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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