Published in: Fractional Calculus and Applied Analysis,vol. 4, No 2 (2001), pp. 153 - 192

http://www.math.bas.bg/fcaa , http://www.diogenes.bg/fcaa

THE FUNDAMENTAL SOLUTION OF

THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION

Francesco Mainardi 1, Yuri Luchko 2, Gianni Pagnini 1

Dedicated to Rudolf Gorenflo,Prof. Emeritus of the Free University of Berlin,

on the occasion of his 70-th birthday (July 31, 2000)

Abstract

We deal with the Cauchy problem for the space-time fractional diffusion equa-tion, which is obtained from the standard diffusion equation by replacing thesecond-order space derivative with a Riesz-Feller derivative of order (0, 2]and skewness (|| min {, 2 }), and the first-order time derivative with aCaputo derivative of order (0, 2] . The fundamental solution (Green function)for the Cauchy problem is investigated with respect to its scaling and similarityproperties, starting from its Fourier-Laplace representation. We review the par-ticular cases of space-fractional diffusion {0 < 2 , = 1} , time-fractionaldiffusion { = 2 , 0 < 2} , and neutral-fractional diffusion {0 < = 2} ,for which the fundamental solution can be interpreted as a spatial probability den-sity function evolving in time. Then, by using the Mellin transform, we provide ageneral representation of the Green functions in terms of Mellin-Barnes integralsin the complex plane, which allows us to extend the probability interpretation tothe ranges {0 < 2} {0 < 1} and {1 < 2}. Furthermore, fromthis representation we derive explicit formulae (convergent series and asymptoticexpansions), which enable us to plot the spatial probability densities for differentvalues of the relevant parameters , , .

Mathematics Subject Classification: 26A33, 33E12, 33C40, 44A10, 45K05,60G52

Key Words and Phrases: diffusion, fractional derivative, Fourier transform,Laplace transform, Mellin transform, Mittag-Leffler function, Wright function,Mellin-Barnes integrals, Green function, stable probability distributions

154 F. Mainardi, Yu. Luchko, G. Pagnini

1. Introduction

A space-time fractional diffusion equation, obtained from the standard dif-fusion equation by replacing the second order space-derivative by a fractionalRiesz derivative order > 0 and the first order time-derivative by a fractionalderivative of order > 0 (in Caputo or Riemann-Liouville sense), has been re-cently treated by a number of authors, see for example Saichev and Zaslavsky[38], Uchaikin and Zolotarev [48], Gorenflo, Iskenderov and Luchko [17], Scalas,Gorenflo and Mainardi [42], Metzler and Klafter [34]. For other treatments of thespace fractional and/or time fractional diffusion equations we refer the reader tothe references cited therein. See below for the restrictions on and .

In this paper we intend to complement the results obtained in [17] by allowingasymmetry in the space fractional derivative. We thus consider the space-timefractional diffusion equation

xD u(x, t) = tD

u(x, t) , x IR , t IR+ , (1.1)

where the , , are real parameters always restricted as follows

0 < 2 , || min{, 2 } , 0 < 2 . (1.2)In (1.1) u = u(x, t) is the (real) field variable, xD is the Riesz-Feller space-fractional derivative of order and skewness , and tD

is the Caputo time-

fractional derivative of order . These fractional derivatives are integro-differentialoperators to be defined later.

The paper is divided as follows. In Section 2 we provide the reader withthe essential notions and notations concerning the Fourier, Laplace and Mellintransforms, which are necessary in the following. In Section 3 we introduce theCauchy problem for the equation (1.1) and find the corresponding fundamentalsolution G,(x, t) (the Green function) in terms of its Fourier-Laplace transformfrom which we derive its general scaling properties and the similarity variablex/t/ . We shall get the fundamental formula

G,(x, t) = t K,(x/t

) , = / , (1.3)

where K,(x) is referred to as the reduced Green function. In Section 4, weconsider the particular cases {0 < 2 , = 1} (space fractional diffusion),{ = 2 , 0 < 2} (time fractional diffusion), and {0 < = 2} (neutralfractional diffusion), for which the fundamental solution can be interpreted as aspatial probability density function (pdf), evolving in time. In Section 5 we showa composition rule for the Green function, valid for {0 < 2} {0 < 1}which ensures its probability interpretation in this range. In Section 6, we providea general representation for the (reduced) Green function in terms of a Mellin-Barnes integral in the complex plane, which enables us to extend the probability

THE FUNDAMENTAL SOLUTION OF . . . 155

interpretation to the range {1 < 2}. In Section 7, we derive for the Greenfunction explicit formulae (convergent series and asymptotic expansions), whoseform depends on the relation between the parameters and . By means of asuitable matching between the convergent and the asymptotic expansion we shallbe able to compute the Green function in all the cases in which it is interpretableas a probability density. Finally, Section 8 is devoted to concluding discussions,and a summary of the results in which we present plots of the Green function fora number of cases.

2. Notions and notations

For the sake of the readers convenience here we present an introduction to theRiesz-Feller and Caputo fractional derivatives starting from their representationin the Fourier and Laplace transform domain, respectively. So doing we avoid thesubtleties lying in the inversion of fractional integrals, see e.g. [39], [21]. We alsorecall the main properties of the Mellin transform that will be used later.

Since in what follows we shall meet only real or complex-valued functions of areal variable that are defined and continuous in a given open interval I = (a, b) , a < b + , except, possibly, at isolated points where these functionscan be infinite, we restrict our presentation of the integral transforms to the classof functions for which the Riemann improper integral on I absolutely converges.In so doing we follow Marichev [32] and we denote this class by Lc(I) or Lc(a, b) .

The Fourier transform and the Riesz-Feller space-fractional derivative

Letf() = F {f(x);} =

+

e+ix f(x) dx , IR , (2.1a)

be the Fourier transform of a function f(x) Lc(IR), and let

f(x) = F1{f();x

}=

12

+

eix f() dx , x IR , (2.1b)

be the inverse Fourier transform.(1) For a sufficiently well-behaved function f(x)we define the Riesz-Feller space-fractional derivative of order and skewness as

F { xD f(x);} = () f() , (2.2)

() = || ei(sign)/2 , 0 < 2 , || min {, 2 } . (2.3)

(1) If f(x) is piecewise differentiable, then the formula (2.1b) holds true at allpoints where f(x) is continuous and the integral in it must be understood inthe sense of the Cauchy principal value.

156 F. Mainardi, Yu. Luchko, G. Pagnini

1

0.5

0

0.5

1

0.5 1 1.52

Fig. 1. The Feller-Takayasu diamond

We note that the allowed region for the parameters and turns out to be a dia-mond in the plane { , } with vertices in the points (0, 0) , (1, 1) , (1,1) , (2, 0) ,that we call the Feller-Takayasu diamond, see Fig. 1.

Thus, we recognize that the Riesz-Feller derivative is required to be thepseudo-differential operator(2) whose symbol () is the logarithm of the char-acteristic function of a general Levy strictly stable probability density with indexof stability and asymmetry parameter (improperly called skewness) accordingto Fellers parameterization [13, 14], as revisited by Gorenflo and Mainardi, see[22, 23, 24].

The operator defined by (2.2)-(2.3) has been referred to as the Riesz-Fellerfractional derivative since it is obtained as the left inverse of a fractional integraloriginally introduced (for = 0 and 6= 1) by Marcel Riesz in the late 1940s,known as the Riesz potential, and then generalized (for 6= 0) by William Fellerin 1952, see [13], [39].

For more details on Levy stable densities we refer the reader to specialistictreatises, as Feller [14], Zolotarev [49], Samorodnitsky and Taqqu [40], Janickiand Weron [25], Sato [41], Uchaikin and Zolotarev [48], where different notationsare adopted. We like to refer also to the 1986 paper by Schneider [43], where hefirst provided the Fox H-function representation of the stable distributions (with 6= 1) and to the 1990 book by Takayasu [45] where he first gave the diamondrepresentation in the plane {, }.

(2) Let us recall that a generic pseudo-differential operator A, acting withrespect to the variable x IR , is defined through its Fourier representation,namely

+ e

ix A [f(x)] dx = A() f() , where A() is referred to as symbolof A , given as A() =

(A eix

)e+ix .

THE FUNDAMENTAL SOLUTION OF . . . 157

For = 0 we have a symmetric operator with respect to x , which can beinterpreted as

xD0 =

( d

2

dx2

)/2, (2.4)

as can be formally deduced by writing || = (2)/2 . We thus recognizethat the operator D0 is related to a power of the positive definitive operatorxD2 = d2dx2 and must not be confused with a power of the first order differentialoperator xD = ddx for which the symbol is i . An alternative illuminatingnotation for the symmetric fractional derivative is due to Zaslavsky, see e.g. [38],and reads

xD0 =

d

d|x| . (2.5)

In its regularized form valid for 0 < < 2 the Riesz space-fractional derivativeadmits the explicit representation(3)

xD0 f(x) =

(1 + )

sin(

2

) 0

f(x + ) 2f(x) + f(x )1+