الفهرس 
Introduction andOnly 14 pages are availabe for public view
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Abstract
Special functions of fractional calculus constitute a very important field of mathematics
according to its vital applications in various branches of science especially physics and
chemistry.
In this thesis, we mainly deal with three special functions; the generalized Mittag-
Leffler function, the generalized M-Series and the generalized K4 – function. We
introduce generalized forms of the three special functions and investigate their various
properties including convergence, differentiation, recurrence relations, integral
transforms representation and their relation with other special functions, moreover we
form some relations connected the generalized Mittag - Leffler function and the
generalized M- Series to Riemann-Liouville fractional calculus operators, Weyl
fractional calculus operators and new fractional integral operators defined in the thesis.
We also establish a new general class of polynomials associated with the generalized
M – Series.
The application part of this dissertation refers to the use of the special functions
mentioned in solving different types of the fractional kinetic equation as a model of
fractional differential and integral equations. We also illustrate the use of the
generalized Mittag – Leffler function to obtain analytical solutions of initial and
boundary value problems associated with fractional nonhomogeneous differential
equations.
This thesis is divided into five chapters
Chapter 1: Introduction and preliminaries
In this chapter, we give introduction and historical review to Mittag - Leffler type
function, M - Series and other special functions related to them. Definitions of fractional
calculus operators such as Riemann-Liouville and Weyl with related formulas are
collected. Integral transforms like Laplace, Beta, Mellin and any other integral
transforms needed during the research work are indicated. This introductory chapter
includes all the related definition, preliminaries and formulas used during this
dissertation.
Chapter 2: On generalized Mittag – Leffler function
In this chapter, we collect and review some results concerning the generalized Mittag-
Leffler function and extend them to obtain new formulas, relations and theorems of that
function. We also recall its relations to Riemann-Liouville fractional integral and
differential operators. A new integral operator containing the generalized Mittag -
Leffler function in its kernel is presented and the composition of the new operator with
Riemann-Liouville fractional integral and differential operators are indicated. Using
Laplace transform method, we give an explicit solutions of general fractional
differential equations including Hilfer fractional differential operator in terms of the
generalized Mittag – Leffler function.
The chapter is also devoted to further properties of the generalized Mittag - Leffler
function with another type of fractional calculus operators called the Weyl fractional
integral and differential operators. We investigate the basic properties of Weyl
fractional integral and differential operators with the generalized Mittag - Leffler function, moreover a new integral operator containing Mittag - Leffler function in its
kernel is established. In addition to that, composition of Weyl fractional integral and
differential operators with the new operator is formed.
The new results of this chapter were published in
” Hindawi Publishing Corporation, Journal of Mathematics” Volume 2013, Article
ID 821762, http://dx.doi.org/10.1155/2013/821762
Chapter 3: On generalized M- Series
In this chapter, we introduce a new generalization of the M-series and examine its
conditions of convergence. Recurrence relations, differentiation, integral transforms
representation and formulas of fractional calculus operators of the series are stated and
proved. A new integral operator containing the generalized M-series in its kernel is
established and the composition of Riemann-Liouville fractional integral and
differential operators with the integral operator defined are demonstrated.
A general class of polynomials associated with the generalized M -Series is
established and its special cases are obtained. We also derive several families of
generating relations and finite summation formulas by employing operational
techniques.
The new results of this chapter were published in
”Asian Journal of Fuzzy and Applied Mathematics”, Vol. 2, No. 5, 2014.
Chapter 4: Generalized fractional kinetic equation in terms of special functions.
In this chapter, we introduce a new generalized 4 K -function and derive some properties
of it. The new 4 K - function is used in solving the generalized fractional kinetic equation
in terms of the generalized Mittag – Leffler function and the generalized M – Series.
We apply two different methods for covering the solutions of the generalized fractional
kinetic equation, one of them based on the fractional differ-integral operator method
while the other based on Laplace transform operator technique.
The new results of this chapter were published in both of
”Journal of Mathematical and Computational Science”, Vol. 4, No. 6, 2014.
”International Mathematical Forum”, Vol. 9, No. 33, 2014.
Chapter 5: Conclusions and recommendations