study of integral transforms and special functions associated with fractional calculus operators and their applications

Abstract

The thesis is organized into five chapters. The first chapter introduces the topic of study and provides a brief overview of the contributions made by previous researchers on the subject. It also includes a summary of the literature review conducted as part of the research work. newlineThe second chapter investigates the compositions and applications of fractional integral operators, specifically focusing on the Pathway Fractional Integral operator and the Extended Unified Mittag-Leffler Function. When applied to power multiples of the Extended Unified Mittag-Leffler Function, this operator results in the multiplication of a beta function with the Extended Unified Mittag-Leffler Function, further transforming into k-Wright functions and hyper geometric forms.The third chapter aims to establish closed-form expressions for the Pathway fractional integral operator and the Marichev-Saigo-Maeda fractional integral and differential operators, particularly involving the product of the Special G-function and the Generalized Mittag-Leffler Function. The results are evaluated in terms of the generalized Wright hyper geometric function.In the fourth chapter we explore various integral transforms including the Euler, Laplace, Whittaker, and Henkel transforms applied to the product of the Special G-function and the Generalized Mittag-Leffler Function. Some results are expressed in terms of the generalized Wright function, with potential applications in various fields of science, engineering, and technology.The fifth chapter establishes the inverse Laplace transform of the generalized Meijer G-function in terms of hyper geometric functions, providing a novel approach to calculating the inverse Laplace transform. The paper also discuss several properties of the generalized Meijer G-function, contributing new insights into the mathematical analysis of special functions and their associated integral transforms.At the end,a list of the author s publications related to the subject matter of this work is provided,along with reprints of t