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CHAPTER – 0
INTRODUCTION TO THE TOPIC
OF STUDY AND CHAPTERWISE
SUMMARY OF THE THESIS
0.1 INTRODUCTION
In this chapter, we give an introduction to the topic of study and brief
survey of the contributions made by some of the earlier workers in this field. A
brief chapter by chapter summary of the thesis is also given.
0.2 MITTAG-LEFFLER FUNCTION
The Mittag-Leffler function has gained importance and popularity due to its
applications in the solution of fractional-order differential, integral, integro-
differential and difference equations arising in certain problems of applied
sciences such as physics, chemistry, biology and engineering [Kilbas et al. (2006)]
This function was introduced by Swedish mathematician Mittag-Leffler
(1903) in terms of the following power series
0
,1
n
n
zE z
n
(0.2.1)
where , , Re( ) 0z . The Mittag-Leffler function (0.2.1) reduces to the
exponential function ze when 1 . For 0 1 , it interpolates between the
pure exponential ze and a geometric function 0
1,
1
n
n
zz
for 1z .
A generalization of (0.2.1) was studied by Wiman (1905), in the form
Chapter 0 2
,
0
,n
n
zE z
n
(0.2.2)
where , , , Re 0, Re 0z .
The Mittag-Leffler function arises naturally in the solution of fractional
order integral equations or fractional order differential equations and especially
in investigations of fractional generalization of kinetic equation, random walks,
Levy flights, super-diffusive transport and in the study of complex systems.
This function also occurs in the solution of certain boundary value problems
involving fractional integro-differential equations of Volterra type [Samko et
al. (1993)]. During last four decades, the interest in Mittag-Leffler type
functions has considerably increased among engineers and scientists due to
their applications in several applied problems, such as fluid flow, rheology,
diffusive transport akin to diffusion, electric networks, probability, statistical
distribution theory etc. For a detailed account of various properties,
generalizations, and application of this function, the works of Dzherbashyan
(1966), Caputo & Mainardi (1971), Scott Blair (1974), Torvik & Bagley
(1984), Gorenflo & Vessella (1991), Kilbas & Saigo (1995), Gorenflo &
Mainardi (1996), Kilbas et al. (2002), Kilbas et al. (2004) and Haubold et al.
(2007), Saxena & Kalla (2008) are worth mentioning.
In Chapter 1, we shall define a Mittag-Leffler type function with four
parameters.
Chapter 0 3
0.3 HYPERGEOMETRIC FUNCTIONS
The study of special functions plays an important role in solving various
problems arising in physics, biology, engineering, chemistry, computer science
and statistics. Due to their great importance and wide applications several
books and a large collection of papers are devoted to their study. The great
mathematicians namely Euler, Gauss, Legendre, Riemann and Ramanujan have
laid the foundations for this beautiful and useful area of mathematics.
A major development in the theory of special functions was the study of
hypergeometric series which was developed by Gauss (1812) and is named as
Gauss hypergeometric function. It is represented by the following series
2
2 1
0
. 1 . . 1., ; ; 1 ...
! 1. 1.2. . 1
n
n n
n n
a b a a b bz a bF a b c z z z
c n c c c
, (0.3.1)
where ,a ,b c and z may be real or complex, 0, 1, 2,...c and n
a is the
pochhammer symbol [Rainville (1960)] defined as
1 2 ... 1n
a a a a a n for 1,n
0
1, 0.a a (0.3.2)
If either a or b is a non-positive integer, the function reduces to a
polynomial.
Chapter 0 4
If we replace z by /z b and let b in (0.3.1) and use the principle of
confluence, we arrive at the following well known Kummer series or confluent
hypergeometric function
2
1 1
0
. 1; ; 1 . ...
! 1! . 1 2!
n
n
n n
a a az a z zF a c z
c n c c c
. (0.3.3)
A natural generalization of Gauss hypergeometric function is the generalized
hypergeometric function p qF which is defined in the following manner
1
1 1
0 1
.....,..., ; , ..., ;
!.....
npn n
p q p q
n qn n
a a zF a a b b z
nb b
, (0.3.4)
where p and q are non-negative integers (interpreting an empty product as unity),
the variable z and all the parameters 1 1,..., , , ...,p qa a b b are real or complex
numbers such that no denominator parameter is zero or a non-positive integer.
The conditions of convergence of p qF are as follows
(i) when p q , the series on the right hand side of (0.3.4) is convergent.
(ii) when 1p q , the series in (0.3.4) is convergent if 1z , divergent if
1z and on the circle 1z , the series is
(a) absolutely convergent, if Re 0w ,
Chapter 0 5
(b) conditionally convergent, if 1 Re 0w for 1z ,
(c) divergent, if Re 1w ,
where 1 1
q p
j j
j j
w b a
.
(iii) when 1p q , the series never converges except when 0z and the
function is defined only when the series terminates.
A comprehensive account of the functions 2 1F , 1 1F and p qF can be found
in the works of Erdélyi et al. (1953), Rainville (1960), Slater (1960, 1966),
Luke (1969) and Exton (1976).
A further generalization of p qF is the Fox-Wright function p q defined as
[Srivastava et al. (1982)]
1 1 1 1
0 1 11 1
, ,..., , ...,
!..., ,..., ,
np p p p
p q
n q qq q
a A a A a A n a A n zz
nb B n b B nb B b B
(0.3.5)
, , ,i jz a b 0, 0,i jA B
1,..., ;i p 1,...,j q
and
1 1
1q p
j i
j i
B A
. (0.3.6)
Chapter 0 6
0.4 MELLIN-BARNES CONTOUR INTEGRALS
In 1946, C.S. Meijer introduced another important generalization of
special functions popularly known as Meijer’s G function in the literature.
Though the G function contains several special functions as its particular
cases, many functions such as Mittag-Leffler function (1903), Wright’s
generalized Bessel function [Wright (1934)], Wright’s generalized
hypergeometric function [Wright (1935)], R -function [Lorenzo & Hartley
(1999)] and several other functions that are useful in the study of fractional
calculus, do not form its special cases. A more general function, known as
H function, which includes all the above mentioned functions as its special
cases, was firstly introduced by Pincherle (1888) in the form of Mellin-Barnes
counter integral. This function was further developed and studied by Fox
(1961). It is defined as
1 1 2 21,, , ,
, , ,
1 1 2 21,
, , , , ,..., ,
, , , , ,..., ,
j j p ppm n m n m n
p q p q p q
j j q qq
a a a aH z H z H z
b b b b
1 1
1 1
11
,2
1
m n
j j j j
j j s
q p
Lj j j j
j m j n
b s a s
z dsi
b s a s
(0.4.1)
where, for details of contour L , various parameters and convergence of (0.4.1),
we refer the book by Srivastava et al. (1982).
Chapter 0 7
The series representation of H function is given by [Srivastava et al.
(1982), p. 12]
,,
, , ,
1 0 1 1
1 1 h r
m nmrm n
p q j j h r j j h r
h r j jj h
H z b a z
1
, ,
1 1
1 ! ,q p
j j h r j j h r h
j m j n
b a r
(0.4.2)
where , /h r h hb r , h j j hb b r and ,j h , 1,..., ,j h m
, 0,1,2,...r .
Lot of research work has been done on the development of H function
and can be referred in the books by Mathai & Saxena (1978), Srivastava et al.
(1982) and Kilbas & Saigo (2004).
0.5 FRACTIONAL CALCULUS
Fractional calculus is a generalization of ordinary differentiation and
integration to arbitrary non-integer order. The idea of fractional calculus has
been a subject of interest not only among mathematicians, but also among
physicists and engineers.
We give here definitions of various fractional integrals and derivatives
that we shall require in subsequent chapters.
Chapter 0 8
(i) The Riemann-Liouville fractional integral of order is defined as
[Samko et al. (1993), p. 33, Eq. (2.17)]
11
, Re 0,
x
a x
a
I f x x t f t dt
(0.5.1)
with
0 .a xI f x f x
Clearly we have
1
,1
a xI x a x a
Re 1. (0.5.2)
(ii) The Riemann-Liouville fractional derivative of order ,
1 Re ,m m m , for a real valued function f x defined on
0, , is defined as [Samko et al. (1993), p. 37, Eq. (2.32)]
11, 1 Re
.
,
xmm
aa x
m
D x t f t dt m mmD f x
D f t m
(0.5.3)
(iii) The Caputo fractional derivative of order , 1 Re ,m m
m , is defined as [Caputo (1969)]
*
1
1 1.
mx
m m
a x a x m
a
dD f x I D f x f t dt
m dtx t
(0.5.4)
Chapter 0 9
Clearly we have
* 1, Re 1.
1a xD x x
(0.5.5)
For the Riemann-Liouville fractional integral and Caputo fractional
derivative, we have the following important relation
1
*
0
, !
km
k
a x a x x ak
x aI D f x f x D f x
k
1 Re ,m m .m
(0.5.6)
(iv) The Riesz-Feller fractional derivative of order , 0 2 , which is
given as a pseudo-differential operator with the Fourier symbol
,k k
is defined as [Feller (1966)]
1 ,d
g x F k g k xd x
(0.5.7)
where g k is the Fourier transform of the function g x , defined as
; .ikxg k F g x k g x e dx
(0.5.8)
(v) The Hilfer fractional derivative of order 0 1 and type 0 1
is defined as follows [Hilfer (2000)]
1 1 1, .t a t a tD f t I D I f t
(0.5.9)
Recently this definition is extended for
1 ,m m ,m 0 1 and is termed as composite fractional
Chapter 0 10
derivative or generalized Riemann-Liouville fractional derivative,
given by [Hilfer et al. (2009)]
1, .
m mm
t a t a tD f t I D I f t
(0.5.10)
The above definition, in the case 0 , gives the Riemann-
Liouville fractional derivative (0.5.3) as
,0 mm
t a t a tD f t D I f t D f t
(0.5.11)
and in the case 1 , it gives the Caputo fractional derivative (0.5.4) as
,1 * .
m m
t a t a tD f t I D f t D f t
(0.5.12)
For 0 1 , it interpolates continuously between these two
derivatives.
The Laplace transform
0
; stL g t s g t e dt
(0.5.13)
of the composite fractional derivative (0.5.10) is given by [Tomovski
(2012)]
, ;tL D f t s
11 1 1
0
; .m
m k m k
a tt ak
s L f t s s D I f t
(0.5.14)
Chapter 0 11
0.6 FRACTIONAL DIFFERENTIAL EQUATIONS
Fractional differential equations have gained considerable importance due
to their application in various fields of science and engineering. In recent years,
there has been a significant development in ordinary and partial differential
equations involving fractional derivatives [see the monographs of Oldham &
Spanier (1974), Samko et al. (1993), Miller & Ross (1993), Podlubny (1999)
and Kilbas et al. (2006)]. Numerous problems in these areas are modeled
mathematically by systems of fractional differential equations.
A growing number of works in science and engineering deal with
dynamical systems described by fractional order equations that involve
derivatives and integrals of non-integer order [Benson et al. (2000), Metzler &
Klafter (2000) and Zaslavasky (2002)]. These new models are more adequate
than the previously used integer order models, because fractional order
derivatives and integrals describe the memory and hereditary properties of
different substances [Podlubny (1999)]. This is the most significant advantage
of the fractional order models in comparison with integer order models, in
which such effects are neglected. In the context of flow in porous media,
fractional space derivatives exhibit large motions through highly conductive
layers or fractures, while fractional time derivatives describe particles that
remain motionless for extended period of time [Meerschaert et al. (2002)].
Recent applications of fractional differential equations to a number of
systems have given opportunity for physicists to study even more complicated
Chapter 0 12
systems. For example, the fractional diffusion equation allows describing complex
systems with anomalous behavior in much the same way as simpler systems.
In Chapter 2, we shall solve fractional free electron laser equations. In
Chapter 3, we shall solve fractional telegraph and fractional Fokker-Planck
equations. In Chapter 4, we shall solve linear fractional reaction-diffusion and
fractional telegraph equations.
0.7 METHODS OF SOLUTION OF FRACTIONAL DIFFERENTIAL
EQUATIONS
Finding accurate and efficient methods for solving fractional differential
equations has been an active research undertaking. In the last decade, various
analytical and numerical methods have been employed to solve linear and non-
linear problems. For example, Integral transform method, in which we use
Laplace, Mellin and Fourier integral transforms to construct explicit solutions
to linear fractional differential equations. Adomian decomposition method
(ADM), introduced and developed by Adomian (1986,1994), which attacks the
problem in a direct way and in straightforward fashion without using
linearization, perturbation or any other restrictive assumption that may change
the physical behavior of the model under discussion. Homotopy perturbation
method (HPM), introduced by He (2004, 2005a, 2005b, 2006), where the
solution is considered as the sum of an infinite series which converges rapidly
to the accurate solution. Variational iteration method (VIM), given by He
Chapter 0 13
(1999), which gives rapidly convergent successive approximations of the exact
solution if such a solution exists. The VIM does not require specific treatments
for non-linear problems as in Adomian decomposition method, perturbation
techniques, etc. Homotopy analysis method (HAM) introduced by Liao
(2003, 2008, 2009). This method is based on homotopy, a fundamental concept
in topology and differential geometry. It is a computational method that yields
analytical solutions and has certain advantages over standard numerical
methods. The method introduces the solution in the form of a convergent
fractional series with elegantly computable terms. Generalized differential
transform method (GDTM) developed by Momani, Odibat and Erturk in their
papers [Momani et al. (2007), Momani & Odibat (2008), Odibat & Momani
(2008) and Odibat et al. (2008)]. This method is a generalization of differential
transform method, proposed by Zhou (1986). GDTM constructs an analytical
solution in the form of a polynomial. Matrix method [Podlubny (2000) and
Podlubny et al. (2009)] which is a numerical method. Unlike other numerical
methods used for solving fractional partial differential equations in which the
solution is obtained step-by-step by moving from the previous time layer to the
next one, here in matrix method, we consider the whole time interval.
In Chapters 2 and 3, we use ADM for solving fractional integro-
differential equations and fractional partial differential equations respectively.
In Chapter 4, we use Laplace and Fourier transforms to solve linear fractional
partial differential equations.
Chapter 0 14
0.8 GENERALIZED TRAPEZOIDAL DISTRIBUTION
Uniform and beta distributions [as given in, Renyi (1970) and Mathai
(1993)] are most commonly studied distributions on finite support. The
triangular distribution has been investigated by Johnson (1997) as a proxy for
the beta distribution, even though its origin can be traced back to Thomas
Simpson (1755). A further generalization of triangular distribution is the
trapezoidal distribution which has been studied by Pouliquen (1970), Powell &
Wilson (1997) and Garvey (2000). Many physical processes in nature, human
body and mind (over time) reflect the form of the trapezoidal distribution. In
this context, trapezoidal distributions have been used in medical applications,
specifically in the screening and detection of cancer [see, e.g., Flehinger &
Kimmel (1987), Brown (1999) and Kimmel & Gorlova (2003)]. Another
domain for applications of the trapezoidal distribution is the applied physics
arena [see, e.g., Davis & Sorenson (1969), Nakao & Iwaki (2000), Sentenac et
al. (2000) and Straaijer & De Jager (2000)]. In the context of nuclear
engineering, uniform and trapezoidal distributions have been assumed as
models for observed axial distributions for burnup credit calculations (see,
Wagner & DeHart (2000) and Neuber (2000) for comprehensive description).
These distributions are important to burnup credit criticality safety analysis for
pressurized-water-reactor (PWR) fuel. Antonia & Goncalves (2006) added
some new applications of triangular and trapezoidal distributions in the genome
Chapter 0 15
analysis, particularly, in the construction of physical mapping of linear and
circular chromosomes.
We shall study here the seven parameter generalized trapezoidal
distribution which is defined by Dorp & Kotz (2003)
-1
-1
-, ,
-
-{( -1) 1}, ,
-
-, ,
-
n
m
x aa x b
b a
c xf x B b x c
c b
d xc x d
d c
(0.8.1)
where
2,
2 ( ) ( 1)( ) 2( )
mnB
b a m c b mn d c n
, .m n (0.8.2)
Generalized trapezoidal distribution herein inherit the four basic
trapezoidal parameters , ,a b c and d and contain two additional parameters m
and n specifying the growth rate and decay rate in the first and third stage of
the distributions, in addition to the boundary ratio parameter . An advantage
of the generalized trapezoidal distribution is in its flexibility which allows us
inter alia to appropriately mimic the great variety of the growth and decay
behaviors.
For 1, 2, 2m n , the generalized trapezoidal distribution reduces to
trapezoidal distribution given in Albert (2002)
Chapter 0 16
, ,
1, ,
, ,
x aa x b
b a
f x B b x c
d xc x d
d c
(0.8.3)
where
2
.Bc d a b
(0.8.4)
Further keeping b c m in (0.8.3), it reduces to triangular distribution,
as defined in Kotz & Dorp (2004)
2, ,
2, .
x aa x m
d a m a
f x
d xm x d
d a d m
(0.8.5)
On taking 1,m n the generalized trapezoidal distribution (0.8.1),
reduces to well known uniform distribution [Springer (1979)]
1, ,
0, otherwise.
a x dd af x
(0.8.6)
Chapter 0 17
0.9 DISTRIBUTION OF ALGEBRAIC FUNCTIONS OF RANDOM
VARIABLES
The problem of deriving the distribution of algebraic functions such as
sum, difference, product, ratio and linear combination of random variables,
where the individual random variable follows a particular probability density
function (pdf), occurs in wide variety of areas. Since 1900, this study received
a great deal of attention and systematic procedures for determining such
distributions have been well developed.
The distribution of sum of random variables has a wide variety of
applications in various fields. The sum of independent gamma random
variables have applications in problems of queuing theory such as
determination of total waiting time, in civil engineering such as determination
of the total excess water flow in a dam. In 1999, Loaiciga and Leipnik derived
the probability distribution of sum of two Gumbel random variables and gave
several examples of its application in hydrology. The works of Witner (1934),
Aroian (1944), Cramer (1946, 1962), Lukacs & Laha (1964), Lukacs (1970),
Moschopoulos (1985), Agrawal & Elmaghraby (2001), Albert (2002), Holm &
Alouini (2004), and Nason (2006) in the study of distribution of sum of random
variables are also worth mentioning.
The distribution of product of random variables is of interest in many
areas of science, engineering, reliability, classification, ranking and selection
Chapter 0 18
and econometrics. For example, in hydrology, stream flow is often defined as a
product of two or more variables, representing the periodic and the stochastic
components. Yang & Nadarajah (2006) added an application in environmental
sciences as, if X and Y denote the drought intensity and the drought duration
then P XY will represent the magnitude of drought. A simple but practical
problem requiring the products of independent random variables concerns
signal amplification. If n amplifiers are connected in series and if iX denotes
the amplification of the thi amplifier, the analysis of the total amplification
1 2... nY X X X is basically a problem in the analysis of products of independent
random variables. The distribution of product of random variables has been
studied by many authors. In this context, the works of Stuart (1962), Springer
& Thompson (1970), Steece (1976), Wallgreen (1980), Bhargava & Khatri
(1981), Tang & Gupta (1984), Malik & Trudel (1986), Glen et al. (2004),
Nadarajah & Gupta (2005), Nadarajah (2005, 2006, 2008), Nadarajah & Ali
(2006), Nadarajah & Dey (2006), Gupta & Nadarajah (2006) and Garg et al.
(2010 ) are worth mentioning.
In Chapter 6, we obtain distribution of the sum of two independent
generalized trapezoidal random variables. In Chapter 7, we derive distribution
of the product of two independent generalized trapezoidal random variables.
Chapter 0 19
0.10 BRIEF CHAPTER BY CHAPTER SUMMARY OF THE THESIS
The work carried out in the thesis is given in Chapters 1 to 7.
In Chapter 1, we introduce and study a Mittag-Leffler type function
, ,E z . This function includes the Mittag-Leffler function defined by
Mittag-Leffler (1903) and its generalization given by Wiman (1905), as its
special cases. Here, we first prove that , ,E z is an entire function in the
complex plane and obtain its order and type. Next, we obtain two integral
representations and Mellin-Barnes contour integral representation of
, ,E z . We further obtain two recurrence relations, differential formula and
fractional integral and derivative of , ,E z . We also obtain Euler (Beta)
transform, Laplace transform and Mellin transform of , ,E z . Finally, we
define an integral operator with , ,E z as kernel and show that it is
bounded on the Lebesgue measurable space ,L a b .
In Chapter 2, we solve non-homogeneous generalized fractional free
electron laser (FFEL) equations in single mode, pulse propagation and
transverse mode cases. We apply Adomian decomposition method to derive the
closed form solutions in terms of confluent hypergeometric functions, Hermite
polynomials and Laguerre polynomials respectively in these cases. The
fractional derivatives considered in all these equations are of Caputo type. As
special cases of our main results, we obtain solutions of four FFEL equations in
Chapter 0 20
single mode case, four FFEL equations in pulse propagation case and three
FFEL equations in transverse mode case.
In Chapter 3, first we obtain solutions of two space-time fractional
telegraph equations using Adomian decomposition method (ADM). The space
and time fractional derivatives are considered in Caputo sense and the solutions
are obtained in terms of Mittag-Leffler type functions. The first, equation
considered here, is a homogeneous space-time fractional telegraph equation.
On specializing the parameters of this equation, we get solution of classical
telegraph equation, solved earlier by Kaya (2000) and space fractional
telegraph equations. The second equation is a nonhomogeneous space-time
fractional telegraph equation. We also obtain solutions of corresponding space
fractional, time fractional and ordinary telegraph equations as its special cases.
Next, we derive closed form solutions of two fractional Fokker-Planck
equations (FFPE) by means of the ADM extended for nonlinear fractional
partial differential equations. The first FFPE is a nonlinear time fractional FPE,
which in the special case yields the solution of a nonlinear FPE studied recently
by Yildrim (2010b). The second FFPE is a linear space-time fractional FPE. On
specializing the parameters, we obtain solutions of corresponding space
fractional, time fractional and ordinary FPE. Two linear Fokker-Planck
equations studied recently by Yildirim (2010b) also follow as its special cases.
In Chapter 4, first we consider a linear space-time fractional reaction-
diffusion equation with composite fractional derivative for time and Riesz-
Chapter 0 21
Feller fractional derivative for space. We apply Laplace and Fourier transforms
to obtain its solution. On specializing the parameters, we obtain solutions of
generalized space-time fractional diffusion equation with composite fractional
derivative for time and Riesz-Feller fractional derivative for space, solved
recently by Tomovski (2012), time fractional inhomogeneous diffusion
equation with composite fractional time derivative, studied by Sandev et al.
(2011) and fractional reaction-diffusion equation with Caputo fractional time
derivative considered by Houbold et al. (2007). Next, we consider a space-time
fractional telegraph equation with composite fractional derivative for time and
Riesz-Feller fractional derivative for space and use Laplace and Fourier
transforms for solving it. On specializing the parameters, we obtain solution of
time fractional telegraph equation with Caputo fractional derivative, studied by
Orshinger & Beghin (2004).
In Chapter 5, we establish some double inequalities involving gamma
functions. First of all, we establish a theorem in which, we consider a function
, , , ,a b c d x in terms of a log-convex function and prove that it increases with
an increase in either of its parameters. As a corollary of this theorem, we obtain
a double inequality involving logarithm of ratio of two gamma functions. We
then establish a lemma, which is required in the proof of the next theorem, in
which we obtain two double inequalities for a finite sum of powers of gamma
functions. In the last theorem, we obtain a double inequality for power of a
gamma function by using log-convex and log-concave properties of gamma
Chapter 0 22
functions. On specializing the parameters in these theorems, we obtain the
results given recently by Neumann (2011).
In Chapter 6, we derive the probability density function (pdf) for the sum
of two independent generalized trapezoidal random variables having different
supports. We consider all possible nine cases with twenty seven sub-cases and
obtain sum of random variables in all of these cases according to their sub-
cases. We apply the technique of Laplace and inverse Laplace transform to
obtain the desired pdf. As an illustration, we further obtain the pdf of sum for a
suitably constrained set of parameters and using MATLAB routines draw
graphs for the pdf with variation in different parameters, occurring in the
definition of generalized trapezoidal distribution. On reducing the generalized
trapezoidal distribution to triangular distribution and uniform distribution, we
obtain the pdf’s of sum of two triangular random variables and two uniform
random variables. These results are same as given in the paper of Garg et al.
(2009) and the book by Springer (1979) respectively.
In Chapter 7, we derive the probability density function (pdf) for the
product of two generalized trapezoidal distribution independent random
variables having different supports. We consider all possible nine cases with
twenty seven sub-cases and obtain product of random variables in all of these
cases according to their sub-cases. We apply the technique of Mellin and
inverse Mellin transform to obtain the desired pdf. As an illustration, we further
obtain the pdf of product for a suitably constrained set of parameters and using
Chapter 0 23
MATLAB routines draw graphs for the pdf with variation in different
parameters, occurring in the definition of generalized trapezoidal distribution.
The result for the product of two random variables with pdf as triangular
distribution, obtained earlier by Glickman & Xu (2008), follows as special case
of our main result.
Following is the list of research papers contributed by the author
having bearing on the subject matter of the present thesis:
1. Solution of space-time fractional telegraph equation by Adomian
decomposition method, Journal of Inequalities and Special Functions
2(1), 1-7 (2011).
2. Exact solution of space-time fractional Fokker-Planck equations by
Adomian decomposition method, Journal of International Academy of
Physical Sciences 15, 1-12 (2012).
3. Fractional free electron laser equation in single mode case with Caputo
fractional derivative using Adomian decomposition method, Journal of
Rajasthan Academy of Physical Sciences 11(2), 125-132 (2012).
4. Multidimensional fractional free electron laser equations with Caputo
fractional derivatives, Journal of Inequalities and Special Functions 4(1),
36-46 (2013).
Chapter 0 24
5. Solution of generalized space-time fractional telegraph equation with
composite and Riesz-Feller fractional derivatives, International Journal
of Pure and Applied Mathematics 83(5), 685-691 (2013).
6. Some inequalities involving ratios and products of the gamma function,
Le Matematiche (Accepted).
7. On a generalized Mittag-Leffler type function with four parameters
(Communicated).
8. Linear space-time fractional reaction-diffusion equation with composite
and Riesz-Feller fractional derivatives (Communicated).
9. On the sum of two generalized trapezoidal distributions
(Communicated).
10. On product of two generalized trapezoidal distributions
(Communicated).
With the hope that most of the subject matter of the present thesis may be
new, original and interesting, it is being submitted for the award of Ph.D.
degree.
Chapter 0 25
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