Applications of Linear Ordinary Frac-tional Differential Equations

As we have already indicated, there are various applications of fractional differen-tial equations in practically all areas of science and engineering.

It must be noted, however, that although there are many applications, thereis still no clear interpretation of the different concepts in the context of fractionalderivatives.

We will not try to present here an exhaustive resource of these applications. Forthat we refer the reader to the works of Carpinteri and Mainardi [132], Podlubny[682], Hilfer [340], Metzler and Klatfer [586], Schumer et al. [747], Zaslavsky[911], Turchetti et al. [834], Sokolov et al [781], and Kilbas et al. [404], where animportant collection of the applications of fractional differential equations may befound.

In this section we will solve some generalized differential models related to thedynamics of a sphere immersed in an incompressible viscous fluid with oscillatoryprocesses that have fractional damping terms. It should be pointed out that the useof damping terms also provides a very important tool in the study of fluid dynamicsacross a porous boundary. Likewise, we will resolve the four-parameter modelproposed by Bagley and Torvik and Caputo and Mainardi (with different fractionalderivatives) for modeling the mechanical behavior, from a rheologic standpoint, ofcertain new materials, especially polymers, which have viscoelastic properties.


Lastly, we will formally consider two fractional models associated with slowdiffusion which are resolved through the method of separation of variables.

We note here the following property which will be of great use in some of theapplications that will be covered in this section.

(Dny)(t) = {Vpaa+y){t) (Vn,p G N; a = - ; p > n; t > a). (7.6.1)

This is an immediate remembering that, if y £ C([a, b]), then

£ (7.6.2)

From this we conclude that any ordinary differential equation can be expressedas a sequential fractional differential equation.

For example, the following differential equation

x'(t) - a2x{t) = 0, (7.6.3)

where x'(t) = (DQ'+ Do'+ x)(t) for t > 0, can be expressed as follows:

{Vl%x){t) - a2x{t) = 0 (a = 1/2). (7.6.4)

By Corollary 7.1, its general solution is given by

x(t)=C1e% + C2e-at (7.6.5)

and since a:(0) < oo, we have Ci = —C\. Therefore,

and since [1 - (-1)-7'] = 0 for j = 2k (k £ N), the solution (7.6.6) takes the form

x{t) = Y, ^f- = Ceah, (7.6.7)

which is the general solution to (7.6.3), as expected.

7.6.1 Dynamics of a Sphere Immersed in an IncompressibleViscous Fluid. Basset's Problem

The dynamics of a sphere immersed in an incompressible viscous fluid is a classicalproblem with numerous applications in engineering and in the study of geophysicalflows. A particularly important problem is the study of a sphere subjected togravity, which was considered by Basset first in [70], and subsequently in [71], whointroduced a special hydraulic force, generally known as "Basset's force". Thisforce was interpreted by Mainardi [518] in terms of a fractional derivative of order1/2 of the velocity of the particle relative to the fluid.

Chapter 7 (§ 7.6) 435

Mainardi et al. [533] introduced a new formulation of that force, called the gen-eralized Basset's force, based on the fractional Caputo derivative CDO+

(0 < a < 1) and on a generalized model of the one originally considered byBasset. That is, when the fluid is at rest and the particle moves vertically underthe effect of gravity with a certain initial velocity F(0+) = Vo:

V'(t) + a (C£>o+ ) (*) + V(t) = 1; V(0+) = V0 {a > 0; 0 < a < 1). (7.6.8)

This problem arises, in particular, from the following fractional relaxation equa-tion, also studied by Mainardi [518], using the Laplace transform [see also Gorenfloet al. [310]]:

U'(t) + a ( ô+ t f) (*) + u(t) = 9{t)\ 17(0+) = Co (0 < a < 1). (7.6.9)

Equation (7.6.9) can be solved explicitly for a 6 Q, without using integraltransforms, with the theory developed in this text and keeping in mind (7.6.1),similar to the method we will use here to solve Basset's original problem:

U'(t) + a{Dy+U){t) + U(t) = 1; (a > 0; t > 0) (7.6.10)

17(0) = 0. (7.6.11)

If we get U'(t) = (V2aU){t) (a = 1/2), the equation (7.6.10) reduces to theform

[L 2a{U)]{xt) = {V2%U){i) + a{D%+U){t) + U{t) = 1. (7.6.12)

This is a non-homogeneous sequential LFDE with constant coefficients, its solution

dependent on the roots of the characteristic polynomial m2 +am + l = 0 associated

with the corresponding homogeneous equation. These roots are the following:

. Ai j 2 = 2 1 1 ^ ^2)i a ^

2. Ai = A2 = - | eR, if a = 2.

3. A2 = Ai, with Ai = -«+»V4-a2; if a < 2.

Therefore, by applying Theorem 7.4, the solution U(t) of (7.6.12) with theinitial condition {7(0) =0 is given as follows:

1. If a > 2,

Jo= f [te^t-"W-' )dridZ = \i1 r

Jo J£ Jo


2. If a = 2,

3. If a < 2

U(t) =Jo

U(t) =

(7.6.14)

(7.6.15)

7.6.2 Oscillatory Processes with Fractional Damping

In 1984, Bagley and Torvik [52] considered the following initial-value problem:

(7.6.16)y"(t) + a (Dl'^y) (t) + by(t) = f(t) (t > 0),

y(0) = 0; i/'(0) = 0,

with a = S^P and b = , and solved this problem by using the Laplace trans-form. This model is associated with the following problem.

Consider a thin, rigid plate of mass m and area s, immersed in a Newtonianfluid of infinite extension with density p and viscoelastic constant fi, and connectedto a fixed point via a spring with spring constant k. In addition suppose that thesurface of the plate is sufficiently large so that the fluid slows the movement of theplate.

Newtonian Viscous Fluidip = densityfj, — viscosityf(t) = displacement of the plate

Figure 1

Another approach, using the Laplace transform, was studied by Mainardi [515]for the following initial-value problem:

V"(t) + aV(0) = Co; V'(0) = Cu

(t) + bV{t) = g{t) (a > 0, 0 < a < 2), (7.6.17)

Chapter 7 (§ 7.6) 437

as a model associated with damping processes with a fractional damping term.

Both models can be solved directly by applying the fractional differential equa-tion theory developed in Chapter 7 of this book.

Nigmatullin and Ryabov [625], Kempfle and Gaul [365], and Beyer and Kempfle[82] also studied the model (7.6.17), but they considered the fractional dampingterm Z)° V or DQ+V instead of CDQ+V, in order to use the Fourier transform.

Here we solve a special case of (7.6.16).

Consider the function

{8, if 0 ^ t 1,0, if t > 1,

and the differential equation

y"(t) + 3 (D$y) (t) + y(t) = /(*) (t > 0), (7.6.19)

with the initial conditions

2/(0) = 0; y ' (0)=0. (7.6.20)

We set y"(t) = yD0^_y\ (t), taking into account that y is differentiate, and

the following Property:Let r\ > 0 and let the functional series {/ n}^_ 0 ê suc^1 that, for t € [a, b]

(D2+fn)(t) < oo (Vn e No). If the series £ £ o / i (0 and T,T=o(D2+fj)(t) areuniformly convergent on any [a + e, 6], e > 0? then

(a < t < b). (7.6.21)

Then the equation (7.6.19) reduces to the sequential LFDE given by

[L4«(»)](*) := {Vta+y) (t) + 3 (Vla+y) (t) + y(t) = f(t) (a = 1/2), (7.6.22)

which can be written in vectorial form as follows:

{D%+Z) (t) = AZ(t) + B, (7.6.23)

with the changes ( ^ + y) (t) = zj+1 (j = 0,1,2,3), in which

0 1 0 0

(7.6.24)A = l I 0 I I I ' S ^ = l n I 5 Z{t) =- 1 0 0 V

In addition, the initial conditions y(0) = y'(0) — 0, knowing that y(t) must bedifferentiate in [0, t], guarantee that

y(0) = (D%+y)(0) = (Dg?i/)(0) = (Do3?2/)(0) = 0 (7.6.25)


and therefore, the desired solution to (7.6.23) must satisfy Z(0) = 0.

Let P be the similarity matrix for the diagonalization of A, that is, it satisfiesthe relationship

Ai 0 0 0° A2 0 00 0 A3 00 0 0 A4

where Xj (j = 1,2,3,4) are the eigenvalues of A. It is directly verified that

(7.6.26)

P =

-3.363 + 0.556i -3.363 - 0.556i-0.911 + 2.073z -0.911 - 2.073i0.822 + 1.260i 0.822 - 1.260i

1 1

- l

-0.0380.114

-0.3381

(7.6.27)

and that its inverse matrix P is equal to

-0.092 - 0.019i -0.050-0.13K 0.123 - 0.172i 0.044 - 0.044z \-0.092 + 0.019i -0.050+ 0.13H 0.123+ 0.172i 0.044 + 0.044i

0.351-0.168

-0.1190.220

0.040-0.287

1.040-0.128

and also that the eigenvalues of A are given by

\i = 0.363 - 0.556iA2 = 0.363 + 0.556iA3 = -2.962A4 = -0.765.

I(7.6.28)

(7.6.29)

If the transform Z(t) = PW(t) is applied to the equation (7.6.23), it reducesto

where

with

B(i) = = bf(t),

(7.6.30)

(7.6.31)

(7.6.32)

0.044 - 0.044i \0.044 + 0.044i

1.040-0.128 /

and with the initial conditionW{0) = 0. (7.6.33)

The only solution to (7.6.30) which satisfies (7.6.33), is obtained by applyingTheorem 7.11 and this solution is given by

W{t) = 8 / e£(

./o= 8 (7.6.34)

Chapter 7 (§ 7.6) 439

Therefore,

t") \

Z(t) = 8P I ei^m = 8PJ" 1

hEQ(X4ta)

(7.6.35)

where Ea(z) is the Mittag-Leffler function (1.8.1), and hence the real part of thissolution has the following form:

y(t) = Re[zi(t)] = 10-2i?e[(-4.8 + 1.4i)£Q(Ai* a)

+(0.9 - 7Ai)Ea(X2ta) + O.M8Ea(X3t

a) - 0.59Ea(X4ta)}. (7.6.36)

7.6.3 Study of the Tension-Deformation Relationship ofViscoelastic Materials

The appearance and proliferation of the so-called "new materials" beginning in themiddle of the 19th century, such as plastics, rubbers, fibers, resins, and polymers,have provided a set of materials whose mechanical behavior cannot be explainedvia the classical rheologic means, such as those contained in the theories of Kelvinor Maxwell.

It should be noted that the field of rheology studies, among other things, theflow and deformation of materials with unusual behavior, particularly the flowof non-Newtonian fluids, the behavior of solids which can flow, or the flow ofsubstances across porous media with fractal geometry.

Viscoelasticity is an intrinsic property of many of the "new materials", andit also constitutes an element of study within rheology. This property describesvarying degrees of behavior between elasticity and viscosity, depending on thematerial (see, for example, Gonzalez [290], Podlubny [682] Mainardi [518, 520],Rossikhin and Shitikova [718], Caputo and Mainardi [129, 130] and Christensen[139]).

Depending on the temperature of the materials, these will achieve, in general,four different states:

1. Vitreous state (in the vitreous region): this region contains organic crystalswhich only allow small deformations;

2. Transition state (in the transition region): this region contains, among oth-ers, linear thermoplastic and reticular polymers (cellulosic materials,polyamides, polyesters . . . );

3. Elastic state (in the elastic region): this region contains elastomeric materials(rubbers, plastics...);

4. Fluid state (in the influence region).


At ambient temperature, a given polymer may be found in one of these fourregions.

In this book we will only consider the models which attempt to explain theviscoelastic deformation of certain materials.

Many rheologic models have been developed to study this behavior. In general,we have the following model:

E da ^ dtaklW = l^ bk^

a)

Next, as an example, we will describe Burger's model.

If a sample of an amorphous polymer is subjected to a constant tension, aninstantaneous deformation will be observed when the force is applied. Ifthe force is immediately relaxed, the deformation disappears. This state iscalled the "elastic domain". Mechanically, this behavior is represented witha spring of modulus E\ which describes the instantaneous elastic response:

a(t) = Exe{t) (Hooke's Law), (7.6.37)

where <r(i) represents the tension (force/surface) and e(t) is the unit defor-mation (AL/L).

Hook Newton

Maxwell

V r l ni

Voigt

t

4-

cr = Ee + rje'

Chapter 7 (§ 7.6) 441

Zener Kelvin

b) As the tension is prolonged, we witness a transition to a "deformation as afunction of time". The deformation slows until it progresses at a constantspeed. This mechanism is represented as being made up of a Voigt typeelement (E2, h2) and which corresponds to a "delayed elasticity". When thetension is removed, the material begins to return to its original shape but therecovery slows and finally disappears before it is completed. Mechanically,this behavior is represented by the parallel grouping of a spring and a piston

a(t) = E2e(t) + (Voigt's model).

c) Lastly, a piston h\ represents the viscous fluid

a(t) = 7]de(t)

dt(Law of Newtonian Fluids).

(7.6.38)

(7.6.39)

The corresponding mechanical representation of Burger's model is shown in thefollowing figure:

E2

1— vh2

hia(t)

Ia = EXE

dt

Figure 2

Therefore, this model has the following mathematical representation associatedwith it:

a"(t) + ai<r'{t) + aoa(t) = b2e"(t) + bie'(t) + boe(t), (7.6.40)


whereEi + E2 Ei Ei -E2

ai = 1- ——; a0 = - 7 - —— (7.6.41)h2 hi hih2

b2 = Ei; bi — —; ; bo = 0. (7.6.42)

Thus the ordinary linear model was widely studied. For example, Caputo-Mainardi [130] and Christensen [139]) have used the following model:

a{t) = e(t), (7.6.43)

where p = q or p = q + 1.

This model represents a series or parallel combination of basic Zener or Kelvintype units.

These models, however, did not adjust themselves well to the behavior demon-strated by many viscoelastic materials. Gerasimov [279] was probably the first topropose, in 1948, a study of models on the anomalous dynamics of the mechanicalbehavior of these viscoelastic materials. He used a generalization of the classicalmodels which implied that the tension could be proportional to the R-L fractionalderivative to the right of D+ of the material deformation, that is,

u{t) = E{D%e){t) (0 < a < 1). (7.6.44)

In 1947, Blair [752] suggested an analogous generalization using the R-L frac-tional derivative Dg+, while Caputo and Mainardi [130, 129] in 1971 and Goren-flo and Mainardi [304] in 1997 suggested an analogous generalization based onthe Caputo derivative to the right CD^_ or that of Caputo CDQ+ considering thedeformation and tension to be negligible for t ^ 0.

These observations were made based on the behavior, experimentally verified,that the modulus of a viscoelastic material in relation to its frequency, that is, themodulus

where a(is) and i(is) represent the Fourier transform (or the Laplace transform,as the case may be) of a(t) and e(t) respectively, adjusts itself better to real powersof order a (0 < a < 1) of the frequency s than to natural powers, as was classicallythought.

This drove many specialists, most notably: Nutting [639], Gemant [277, 278]Blair [752, 753], Blair and Caffyn [754], Rabotnov [692, 693] and Meshkov andRossikhin [573] before 1950; between 1966 and 1969 Caputo [120, 121, 122], Ca-puto and Mainardi [130, 129] (who, in addition, suggested the use of the fractionalderivative in seismological and metallurgical dissipation models); and, in the last25 years, Lokshin and Suvorova [497], Caputo [123, 124, 125], Mainardi [516],Mainardi and Bonetti [523], Smith and Vries [772], Scarpi [736], Stiassnie [803],Bagley and Torvik [50, 51, 52, 53], Rogers [714], Koeller [431, 432] Koh and Kelly

Chapter 7 (§ 7.6) 443

[433], Friedrich [266], Nonnenmacher and Glockle [635], Glockle and Nomenmacher[285], Makris and Constantinou [539], Fernander [257], Pritz [688], Rossikhin andShitikova [718], and Lion [490], to propose the following fractional model whichgeneralizes (7.6.43)

p p

va" (*) + J2 akDaka{t) = ] T bkD0ke{t) (7.6.46)

fc=l k=0

with ak,bk,a.k,Pk £K (k = 0,1,...,p) and where Da represents the appropriatefractional derivative such that the Fourier transform of (7.6.46) becomes

p

\(is)an +2_âk(is)L A=l

&(is) =u=o

i(is). (7.6.47)

Along these lines, Caputo and Mainardi proposed in [130] the following model,based on four parameters, as the most adequate for representing the viscoelasticbehavior of certain materials from a rheologic point of view:

[1 + bDa] a(t) = [Eo + EiD"} e{t) (0 < a < 1). (7.6.48)

Between 1979 and 1986, Bagley and Torvik [50, 51, 52, 53, 54] and Bagley [45]made a series of fundamental contributions which would consolidate this as thedefinitive model showing in 1986 [54] that, with the following restrictions

Eo ^ 0, El > 0, b 0; Ex ^ bE0, (7.6.49)

the model satisfies the First and Second Laws of Thermodynamics.In [54], Bagley and Torvic also showed experimentally that this model is in

close agreement with the behavior of over 150 viscoelastic materials.It must be noted that although Bagley and Torvik suggested the R-L fractional

derivative -DQ+, their work does not make clear which Da fractional derivative isbest suited to their model, since their only consideration was that the Fouriertransform should satisfy the relation

T{Daf{t)){s) = (is)af(is), (7.6.50)

where f(is) = J"(/(i))(s), and with this sole condition there are several definitionsof a fractional derivative which can be of use, such as that of Griinwald-LetnikovGD°l, that of Liouville £>" or that of Caputo-Liouville CD^_, over infinite intervals,or that of R-L Dg+f (if f(t) = 0 for t < 0) or D%+f (if f(t) = 0 for t < a), orthose corresponding to Caputo CDO+ and cDa+.

Here we present, as an example of the application of the theory developed inthe previous Chapters, the explicit solutions of diverse problems based on the four-parameter model (7.6.48). We propose a generalization of the same type whichis perhaps more representative of the behavior of certain viscoelastic materialsand whose explicit solution follows from the application of the theory presentedpreviously.


It must be pointed out that the methods used for obtaining solutions do notrequire the use of an integral transform, which in a way provides a certain degreeof flexibility .

Proceeding from the four-parameter model (7.6.48), generally accepted,

a{t) + b{Daa){t) = Eoe{t) + E1(Das)(t) (0 < a 1), (7.6.51)

with b ^ 0, Eo ^ 0, Ei > 0, b ^ ^ , and where Da represents an adequatefractional derivative, we find that (7.6.51) can be represented as follows:

a) For a known tension a(t),

(Daz)(t) + Xz(t) = f(t), (7.6.52)

with z(t) = £<r(t) - e(t), \=§£,A = i , and f(t) = Aa(t).

The general solution to (7.6.52), when Da = D"+, if a is continuousand integrable in (a, t), is given by

z(t) = Ce-X^ +A f e-A<*-«MOde, (7-6-53)J a

where C is an arbitrary real constant.Therefore the general solution to the equation (7.6.51) has the form

e(i) = —cr(i) - Cea 1 -^— / ea

x CT(O^- (7.6.54)

Moreover, if C = 0 and <r € C([a, i]), then e(a) = -J-cr(a). In particular,e(a) = <7 (a) = 0, if C = 0 and a € C([a, £]), while for the case when

l im ( ^ a +( i -a) M* ) = *i» ; (7.6.55)

the solution to the equation (7.6.51) is given by (7.6.54), where

C = T(a)[~k1-k2\. (7.6.56)

In addition, by application of Theorem 7.13, we derive the generalsolution to the equation (7.6.52) with Day = cD™+y:

(t)=J a

(7.6.57)

If z(a) = k, then the corresponding solution to the equation (7.6.51) has theform

— \h I At- 4- h 17 (\ ^Si

Chapter 7 (§ 7.6) 445

b) For a known unit deformation e(t),

(Day)(t)+jy(t)=g(t), (7.6.59)

with y(t) = fe(t) - a(t), 7 = I, B = * , and g(t) = Be(t).

If e(t) is continuous and integrable in (a, t), then the general solutionto the equation (7.6.59) with Day = D"+y is given by

f e-./a

y(t) = Ce-^'-") + B f e-**-« e(Od£- (7.6.60)

Therefore the general solution to the equation (7.6.51) can be represented asfollows:

a(t) = e(t) - Ce~^-a) - ^ ° f' e ^ ^ R . (7.6.61)

In addition, if C = 0 and e e C([a,i\), then <j(a) = ^£(0). Inparticular, s(a) = cr(a) = 0, if C = 0 and e G C([a, £]), as expected.

In the case when the conditions in (7.6.55) are satisfied, the solutionto the equation (7.6.51) is given by (7.6.61) with

C = r(a) (^-k2 - k^\ . (7.6.62)

Moreover, applying Theorem 7.13 to the equation (7.6.59) withDay = cD^+y, we obtain the following general solution:

V(t) = f eZ(t-?) [Be(0 - jk] d£ + k,J a

(7.6.63)

if y(a) = k, and the corresponding solution to the equation (7.6.51) has theform:

a{t) = fe(t) - I* el^) [ ( ^ ^ ) e(0 - 7*] <% + k. (7.6.64)

As we have already noted, the model (7.6.51) is the one proposed by Torvik-Bagley with Da = D%+. Therefore in the case a = 0, (7.6.54) or (7.6.58) representthe solutions of this model when the tension a(t) or the deformation e(i), respec-tively, are known.

It seems natural, therefore, to propose the following more general "newrheologic models" along the lines of (7.6.46) based on the R-L sequential frac-tional derivative £>"+ and on that of Caputo CD"+.

Let T>iay (0 < a < 1) be the R-L or Caputo sequential fractional derivative.


Model A.n - l

(Vnaz)(t) = Ka(t),3=0

(7.6.65)

(7.6.66)if a{t) is known, with

z(t) = 0a{t) - e(t)

and with the parameters a, K, /?, and A^ e l ( j = 1,2,..., n — 1), or

(2?™*) W + 53 tft(^'a*)(* ) = ^ W , (7-6.67)i=o

if e(i) is known, withz(t) = fe(t) - a(t) (7.6.68)

and with the parameters a, E, 7 and rjj £ M. (j = 1, 2,..., n - 1).

The general solution to these models could be explicitly obtained with-out difficulty by applying the theory developed above. For example, themodel given by (7.6.65) can be expressed as follows:

{DaZ){t) = AZ{t) + B(i), (7.6.69)

where Day is the Riemann-Liouville derivative D"+y or the Caputo deriva-tive cD"+y and

/ 00

A =

01

00

B(t) = K

I 0\ i - A 2

0 \00

a(t) J

and Z(t) =

0 \0

12 —An_i J

Zl(t) \

(7.6.70)

\ Zn(t)

(7.6.71)

with Zj(t) = (Vij-I)az)(t) (j = l,...,n) or (zj{t) = (CD(J+ 1)az)(t)).

According to Theorem 7.11, the general solution to the system (7.6.69) isrepresented as follows:

Z{t) = e^-^C + [ e^-^BiOdt, (7.6.72)J

where z\(t) = z(t) = (lcr{t) - s(t) (/3 G E) and C is an arbitrary constantvector.

This representation suggests a new generalization of the model pre-sented in Model A.

Chapter 7 (§ 7.6) 447

Model B.

where

B(*) =

A = (0^)4^=1

and

\ <?n\

*l(t) \

\ Zn{

(7.6.73)

(7.6.74)

(7.6.75)

and Zj(t) = (v[J+

1)az) (t) (j = 1,2,..., n - 1). The general solution to thesystem (7.6.73) can also be easily expressed by

a

(7.6.76)

The function (7.6.76) is also the solution to the system of the form(7.6.73), with the Caputo fractional derivative CD"+Z instead of the Riemann-Liouvill e one D%+Z.

Model C.(cDa

a+Z)(t) = AZ(t) + B(t), (7.6.77)

where A, B(t), and Z(t) are given in (7.6.74)-(7.6.75). The general solutionto the equation (7.6.77), also obtained in Theorem 7.13, has the form

Z{t) = f e^(*"« [B(0 + AY (a)] d£ + Y(a), (7.6.78)J a

In particular, the solution which satisfies Y(a) = b is given by

Z{t) = / e£('-«> [B(0 + Ab] ^ + 6. (7.6.79)J a


Chapter 8

FURTHE RAPPLICATION S OFFRACTIONA L MODEL S

This chapter is devoted to some aspects of differential equations of fractional or-der and their applications. The first part supports and explains the emergence offractional differential equations as a useful tool for the modeling of many anoma-lous phenomena in nature and in the theory of complex systems. This conclusionis already well known to many applied researchers, but it is not so evident tomany mathematical analysts and researchers. The main reason for the utility ofsuch fractional derivative operators is the strong relationship between these oper-ators and fractional Brownian motion, the continuous time random walk (CTRW)method, the Levi stable distributions, and the generalized central limit theorem.Moreover, the fractional derivative operators allow for the representation of thelong-memory and non-local dependence of many anomalous processes.

In the second part of this chapter, some useful properties of various fractionalintegral and fractional derivative operators are presented. We give examples ofsimple fractional models with boundary conditions in one dimension in connectionwith anomalous diffusion, and explain why several specific fractional derivativeoperators are suitable in the sub-diffusive case. We introduce here a new modeluseful for simulating both sub-diffusion and super-fast diffusion processes. Such amodel involves a generalized Liouville fractional derivative operator over the timevariable.

8.1 Preliminary Review

In the following subsections we present the reader with a historical review of theapplications of the fractional calculus to complex systems, which is not intendedto be exhaustive. In the last subsection we introduce the properties of certain

449


fractional operators which will be useful in later sections.

8.1.1 Historical Overview

Questions as to what we mean by, and where we could apply, the fractional calculusoperators have fascinated us all ever since 1695, when the so-called fractionalcalculus was conceptualized in connection with the infinitesimal calculus [717][643]. We should like to recall the famous correspondence between Leibniz andL'Hopital. Many great mathematicians have been interested in these questionsduring the last 170 years, and some remarkable applications of fractional calculushave emerged as a result. A reasonably adequate study of fractional calculus canbe found in [729] and some of its applications in [643] and [682].

The interest in the study of the fractional differential equations (FDE) as aseparate topic arose some 40 years ago. Questions about the existence of solutionsto Cauchy type problems involving the Riemann-Liouville operators D%+ (a G Cand a e R) and some other problems have attracted the attention of severalmathematicians. A systematic and rigorous study of some problems of this kindinvolving FDE can be found in recent literature [see [407] and [408] for a review].

Although numerous theoretical applications of the fractional calculus operatorshave been found during their long history, many mathematicians and applied re-searchers have tried to model real processes using FDE. However, except possiblyin a handful of applications, the literature on fractional calculus does not seem tocontain results of far-reaching consequences. This may be due, in part, to the factthat many of the useful properties of the ordinary derivative D are not known tocarry over analogously for the case of the fractional derivative operator Da, suchas, for example, a clear geometric or physical meaning, product rules, chain rules,and so on.

So then what are the useful properties of these fractional calculus operatorswhich help in the modeling of so many anomalous processes? From the point ofview of the authors and from known experimental results, most of the processesassociated with complex systems have non-local dynamics involving long-memoryin time, and the fractional integral and fractional derivative operators do havesome of those characteristics. Perhaps this is one of the reasons why these frac-tional calculus operators lose the above-mentioned useful properties of the ordinaryderivative D.

It is known that the classical calculus provides a powerful tool for explainingand modeling important dynamic processes in many areas of the applied sciences.But experiments and reality teach us that there are many complex systems innature with anomalous dynamics, such as the transport of chemical contaminantsthrough water around rocks, the dynamics of viscoelastic materials as polymers,atmospheric diffusion of pollution, cellular diffusion processes, network traffic, sig-nal transmission through strong magnetic fields, the effect of speculation on theprofitability of stocks in financial markets, and many more.

In most of the above-mentioned cases, these kinds of anomalous processes havea macroscopic complex behavior, and their dynamics can not be characterized by

Chapter & (§ 8.1) 451

classical derivative models. Nevertheless, the heuristic explanation of the corre-sponding models of some of these processes are often not so complicated when toolsfrom statistical physics are applied. For such an explanation, one must use somegeneralized concepts from classical physics such as fractional Brownian motion,the continuous time random walk (CTRW) method involving Levi stable distri-butions (instead of Gaussian distributions), the generalized central limit theorem(instead of the classical limit theorem), and non-Markovian (and thus non-local)distributions (instead of the classical Markovian ones). From this approach it isalso important to note that the anomalous behavior of many complex processesincludes multi-scaling in the time and space variables, and therefore also in thefractal characteristics of the media.

The above-mentioned tools have been used extensively during the last 30 years.But the connections between these statistical models and certain FDE involvingthe fractional integral and derivative operators (Riemann-Liouville, Caputo, Liou-vill e or Weyl and Riesz) had not been formally established until the last 15 yearsand, more intensively, during the last 5 years, as we will show later. The mentionedconnections have been obtained in several ways. One of the most used approachesis based on generalizing the propagator of the model through the use of Fourierand/or Laplace transforms. In this approach the characteristic function of thecorresponding probability density function (pdf), which contains the exponentialfunction ex in the classical case, is replaced by the Mittag-Lefner, Wright, or Foxfunctions which satisfy some of the main properties of ex. Using this concept,many researchers have obtained useful fractional dynamic models where fractionalderivative operators in the time and space variables assume the long-memory andnon-local dependence characteristics of the anomalous processes [67], [66], [534],[586], [429], [512], [135] and [873]. At present, the more commonly studied modelsare connected with the anomalous diffusion processes. These models have gener-ated the fractional versions of the diffusion and advection-diffusion equations, theFokker-Planck and Kramer equations, and some others [see [586] for a review].

Here we should point out that the above-mentioned fractional integral andfractional derivative operators have allowed for the modeling of some special situ-ations. But there are many other types of fractional calculus operators, fractionalpseudo-differential operators, and perhaps other singular operators that could helpus to remodel some of the anomalous processes. In this regard, we shall use a gen-eralized Liouville operator to obtain a new model for super-diffusion processeswhere the fractional derivative operator only works over the time variable.

We may conclude from the foregoing presentation that it is important to workout a rigorous theory of FDE analogous to that for ordinary differential equations(ODE). In fact, there are many similar and new open questions in this subject,the answers to which can help in the development of the applied sciences. Let usmention here, as an illustrative historical example, that the delta function, intro-duced by Dirac at the end of 19th century, has been used formally by physicistsin many of their ordinary derivative models for a long time. And their investiga-tions became rigorous only about 50 years later after using the distribution theorypresented by Laurent Schwartz in 1944 (see, for example, [750]).


This section is organized as follows. The first part given in Subsection 8.1.2is dedicated to an explanation of FDE as an emergent topic in the developmentof complex system modeling. In Subsection 8.1.3 some properties of several Thesecond part includes a mathematical justification of new models for super-diffusionprocesses in terms of Liouville-type fractional derivatives in the time variable. InSection 3 some properties of several fractional integrals and derivatives, consider-ing i Chapter 2, are presented. Subsection 8.2 includes a mathematical justificationof a new models for superdiffusion processes in terms of Louiville-type fractionalderivatives in the time variable. Subsection 8.3 deals with the study of diffusionmechanisms with internal degrees of freedom. Finally, in Section 4 examples ofsimple partial FDE with boundary conditions are discussed in connection withanomalous diffusion equations. Here, a new model useful in simulating both sub-diffusion and super-fast diffusion processes is introduced. Such a model involvesa generalized Liouville fractional derivative operator in the time variable. In thiscontext we note also that some earlier authors have studied only the models ofsuper-diffusion processes involving the Riesz fractional derivative in the space vari-ables.

8.1.2 Complex Systems

The asymptotic approach to equilibrium of physical systems is captured by thesecond law of thermodynamics, which states that the entropy of an isolated systemeither remains fixed or increases over time. All physical states in equilibrium arestatistically equivalent and this is the endpoint of all physical activity. There aremany systems in nature which apparently contradict the second law of thermo-dynamics. This apparent conflict with physical law arises mainly because thereare no systems which are completely isolated [see [868], [58], [78], [869], [771], and[873]].

Schr5dinger discussed the idea that macroscopic laws are the consequences ofthe interactions between a large number of particles. Therefore, such laws canonly be described by means of statistical physics. For this reason, the Gaussian ornormal probability distribution, Brownian motion, and the classical central limittheorem are important and useful. These tools have permitted important advancesin explaining many natural processes stemming from the classical physical modelswhich are based upon the use of the infinitesimal calculus, and which involve theordinary and local derivative operator D. One relevant example of such utility isclassical diffusion and Fokker-Planck equations [66], [586], and [62].

Complex systems are very familiar and occur often in nature. Examples ofthis kind of dynamic process appear throughout science. In the majority of suchsystems, classical models and analytical functions are often not sufficient to de-scribe their basic features. A familiar case is the so-called Brownian motion, whichdescribes the erratic path of a particle through space.

This Brownian motion in nature could be modeled as a random walk on a two-dimensional plane, where the size of the jumps and the wait time before each jumpare random and not uniform. This is the background of the so-called "continuous

Chapter 8 (§ 8.1) 453

time random walk" (CTRW) method. Such a method is the basis for a heuristicexplanation of the physical behavior of normal and anomalous diffusion processes.

For example, we can see macroscopic dynamic processes of normal diffusionthat are modeled by a diffusion equation or by the Fokker-Planck equation, withfixed parameters, by using the CTRW approach. That is, the above processes canbe seen as the mean line of the CTRW method applied to a set of microscopicparticles whose motions are random in the time-wait and in the space-jump, andthese random motions are governed by characteristic distributions. When bothrandom variables are independent and their pdf's are Gaussian and Markovian(local process in the sense that the entire past of the system does not influencethe future and only the present determines the immediate future), we can applythe classical central limit theorem. But when, for instance, the wait-time follows aLevi stable distribution instead of a Gaussian one, we can describe more complexprocesses such as sub-diffusion and super-diffusion by using the CTRW methodand the generalized central limit theorem.

The CTRW method can be characterized by the moments of the random meanmotion. If the process is non-local or has a memory wait-time, then the secondmoment of the random variable X of the jumps < X2 > is proportional to a powerta of order a of the time when the time is sufficiently large. This type of stochasticmodel lets us characterize the sub-diffusion, normal-diffusion, and super-diffusionprocesses as the cases where 0 < a < 1, a = 1, and a > 1, respectively.

On the other hand, when the random variables of the jumps in space do notfollow a Gaussian distribution, then a certain relationship with the so-called fractalmedia behavior of many complex phenomena can be obtained.

We should point out that close connections exist between fractional differen-tial equations and the dynamics of many complex systems, including anomalousprocesses, CTRW method, fractional Brownian motion, fractional diffusion, andfractal media.

Anomalous diffusion is perhaps the most frequently studied complex problem.Richardson [705] was probably the first to consider the existence of such anomalouskinetics in connection with turbulent diffusion in the atmosphere. Later on, otherauthors observed the same phenomena in other areas, such as chemical and plasmaphysics [835] and [445]. The characterization of the Levi stable distributions interms of the Fourier transform of their probability density functions (pdf), thegeneralized central limit theorem, and fractional Brownian motion were introducedby Levi [479] and [480]. The last concept was developed by Mandelbrot andVan Ness [548]. Subsequently, Mandelbrot [546] found some connections betweenfractals and the geometry of complex systems in nature.

The CTRW method was probably first introduced by Gnedenko and Kol-mogorov in 1949 in the Russian version of their classical book [288]. Later on,such an approach was applied by Montroll and Weiss [608] to study the mechan-ical properties of lattices. A good introduction to the above statistical tools canbe found in the book by Feller [256].

Oldham and Spanier [643] wrote the first book dedicated specifically to the


subject of fractional calculus and its applications. This book contains a verygood history written by B. Ross of the developments of fractional integration andfractional differentiation and their applications from 1695 to 1974, and also someof their applications to the chemical and physical sciences. In 1987 Samko et al.published a book in Russian [729], where one can find an encyclopedic, detailed,and rigorous study of the fractional calculus operators. The book by Miller andRoss [603] presents the first introduction to fractional differential equations [seealso a more recent book by Podlubny [682]].

The first Ph.D. thesis to apply fractional differential equations in modeling thebehavior of viscoelastic materials was written by Bagley, advised by Torvik [45], atthe Air Force Institute of Technology and Material Laboratory of Ohio. After thatmany researchers paid attention to the application of fractional differential equa-tions to viscoelastic materials and other complex processes, including anomalousdiffusion phenomena.

We shall classify a short selection of relevant applications in selected fields.Additionally, we will provide some survey-type references on the topic of thispresentation which will give the reader an idea of the evolution of the mathematicalmodeling of some Complex Systems. We begin with the articles published in theperiod 1970-1990. Perhaps [641] [see also [643]] was the first to introduce thefractional differential equation as a model for certain chemical processes.

The CTRW method was applied to some physical problems in [764], [767],[422], and [866]; the last paper contains a survey of the results in this approach.Anomalous Processes in Amorphous and Disordered Media were studied in [739],[672], and [673]. Models of Viscoelastic Materials in terms of fractional differentialequations were discussed in [803], [51], [52], [431], and [54]. Problems involvingthe Kinetics of Anomalous Diffusion were considered using random walk methodsin [332], [333], [420], [765], and [99]. A generalization of the entropy conceptwas presented in [824] and [825], and this concept has a close connection withthe behavior of some complex systems. Fractional diffusion and fractional waveequations, generalizing the classical ones, were investigated in [746] and [267].

Here, we present some of the many articles devoted to fractional models indifferent applied sciences from 1991, Material Theory: [284], [537], [538], [775],[240], [761], [92], [770] [594], and [39]; Transport Processes, Fluid Flow Phenom-ena, Earthquakes, Solute Transport: [433], [358], [61], [80], [660], [540], [2], [165],[659], [747] [748], [458], [570], [201], [59], and [351]; Chemistry, Wave Propaga-tion and Signal Theory: [326], [730], [348], [329], [806], [203], and [203]; Biology:[287], [334], [33], [336], [813], [465], [838], [337], [854], and [468] ElectromagneticTheory: [259], [241], [242], [244], [245], [247], [86], and [840]; Thermodynamics:[142], [575], [776], [780], and [452]; Mechanics: [708], [718], and [676]; Geologyand Astrophysics: [536], [460], and [135]; Economics: [58], [534], [873], [457], and[414]; Control Theory: [653], [678], [814], [668], [691],[815],[11], [204], and [598];Chaos and Fractals: [911], [648], [918], [360], [463], [464], [649], [826] [922], [564],and [915].

Models of some anomalous processes in terms of fractional differential equations

Chapter 8 (§ 8.1) 455

Table 8.1: Evolution in the Number of Publications in Connection withModels of Complex Systems Considered by JCR

Descriptive PhrasesFractional Brownian MotionAnomalous DiffusionAnomalous RelaxationSuper-Diffusion or Sub-DiffusionAnomalous Dynamics orAnomalous Processes orFractional Models orFractional Relaxation orFractional KineticsFractional Diffusion Equation orFractional Fokker-Planck Equation

45-803

116100

18

0

81-906

1081418

28

2

91-961585202330

79

19

97-03507104751202

105

129

Total674179198250

230

150

were discussed in [270], [260], [286], [581], [140], [723], [67], [305], [310], [711], [65],[66], [582], [512], [778], [834], [8], [327], [328], [327], [355], and [593].

Note also that the applications of the described fractional calculus methodsto modeling anomalous processes in the above and other sciences can be found inthe following books and survey papers: [666], [901], [620], [865], [352], [626], [766],[352], [444], [100], [132], [67], [917], [547], [665], [751], [77], [340], [586], [638], [658],[338], [345], [871], and [916]. See also [740], [768] and [421], where one may findan introduction to anomalous phenomena.

We have mentioned a series of papers and books devoted to the modelingof some complex systems through the use of different tools, including fractionaldifferential equations. We chose only some of the many publications in this field,which appeared basically during the last few years. In Table 8.1 we show theevolution in the number of these publications. We have used for this only thosepapers in scientific journals which are included in the lists of the Science CitationIndex and Social Science Citation Index of the Institute for Scientific Information(JCR) from 1945 to 2001.

We have not included here those papers that are devoted to applications ofthe CTRW method, although they have close links with the fractional derivativemodels we have considered in this table.

We can conclude from this study that the number of publications devotedto the modeling of anomalous processes has considerably increased during thelast 10 years. We also note that the use of the fractional derivative operatorsplays an important role in obtaining the corresponding derivative models of suchprocesses. These models include non-local and long-memory properties and arestrongly related with the CTRW method.

We note that, during the last few years, there has been a special interestin the modeling of anomalous diffusion processes involving ultra-slow diffusion.But the anomalous phenomena, such as super-diffusion, super-conductivity, wave


propagation, and others, have not been studied thoroughly.

8.1.3 Fractional Integral and Fractional Derivative Ope-rators

In this section, we recall some properties of the known fractional integral andfractional derivative operators from Chapter 2. Let / be a real function, a G Cand n G N = {1,2,3, . The classical Liouville (or Weyl) fractional integral 7"/of order a with ^(a) > 0 is defined by

where T(a) is the Euler Gamma function. The corresponding fractional derivativeD°Lf of order a (9S(a) ^ 0; a 0) has the form

(D1f)(x) = (-D)n(r-af)(x) (x>a - o o ), (8.1.2)

with Dn = (d/dx)n and n = pH(a)] + 1, where pH(a)] means the integer part of<R(a).

The Liouville fractional calculus operators I" and £>" of exponential functionyield the same exponential function, both apart from a constant multiplicationfactor.

Lemma 8.1 Let a, A 6 C (SH(a) > 0; <R(A) > 0). Then

{I1e~xt){x) = \-ae~Xx and (D1e-Xt)(x) = \ae~Xx. (8.1.3)

Now we present the so-called fractional integrals and fractional derivatives of afunction / with respect to another function g. Such operators play the main rolein the new model of sub- and super-diffusion processes. We shall introduce theseprocesses in the next section.

Let g(x) be an increasing and positive monotone function on (a, oo), havinga continuous derivative g'(x) on (a, oo). We define the general fractional integraloperator I".<gf of a function / with respect to a function g, of order a (SH(a) > 0)

If g' (x) 0 (a < x < oo), then the operator I^.g can be expressed via the fractionalintegral 7" by

IZ.J = QgltQ;1/, (8.1.5)

where Qg is the substitution operator (Qgf)(x) = f[g{x)] and Qj1 is its inverseoperator. When g'(x) 0 (a < x < b), the corresponding fractional derivativeD1.gf of a function / with respect to g of order a (fH(a) ^ 0; a 0) has the form

{I n_7gaf){x) (8.1.6)

Chapter 8 (§ 8.1) 457

where D = djdx and n = [£H(a)j + 1.

When a — n G N, the Liouvill e derivative (8.1.2) and the generalized derivative(8.1.6) have the following forms:

(Dn_f)(x) = (-ir/ (n)(* ) and {Dn_.J)(x) = ( - ^ y £ ) /(*) (8-1-8)

In particular, (Dlf)(x) = -f'(x) and (Dl.gf)(x) = -f(x)/g'(x).

The following properties will be important in the development of new dynamicmodels of super-diffusion and sub-diffusion (see Properties 2.19 and 2.21 in Sub-section 2.5):

(I%e-X3^)(x) =\-ae-

X9 (8.1.9)

and(D°.ge-XB^)(x) = Xae-X^xl (8.1.10)

where a 6 C (?R(a) > 0) and A > 0.In particular, when a = 0 and g(x) = xa {a > 0), (8.1.4) and (8.1.6) yield the

following special cases of the fractional integral and fractional derivative operatorsof order a with respect to the function x":

when <H(a) > 0, and

when fR(a) 0 (a / 0) w«i/i n = [fR(a)] + 1.

We note that I".<x*f is related to the fractional integral /f / by

and also that it is connected with the so-called Erdelyi-Kober type operator apartfrom a power multiplier factor (see Section 2.6 for details).

Lemma 8.2 Let a > 0, A > 0, and a e C (9t(a) > 0). Then

{I1.x<,e-xt°){x)=\-ae-Xx'r and (DZ.x*e-xt')(x) = \ae-Xx'. (8.1.14)

Remark 8.1 The formula (8.1.10) and the second relations in (8.1.3) and (8.1.14)are valid for a 6 C with 9t(a) = 0 and a ^ 0.


The Riesz integral and Riesz derivative operators Ia and Da of order a G C{9\(a) > 0) in the n-dimensional Euclidean space Rn deserve special attention.These operators are defined as negative (—A)~a/2 and positive (—A)a/2 powersof the Laplace operator

(8.1.15)

They can be represented in terms of the direct Fourier T and inverse Fourier T~x

transforms by

laj = (—A)~a/2/ = J-~1\x\~aTf\ Daf = (—A)a/2f = J-~l\x\a!Ff, (8.1.16)

where x = (xi, , xn) G M" and \x\ = [x\ + + xl)When 0 < a < n, the Riesz fractional integration Ia can be represented for

sufficiently good functions / by the Riesz potential, given (for x G E") by

We note that the Riesz potential (8.1.17) and the Riesz derivative operator Da,expressed explicitly as a hypersingular integral by E. M. Stein in 1961, have beensufficiently studied (see, for example [707], [728, 727] and Sections 25-27 in [729]).In connection with the fractional diffusion processes such a multidimensional Rieszfractional differentiation was used by Kochubei in [424], [425], [427], [428] and [429].Although such a Riesz derivative operator was used by many applied researchers,they only used its characterization in terms of Fourier transforms.

We conclude this section by noting that a great number of other fractionalcalculus operators of one and several variables exist which preserve different mem-ory properties. Some fractional calculus operators, such as the Riemann-Liouville,Griinwald-Letnikov, Marchaud, Erdelyi-Kober, Hadamard, etc., can be found inChapter 2. We also mention another smooth fractional calculus operator, knownas the Caputo derivative (see, for example [305] and [310]), which is often usedin many applied fractional calculus models, explicitly or implicitly (see for de-tails, Section 2.4). Furthermore, it is possible to introduce or define new singularoperators with long memory and non-local properties.

8.2 Fractional Model for the Super-DiffusionProcesses

As mentioned in the preceding section, many papers published during the lastten years presented fractional calculus models for the kinetics of natural anoma-lous processes in complex systems, which maintain the long-memory and non-localproperties of the corresponding dynamics. Special interest has been paid to theanomalous diffusion processes, that is, super-slow diffusion (or sub-diffusion) andsuper-fast diffusion (or super-diffusion) processes. Nigmatullin in [622] and [621]

Chapter 8 (§ 8.2) 459

was probably the first who considered the following diffusion equation with mem-ory:

U ^ ' l ' = f K(t- T)AU(X, r)dr (t>0; xe M.n), (8.2.1)

where A is the Laplace operator (8.1.15). In the particular case whenK(t) = p25(t — a), 8(t - a) being the Dirac delta function, this relation repre-sents the classical diffusion equation

dU(x,t) _ 2

dt= p2AU(x, t) (t > 0; x e M.n). (8.2.2)

As we have seen in the first part of this presentation, there are many fractionalcalculus models for the sub-diffusion processes, without and with an external forcefield, such as the Fractional Diffusion Equation and the Advection-Diffusion orthe Fokker-Planck Equations, respectively. All these models include the Riemann-Liouvill e and Caputo fractional derivative operators acting on the time variable,and the Liouvill e (or Weyl) and Riesz Da fractional derivative operators actingon the space variable.

The super-diffusion processes, however, are studied less frequently. Further-more, only a few papers include fractional calculus models, where the authorsintroduce the Riesz derivative operator Da acting on the space variables by usingits definition, which is given in (8.1.16) in terms of the direct and inverse Fouriertransforms (see, for example, [250], [872], [474], and [135]).

We propose here a (presumably new) fractional derivative model, including thegeneralized Liouvill e fractional derivative operator D^.g, which acts on the timevariable. This model allows us to have strong control of both the sub- and super-diffusion processes. It also has great flexibility due to the free function g, as wellas many other advantages from a purely technical point of view in the sense thatwe can use the same methods as in the ordinary case (that is, those based uponintegral transforms and separation of variables) to solve boundary-value problemsassociated with the fractional calculus models.

First we consider our fractional calculus model in the one-dimensional case.We begin with a scheme for the solution of the classical problem associated withthe one-dimensional heat equation

-=P2-Û(x,t) ( p > 0 ; t>0; 0 < x < I), (8.2.3)

with the boundary conditions

U{0,t) = U{l,t) = 0 (t>0), (8.2.4)

(/ being a sufficiently smooth function on [0,/]) and with the initial condition

U{x,0) = f{x) {0<x<l). (8.2.5)

We note that the parameter p2 can characterize the thermal or diffusion con-stant of the media.


Using the method of separation of variables, we seek the solution to the problem(8.2.3)-(8.2.5) in the form U(x,t) = X{x)T(t). Then

Y"(v\ T'(t\A [X) 1 {1) 2 /o o a \

= x ( 8 2 6 )

Therefore, X(x) is a solution of the Sturm-Liouville problem

X"[x) + XX(x) = 0, X(0)=X(l) = 0, (8.2.7)

for which the eigenvalues and the corresponding eigenfunctions are given by

An = a2; Xn(x) = sin(az); a = ^ (n G N). (8.2.8)

On the other hand, T(t) is a solution of the differential equation

T'{t) + (ap)2T(i) = 0, (8.2.9)

given byTn{t) = exp [-{ap)2t] (n £ N). (8.2.10)

Finally, the solution of the problem modeled by (8.2.3) to (8.2.5) is obtainedas a series solution whose coefficients are found from the following Fourier devel-opment of the function / involved in the initial condition (8.2.5):

oo

U(x, t) = J2 cnXn(x)Tn(t), (8.2.11)n=l

where

cn=j[ sin(ax)f(x)dx (n 6 N), (8.2.12)» Jo

and Xn(x) and Tn(t) are given in (8.2.8) and (8.2.10), respectively.Thus we need only show the convergence on [0,1] x [0, oo) of the series for U(x, t)

in (8.2.11) and of the series for Uxx(x,t) and Ut(x,t) obtained from (8.2.11) byterm-by-term differentiation with respect to x and t, respectively.

Now we can see more clearly that a sub-diffusive process, associated withthe above problem, can be obtained by replacing the factor Tn(t) in the solu-tion (8.2.11) by some other function Tn(t, a) depending on the time t and a newparameter a. Such a new parameter gives us the possibility of controlling thedecay of the process in the time variable; for instance, we have

] (n G N). (8.2.13)

In this case, (8.2.11) is extended to the form

oo

U(x, t) = J2 cnXn(x)Tn(t, a), (8.2.14)n=l

Chapter 8 (§ 8.2) 461

where Xn{i) and Tn(t,a) are given in (8.2.8) and (8.2.13), respectively. Thus asub- or super-diffusive process can be controlled via the function g(t, a).

For example, if g{t,a) = ta, we have sub-diffusion kinetics when 0 < a < 1and super-diffusion kinetics when a > 1. This is in agreement with the CTRWmethod.

In the above-mentioned way, by choosing a function g(t, a), we can obtain othersuitable solutions to our problem so as to control a very strong super-diffusionprocess. For example, if we take g(t,a) = exp (ta) with a > 1, perhaps (8.2.14)would yield a new model for the dynamics of the properties of super-conductivityof some materials under certain boundary conditions.

Now we can express the corresponding differential model which has the solu-tion (8.2.14). The problem then becomes finding some derivative operator Da

(fractional, singular with a memory, or others) with the following property:

*), (8.2.15)

whereT(t, a) = Tn(t,a) = exp \-{&p)2'ag{t,a)] (n e N). (8.2.16)

Such a problem is solved by taking D" = D" ,t ,, where D™_ ,t ,f, as a func-tion of t, is the fractional derivative of a function / with respect to g(t, a) definedin (8.1.6). Indeed, if we suppose that g(t,a) (for any fixed a) is an increasingand monotonic function of t 6 R+ = (0, oo) having the continuous derivative

7 0, then, in accordance with (8.1.10),

(Da_.g(ta) exp [-(ap)2/«5(r, a)] ) (t) = (ap)2 exp [-(ap)2/Q9(i , a)] , (8.2.17)

and hence (8.2.15) holds true for Da = 2?". ,t ,. Therefore, the new fractionaldiffusion model can be presented by the fractional differential equation

D°^.g(ta)U{x,t) = p2—U(x,t) (a > 0; t > 0; xe M). (8.2.18)

If we additionally suppose that limt_>0+ g{t, a) = k(a) £ E for any a > 0, thenthe explicit solution to the problem modeled by (8.2.18), (8.2.4), and (8.2.5) hasthe form (8.2.14)

U(x,<) = ^ r ^ cns i n ( a x ) e xp [-(ap)2/Q5(i,a)] , (8.2.19)

^ ' ln = l

provided that g(t,a) is chosen so that U(x,t), D^_ ,t aM(x,t), and Uxx(x,t) areconvergent series on [0,/] x [0, oo), cn being given in (8.2.12).

For example, when g(t,a) = ta (a > 0), we can prove the above convergencejust as in the ordinary case when a = 1. In this case, D°_. ,t > = -D".(Q is thefractional derivative (8.1.12), and the equation (8.2.18) takes the form

d2

D°.taU(x,t) = p2—1U(x,t) (a > 0; t > 0; xe M), (8.2.20)


and the explicit solution of the problem modeled by (8.2.20), (8.2.4), and (8.2.5)is represented by

U(x,t) = 2_^cnsin(a:c) exp \-(a.p)2/ata . (8.2.21)n=l

In a similar manner, we can consider the case when g(t, a) = exp (ta).The model (8.2.18) can be extended to the space variable x 6 K™ if we replace

the partial derivative d2 jdx2 by the Laplace operator (8.1.15) with respect to x:

D°.g{tjQ)U{x,t) =p2AslI{x,t) ( a > 0 ; t > 0 ; i e I " ). (8.2.22)

In particular, the model (8.2.20) is extended to the form

DZ.taU(x,t) = p2AsU{x,t) (a > 0; t > 0; x <E E"). (8.2.23)

Analogous to (8.2.19) and (8.2.21), the explicit solutions to these equationsunder the corresponding boundary and initial conditions can be also derived.

It seems that the above approach, based on the use of the generalized fractionalderivative (8.1.6), can also be applied in order to construct the corresponding gen-eralized fractional calculus models for the Fokker-Planck and advection-diffusionequations. We think that these models will be compatible with those for the cor-responding generalizations including, for example, the Riesz operator Da actingon the space variable.

We note that there are many more possibilities for controlling the exponential-type decay of the fundamental solutions of the generalized fractional diffusionequations, and that (in this way) the same mathematical technique may be used tostudy new fractional calculus models for the control of anomalous wave processes.

8.3 Dirac Equations for the Ordinary DiffusionEquation

In this section we will apply the results obtained in Section 6.4 to the study ofdiffusion mechanisms with internal degrees of freedom.

If, in Example 6.11, we set A = — 1 and A = 1, we obtain the following solutions:

r-+oo

and

Jto the respective Cauchy type problems

I p + OO

u(x, t) = —J E1/2 (tat1/2) {Txf) {a)e-^xda (8.3.1)

1 p+°° , .u(x, t) = —J E1/2 (-iat^2) {Txf) (a)e-^xda, (8.3.2)

(cDl^tu){x,t) + dutf>V = o, u{x,0) = f(x) ( i £ l ; t > 0) (8.3.3)OXOX

Chapter 8 (§ 8.4) 463

and

( c D l £ t u ) ( x, t) - ^ ^ = 0, u(x,0) = f(x) ( x e R ; t > 0 ). (8.3.4)

Finally, we will apply the above results to equations of the forms (8.3.3) and (8.3.4)which arise in the study of diffusion mechanisms with internal degrees of freedomrelated to the square root of the one-dimensional diffusion equation Ut — uxx — 0[see Vazquez [842] and Vazquez and Vilela Mendez [845]].

I In -l In

If we use the property dt dt u = dtU, which holds for "suitable" functionsu(x,t), then ut — uxx = 0 can be rewritten in the following form [see Vazquez[842]]:

j (8.3.5)

where A and B are 2 x 2 matrices satisfying the conditions

A2 =1, B2 = -I, AB + BA = 0, (8.3.6)

and / is the identity operator. One of the possible choices, according to Pauli's

algebra, is A = I 1 and B = I j , and, therefore, the system (8.3.6)

is reduced to the following Dirac type equations:

= 0, (8.3.7)

*) = 0. (8.3.8)ox

Now, if we substitute dj/2 = cDl^t, then the equations (8.3.7) and (8.3.8)coincide with the equations (8.3.3) and (8.3.4), respectively. Hence (8.3.1) and(8.3.2) are solutions of the equations (8.3.7) and (8.3.8) with the initial conditionu{x,0) =f(x).

8.4 Applications Describing Carrier Transport inAmorphous Semiconductors with MultipleTrapping

The physical analysis of dispersive transport was pioneered in the 1970s by Scherand Montroll [739] and Pfister and Scher [673]. At that time, the main interestof research was the operation of photocopy machines. In order to create theprinting image, carriers photogenerated by a flash of light in a sheet of organicconductor must be separated by drift under a strong applied electrical field. It wasobserved that the photocurrents departed markedly from that expected of gaussiantransport; instead, the photocurrents displayed a power law time dependence both


at short and at long times with respect to the typical transit time of a carrierthrough the film. Such behavior was attributed to the strong disorder of theamorphous organic conductor, and the formalism of the CTRW was developedto account for it [739] and [673], assuming a wide distribution of wait times foran individual carrier to hop from one site to another. Such a distribution wasinterpreted in terms of a significant number of traps for the carriers in the material[see [744] and [637]]. In the early 1980s, the formalism of multiple trapping (MT)was developed and applied in a variety of transport experiments [see [818], [817],[646], and [596]]. The main assumption of MT is the distinction between a bandof transport states, where carriers can move freely, and a distribution of bandgap(trap) states, where carriers remain immobile. Both kinds of states are able toexchange carriers, so that these can be trapped from or released to the transportstates [see [818], [817], [646]]. Obviously, the rate of transport is reduced when thefraction of trapped carriers increases. A formal connection was also establishedbetween the MT model and the CTRW model [see [744] and [637]]. The CTRWcan be formulated rigorously in terms of a fractional diffusion equation (FDE) [344]and [343]:

^ [ ^ \ (8.4.1)

This FDE has been amply studied in the literature ([344], [347], [140], [575],[586], [336] and [62]). Another possible FDE that generalizes the ordinary Fick'slaw is based on the replacement of the time derivative in the ordinary diffusionequation with a derivative of non-integer order:

^ ^ (8.4.2)

Here, / is a probability distribution and Ca is the fractional diffusion coefficient,which takes the form

Ca = Ko Txa-

a (8.4.3)

with respect to the ordinary diffusion coefficient KQ (in cm2/s) and a time constantra. The fractional Riemann-Liouville derivative operator of order a(0 < a ^ 1),tD%+, is defined (see (2.9.3) and (2.9.9)) as follows:

(tD2+f)(x,t):=^tIla-

a, (8.4.4)

where

^ f \ l f { x , y ) dy (8.4.5)

is the partial Riemann-Liouville fractional integral operator of order a. The frac-tional time derivative can be written by

| / ' i ^ * (8-4.6)

Chapter 8 (§ 8.4) 465

and its Laplace transform is

(CtD%+f)(s) = Saf(x,s)-fo (8.4.7)

where [see [389]]

/o = (/o+,?/)(a;.f )|t=0 = l i n i / - a/ ( M ) (8.4.8)

The equation (8.4.2) was discussed as a natural generalized diffusion equation,suggesting that it could describe anomalous diffusion processes in [344], [575], [62],[360], [342], [143] and [65]. However, those possible applications have been scarcelydeveloped because of a problem of physical interpretation, owing to the fact thatthe function f(x,t) is not a normalized function ([575] and [62]) and the initialcondition could not be the usual one. Indeed, it was found in [344] that f(x,t) isdivergent for t —¥ 0 , as described in the context of the Cauchy problem by Kilbaset al. in ([389], Lemma 9). Hilfer related the nonlocal form of the initial conditionin [344] to the fractional stationarity concept [see [342] and [65]] which establishes,in addition to the conventional constants, a second class of stationary states thatobey a power law time dependence. Ryabov in [720] discussed the solution andinitial conditions of (8.4.2) in different dimensions and concluded that "virtualsources" of the diffusion agent are required at the origin, meaning that some massis injected at the origin. However, this injection is not defined by the boundaryconditions of the problem. Thus, Ryabov in [720] concluded that the presence ofsuch "virtual sources" in solutions is physically meaningless and that this contra-diction needed a resolution. The first interpretation of (8.4.2) in terms of carriertransport was provided by Bisquert, who showed [86] that the FDE (8.4.2) de-scribes the diffusion of free carriers in MT with an exponential distribution ofgap states. This seems contradictory, because earlier papers had demonstrated anequivalence of MT with CTRW [see [744] and [637]] as already mentioned, while(8.4.1) and (8.4.2) are not equivalent. However, the earlier results [744], [637],referred to the total injected charge in MT, which indeed undergoes a CTRW, asconfirmed by the analysis of Bisquert in [86]. In contrast, the equation (8.4.2)describes only the diffusion of carriers in transport states [see [86]]. The motion ofthe total charge is an important magnitude for those where the measured currentis a displacement current, corresponding to standard time-of-flight measurements.However, in the 1990s another kind of system became highly visible: nanostruc-tured semiconductors permeated with an electrolyte, particularly in connectionwith dye-sensitized solar cells [see [645] and [89]]. In later systems, electrical fieldswere shielded at the nanoscale in the transport layer, so that measured photocur-rents corresponded with the arrival of diffusing carriers at an extracting contact[see [160], [87] and [840]]. In this case, the magnitude of physical interest is thediffusive flux of free carriers. Hence both equations (8.4.1) and (8.4.2) are legit-imate interpretations of MT transport, but the one that applies depends on thekind of measurement. In further work [88], Bisquert clarified the question of thenon-conserved probability density of equation (8.4.2). This is a natural feature,because the equation (8.4.2) refers to the transport of a fraction of the injectedcarriers, those in transport states, which may decay to trap states. Hence, thedensity of carriers described by (8.4.2) is not conserved. Indeed, considering the


expression of the FDE (8.4.2) in Fourier-Laplace space (with x —> u variables andt —> s), then

/(".* ) c-i{ 2 (8-4-9)TS sa + K0u

2

where /o,a is a constant. The case g->0 describes a situation in which the carrierdensity is homogenous, i.e., there is no diffusion at all, which gives

f(u,O) = fo,Qr^-1u-a. (8.4.10)

Therefore, the time decay of the probability in spatially homogeneous conditionsis given by

/ (<M) = r ^ * - (8-4>11)

The last result, the decay law for (8.4.2), shows that the number of particlesdecreases with time. The decay law in (8.4.11) can be explained physically bythe heuristic arguments for the evolution of free carrier density in MT providedby Tiedje and Rose in [818] and [817], and by Orenstein and Kastner in [646].In brief [for details see [88] and [90]], photogenerated carriers in a semiconductorwith an exponential distribution of traps will be rapidly trapped. Subsequently,the carriers in shallow traps (close to the transport level) will be released andretrapped, so that a time-dependent demarcation level exists which separates thefrozen charge below it from the thermalized charge above it, as indicated in Fig.1. This demarcation level plays the role of a Fermi level, so that in effect thedensity of free carriers decreases with time. These arguments explain and give asimple interpretation for the temporal decay of the probability of the FDE, andindeed (8.4.11) is confirmed by the experimental results of transient photocurrentsin conditions of homogeneous light absorption [see [646] and [289]].

It has been stressed in the literature [see [344] and [389]] that the equation(8.4.2) requires an initial condition for the Green function of the form

tIâ[f(x,0+)] = fo,aS{x), (8-4.12)

where S(x) is the Dirac measure at the origin and /a,o is a constant. Indeed, inthe study of the Cauchy problem for (8.4.1), it has been shown rigorously in [389]that the initial condition f(t) satisfies the condition

lim ^- " / ( f ) < oo. (8.4.13)

It is clear that the temporal dynamics in the FDE imply the divergence of f(t)as t —¥ 0 [see [344] and [389]]. However, it must be recognized that in the MTinterpretation discussed above, the decay law cannot be extrapolated to t = 0,because the equation (8.4.11) makes no sense without a minimal time for initialthermalization such that E < Ec (see Fig. 1).

In summary, the equation (8.4.2) describes a dissipative dynamics in which thetotal energy (determined by the electrochemical potential of the electrons Ed(t))decreases with time while the number density is conserved. The relaxation in

Chapter 8 (§ 8.4) 467

the full energy space is not considered explicitly in (8.4.2), which only containsthe resulting evolution of the carriers in extended states, in correspondence withthe requirements of the experimental techniques that monitor carrier transport bydetecting only the free carriers.

Example 8.1 Let us find the solution to the following model [see [90]]):

S+tt u(x, t) = vd2f* 2't] ( t > 0 ; 0 < x < l ) (8.4.14)

( 8-415)u(l,t) = K

1

where e << I, y

To solve it we will not use the usual and powerful technique of integral trans-forms; instead, we will reduce the problem (8.4.14)-(8.4.15)-(8.4.16) to one withhomogeneous boundary conditions via the usual change

u(x,t) =

Our problem thus becomes the following:

Kr2

+ V(x,t). (8.4.18)

tDS+V(x, t) = aVxx + f- (8.4.19)

) = 0; V(0,

= hcix) -

Vx(0,t) = 0; V(0,l) = 0 (8.4.20)1 TCI

Vm*-°V{x,t) = hcix) -

Now we look for a solution V of (8.4.19) in the form

V(x,t) =oo

n = l

which satisfies (8.4.20) if T0(l) = 0.Substituting (??) into (8.4.19) we have

n = l

and, therefore, since TQ = , it holds that

{^TT) +lta~l (8A24)


Since T0(l) = 0, then

T<" | £ ' ( 8 - 4 2 5 )

Additionally, (8.4.23) requires that the following relation must be satisfied:

(8.4.26)

and imposing the initial condition (8.4.21) it holds that

^ - x) (8.4.27)= 2/ +n=l

Therefore, any function Tn(t) is established as a solution of the Cauchy typeproblem

2 t)=0 (8.4.28)

(lh+aTn) (0) = Jim t^T^t) = j, (8.4.29)

By Example 4.2, the unique solution of this problem is given by

- (nGN) (8.4.30)

Hence,in accordance with (8.4.18), (8.4.22), (8.4.25) and (??), the solution toof the initial problem (8.4.14)-(8.4.16) has the form:

. Kx2 K[ta-lta-1} T{a) ,-vi^ft friTT \

u(x,t) = -jr+ r ( a + 1} + — E ^ 2 l ) cos("JT X) (8-431)

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Subject Index

Q-Exponential functions, 50-51, 53Q-Wronskian of n functions, 395Absolutely continuous function, 1-3, 92Amorphous polymer, 440Analytic continuation, 27, 86, 120Asymptotic behavior, 28-29, 31, 33-35,

39-41, 43, 54-56, 61-62, 349

Banach space, 4, 5, 360-361Basset's problem, 434-435Bessel function of the first kind, 17, 32-35,

55, 65-66, 342Beta function, 26Binomial coefficients, 26Burger's model, 440-441

Caputo fractional derivatives, 91-93, 97,99, 230, 312, 322-323, 418

Caputo sequential fractional derivative, 421,445

Cauchy type problem, 135-140, 144-145,147-148, 150-160, 162-177, 179-186, 191-192, 195-198, 211-219,222-228, 234-238, 251-256, 266-270, 275, 277, 282, 309-311, 327,349, 353-354, 356, 358-359, 362,364-366, 368, 370-371, 373, 380,388, 407, 415, 450, 462, 468

Commutative property, 11, 13, 19-20, 22-23

Compact support, 7-9, 83, 89, 131, 355Complex conjugate solutions, 402, 431Composition relations, 74Continuous embedding, 5Continuous global solution, 411, 414

Delayed elasticity, 441Differential operator, 380Dirac function, 7, 9, 16, 23, 44, 133, 350,

451, 459, 462-463, 466

Elastic region, 439Elastic response, 440Erdelyi-Kober type fractional operators,

105-106, 108-109, 143-144, 270,352, 457

Euler integral, 24, 26-27Euler Gamma function, 24-27, 36, 38, 69,

289, 368, 372, 388-389, 456Euler psi function, 16, 23, 26,Euler transformation formula, 28Exponential weight, 88

Fourier convolution operator, 11, 13, 127,341

Fourier convolution theorem, 12, 13, 128Fourier integrals, 10, 13Fourier transform, 10-18, 20, 44, 90, 121,

127-128,131-133, 341-342, 353-356,358-359, 361-364, 366-368,374, 376-381, 383-384, 388, 437,442-443, 453, 458-459

Fractal, 349, 350, 357, 439, 451, 453-454Fractional damping, 433, 436-437Fractional derivative, 69-80, 83-105, 108-

112, 115-127, 130-131, 135, 138,140-145, 162-163, 172, 176, 182,199, 205, 208, 212, 221-222, 230,234, 238-242, 245-247, 251, 256-257, 261, 264-279, 283, 295, 311-312, 322-323, 329-331, 336, 341,344-355, 358-362, 373,379-380,384, 393, 409, 415-418, 426, 431-434, 442—452, 455-462

Fractional Green function, 359, 373, 379,393, 403, 407, 409, 412

Fractional integral, 69, 74-80, 83-84, 86-87, 90, 99-100, 103-106, 108-114, 116-117, 119, 123-127, 154,157, 163, 168, 189, 238, 261-263, 272, 276, 352, 359, 449-452, 456-457, 464

Fractional integration by part, 76, 83, 107Fractional relaxation, 435, 455Fractional sequential derivative, 394-395,

397Frobenius method, 394, 415, 417, 429Fundamental system of solutions, 283-285,

288-291,293-294, 312-316, 319-321, 396, 399-400, 402-403, 410


Gauss hypergeometric function, 27-28, 30,132, 143, 350

Gauss-Legendre multiplication theorem, 25Gauss-Weierstrass transforms, 130Generalized functions, 6-10, 14-16, 23, 133,

143, 351Generalized hypergeometric functions, 27,

30-32, 45, 58, 65, 353Generalized Stirling numbers, 115Griinwald-Letnikov fractional derivatives,

121-122, 443, 458

fl-Function, 58-65, 352-354, 368-369, 371-372, 379, 388

Holderian functions, 129Hadamard type fractional derivatives, 111,

113, 115-116, 119-120 120, 122,123

Hadamard type fractional integral, 110-114, 116, 119

Hardy-Littlewood theorem, 72, 82, 88, 128Homogeneous linear FDE, 132, 142, 144,

197, 224-225, 231-232, 235-236,239, 242, 252, 256-257, 276, 280-281, 283, 295, 302-303, 309, 311-312, 322-323, 326, 359, 393-394,396-397, 399-400, 403, 406-410,415, 424, 427, 435, 466-467

Hypergeometric series, 27, 29-30Hypersingular integral, 130-132, 458

Incomplete gamma functions, 27, 289Incompressible viscous fluid, 433-434Instantaneous deformation, 440Integral representations, 29, 34-35

Kummer hypergeometric functions, 29-30,65

Kummer confluent hypergeometric func-tion, 29, 45

Laplace convolution theorem, 20Laplace transform, 18-19, 23, 31, 36, 42,

44, 47-48, 50, 52, 55, 58, 84,98, 140, 279-284, 287, 291, 295,303-304, 306, 311-312, 315,322-323, 329, 336, 340, 350, 352-353, 356-357, 362-364, 366-370,373-377, 380-381, 384, 393, 400,402-405, 435-436, 442, 451, 465

Laws of Thermodynamics, 443

Lebesgue measurable functions, 1-3, 79,151

Left-sided and right-sided Caputo fractionalderivatives, 91

Left-sided difference, 121Left-sided Griinwald-Letnikov fractional

derivative, 122Left-sided Liouville fractional derivative,

88, 336, 338Legendre duplication formula, 25, 390Linearly independent solutions, 28, 34, 36,

244, 281, 283, 286, 288-91, 293,294-295, 315-316, 318, 320, 401,408, 429, 432-433

Liouvill e fractional derivatives, 80, 83, 87,89-90, 101-103, 105, 127, 239,245, 257, 266, 270, 279, 283, 295,322-323, 329

Liouvill e fractional integrals, 78-79, 83-84, 86-87, 90, 100, 106, 127, 163,189, 261, 263, 359

Liouvill e left- and right-sided fractional op-erators, 80, 87, 338

Lizorkin space, 9, 128, 131Logarithmic derivative, 26

Marchaud fractional derivatives, 122, 458Mellin convolution operator, 22-23, 330-

331Mellin convolution theorem, 23Mellin transform, 18, 20-24, 31-32, 34,

36-39, 41, 44, 46, 48, 54, 57, 84,99, 104, 109, 119-120, 283, 329-330, 332, 337, 346, 352-353

Mellin-Barnes contour, 27, 30, 33-35, 38-39, 41, 43, 46-48, 54-55, 57-58

Minkowski inequality, 113Mittag-Leffler Functions, 40-42, 44-46, 48-

50, 55, 66-67, 78, 86, 98, 137,141-142, 144, 223, 226, 237-239,250, 265, 270, 272, 280, 284, 295,308-309, 312, 355, 361, 381, 383,394, 416, 439, 451

n-dimensional Fourier transform, 12, 17rz-dimensional Laplace transform, 13, 19n-dimensional Mellin transform, 22Non-Sequential Linear FDE, 407, 433Non-Constant or Varible Coefficients lin-

ear FDE, 394, 409, 415, 417

Subject Index 523

Non-Homogeneous linear FDE, 132, 245,251, 256, 279-281, 295, 302-303,310-311, 322-323, 327, 329, 341,344, 346, 392, 394, 400, 403-410, 412, 435

Operational decomposition method, 142,405

Operational methods, 141, 260, 270, 402-404, 413

Partial Liouville fractional derivatives, 348Partial Riemann-Liouville fractional op-

erators, 78, 123-124, 273, 348,350-351, 355, 359, 362, 464

Pochhammer symbol, 24, 27, 45Positive monotone function, 99, 456Purely imaginary order, 71, 80, 87

Riemann-Liouville fractional derivatives,70-71, 90-95, 93-95, 124-126,138, 143, 222, 242, 251, 348, 351

Riemann-Liouville fractional integration,71, 95, 136, 187, 323

Riesz, 127-131, 279, 341, 344, 359, 458-459

Right-sided mixed Riemann-Liouville frac-tional integrals, 124-126

Right-sided partial Riemann-Liouville frac-tional derivatives, 124

Schwartz space, 8, 10, 354Semigroup properties, 51, 53, 73, 75, 83,

96, 101, 107, 114, 121, 128, 261Sequential, 138, 140, 270, 282, 393-397,

407-408, 410, 412, 415, 421, 427,431, 433-435, 437, 445,

Sequential FDE, 282, 393-397, 427, 433-435, 437

Singular Points for linear FDE, 416-417,424-427, 429-431

Sobolev theorem, 128Stirling, 25, 115, 475, 521Substitution operator, 100, 456Systems of linear FDE, 294, 393, 474

Tempered distributions, 8Transition region, 439Truncated hypersingular integral, 132

Viscoelasticity, 347, 439Vitreous region, 439

Weighted Lp-space, 1, 129Wright Function, 42, 44, 47, 54-58, 67,

286, 291, 297, 299, 303, 305, 314,319, 331, 336, 356, 364, 366, 374

Young theorem, 11, 13

Viscoelastic material, 442, 519

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integral transform method for explicit solutions to fractional differential equations