A Study on Certain Special Functions Fractional Calculus of Mathematics and Its Applications

Abstract

ABSTRACT newlineThis thesis explores the study of special functions, fractional calculus, and their newlineapplications within the realm of mathematics, providing an in-depth analysis of the newlineinterplay between these two areas. Special functions are mathematical constructs that newlineextend the basic functions to solve complex problems in a variety of fields, from newlinephysics to engineering. These functions, including Gamma, Beta, and Bessel newlinefunctions, have widespread applications in differential equations, quantum mechanics, newlinestatistical mechanics, and other disciplines. This study aims to explore the properties, newlinetransformations, and applications of these functions in the context of fractional newlinecalculus. newlineFractional calculus is a branch of mathematical analysis that generalizes the concept newlineof differentiation and integration to non-integer orders. Traditionally, calculus has newlinebeen limited to integer-order derivatives and integrals, but fractional calculus provides newlinea more versatile framework that captures the memory and hereditary properties of newlinevarious phenomena. In this thesis, we investigate the fundamental principles of newlinefractional calculus, focusing on the Riemann-Liouville and Caputo definitions of newlinefractional derivatives and integrals, which have become central in modern newlineapplications. newlineOne of the core aims of this research is to explore how fractional calculus can be newlineapplied to special functions, enriching their properties and expanding their potential in newlinesolving real-world problems. The thesis examines fractional differential equations newline(FDEs) and their solutions using special functions. The role of special functions such newlineas Mittag-Leffler functions, which arise naturally in the solutions of fractional newlinedifferential equations, is discussed in detail. These functions have gained significant newlineattention due to their ability to describe processes with memory effects, anomalous newlinediffusion, and other non-local behaviours. newlineAdditionally, the thesis delves into the application of fractional calculus and special newlinefunctions in various domains, such as signal processing, control theory, physics, and newlineengineering. In signal processing, fractional-order filters are explored for their ability newlineto provide more flexible and precise filtering characteristics. In control theory, newlinefractional-order controllers are presented as an alternative to traditional controllers, newlineoffering enhanced performance in systems exhibiting long-term memory effects. The newlineapplication of these techniques to real-world problems, including viscoelastic newlinematerials, anomalous diffusion in porous media, and electrical circuits, is also newlinethoroughly examined. newlineThe research highlights the connection between fractional calculus and complex newlinephysical systems, where standard integer-order models often fail to adequately newlinedescribe real-world phenomena. In this regard, fractional models provide a more newlineaccurate and generalized description, particularly in systems involving memory, delay, newlineor non-local interactions. By leveraging special functions and fractional calculus, this newlinework contributes to the development of mathematical tools capable of addressing newlinecomplex, real-world problems across multiple disciplines. newlineIn conclusion, the thesis presents a comprehensive study of special functions and newlinefractional calculus, demonstrating their profound impact on mathematical modelling newlineand their practical applications. Through a systematic exploration of fractional newlinedifferential equations, the study highlights the importance of these mathematical tools newlinein advancing the understanding and solving of modern problems in various scientific newlineand engineering fields. The findings suggest that the synergy between special newlinefunctions and fractional calculus holds significant promise for future research and newlineapplications, paving the way for new insights and innovations. newline