THEORETICALSTUDYOFCERTAINPROBLEMS OFSPECIALFUNCTIONSANDFRACTIONAL CALCULUSOPERATORS

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 theirrelation with other special functions, moreover we form some relations connected the generalized Mittag - Leffler function and the generalized M- Series toRiemann-Liouville fractional calculus operators, Weylfractional calculus operators and new fractional integral operatorsdefined in the thesis. We alsoestablish a new general class of polynomialsassociated withthe generalized M – Series. The application part of this dissertation refers to the use ofthe 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 obtainanalytical solutions of initial and boundary value problems associated with fractionalnonhomogeneous differential equations.

This thesis is divided into five chapters

Chapter 1: Introduction and preliminaries In this chapter, wegive introduction and historical review to Mittag-Lefflertype function, M -Series and other special functions related to them. Definitions of fractional calculus operatorssuch 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 workare indicated.This introductory chapter includes all the related definition, preliminaries and formulas used during this dissertation. Chapter 2:On generalized Mittag – Lefflerfunction In this chapter, we collect and reviewsome resultsconcerningthe 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 operatorsareindicated.Using Laplace transform method, we give an explicit solutions of general fractional differential equationsincluding Hilferfractional differential operator in terms of the generalized Mittag – Leffler function. Thechapter is alsodevotedto further properties of the generalized Mittag - Leffler function with another type of fractional calculus operatorscalled the Weyl fractional integral and differential operators.We investigate the basic properties of Weyl fractional integral and differential operators with the generalized Mittag - Lefflerfunction, moreover a new integral operator containing Mittag - Lefflerfunction 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 introducea new generalization of the M-series and examine its conditions of convergence. Recurrence relations, differentiation, integral transforms representationand formulas of fractional calculus operators of the series are stated and proved.A new integral operator containing the generalized M-seriesin its kernel is established and the composition of Riemann-Liouville fractional integral and differential operators with theintegral 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 formulasby 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 introducea new generalized -function andderive some properties of it. The new -function is used insolving the generalized fractional kinetic equation in terms of thegeneralized Mittag– Lefflerfunction and the generalizedM – 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 otherbased 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. Chapter5:Conclusions and recommendations In this chapter,we submitconclusions obtained during the work, comments and plans for futurework.