On Some Problems in the Fractional Calculus

Abstract

Introduction In this thesis, a study will be made about the use of fractional calculus in solving several problems of mathematical and physical importance involving ordinary and partial differential equations of fractional-order. 'These equations are associated with distinct types of physical phe• nomena such as: relaxation process, oscillation process, diffusion process and wave propagation ct.c. Conscqnucntly, they are of fundamental importance in many branches of physics. They arc also of considerable significance from a mathematical point of view. In the last few ycars thc applications of fractional calculus has continued to develop rapidly. Nowadays, there exists a great number of articles entirely devoted to the applications of fractional calculus. Some of the main directions are: (i) Differential and integral equations (i) Partial differential equations (iii) Special functions, and other branches of analysis. The first Chapter consist.s of a sequence of definitions and preliminary properties intcrleavened with many examples which will illustrate the main ideas. We begin by reviewing the most basic and known definitions of fractional integral (Riemman-Lionville and finite Weyl opcrators) and its properties, and compare between two definitions of fractional derivatives (lUiemman-Liou ville and El-Saycd approaches) from the viewpoint of formulation and the use in applicd problems. Also the negative-direction fractional calculus will be considered and some of its properties. After introducing definitions and preliminary properties we briefly cover the background material which