Investigation on Fractional Integration Fractional Differentiation and Fractional Differential Equations Involving Special Functions and Hypergeometric Functions

Abstract

Fractional Calculus have applications in diverse and widespread fields of engineering newlineand sciences such as electromagnetics, viscoelasticity, fluid mechanics, newlineelectrochemistry, biological population models, optics, and signals processing. It has newlinebeen used to create physical and engineering models for processing the best newlineexplanation through fractional differential equations. The fractional derivative models newlineare used for accurate modeling of those systems that require accurate modeling of newlinedamping. In these fields, various analytical and numerical methods including their newlineapplications have been proposed in recent years. In spite of fractional inequalities newlinehave many applications, the most useful ones are in fractional boundary values newlineproblems and fractional partial differential equations creating uniqueness of solutions. newlineThese considerations have led various researchers in the field of integral inequalities newlineto explore certain extensions and generalizations involving fractional calculus newlineoperators. In present work, investigation related to fractional integral and derivative newlineformulae, solutions of fractional kinetic equations and study of inequality are newlinepresented. The chapter wise details are listed as follows : newline1. Introduction and Preliminaries : This chapter is preliminary in nature and newlinecontains a brief introduction to various topics studied in the thesis. In this chapter newlinebasic theory of special functions, generalized polynomials, Integral transforms, newlinefractional calculus operators and generating functions is given. newline2. Literature Review : In this chapter, review of past literature is presented. To find newlinethe gap between past and present work a number of research papers are studied and newlinediscussed here. newline3. Solutions of Fractional Kinetic Equations Involving Generalized Multiindex newlineBessel Function : In this chapter, define a new generalized form of the fractional newlinekinetic equation involving generalized multi-index Bessel-Maitland function is newlinedeveloped. The manifold generality of the generalized multi-index Bessel- newlineMaitland func