A study of generalized Riemann Liouville operators and their applications

Abstract

One of the classical streams of Mathematics is Fractional Calculus which is connected to the newlinestudy of Fractional order integrals as well as derivative operators which encompasses both newlinethe domains be it real or complex. This study also comprises their applications in the newlineassociated fields. Today, Fractional Calculus is a known field of mathematical studies that newlinehas emanated from the usual definitions of the calculus integral and derivative operators more newline newlineSchool of Basic and Applied Sciences, Poornima University, Jaipur newline newlineNovember 2019 newline newlineor less in the similar way as the Fractional exponents being an outgrowth of exponents with newlineinteger value. The fundamentals of Fractional Calculus (Fractional derivatives and Fractional newlineintegrals) are quite traditional. A Fractional derivative may be simply defined as an operator, newlinethe function of which is to generalize an ordinary derivative. Fractional differential equations newlinethat have involved both the real and the complex order derivatives definitely have created a newlineniche for modelling the unusual dynamics of several processes that have relation with newlinecomplex systems in the areas of engineering as well as science. Fractional calculus may be newlineconsidered as generalization of conventional calculus due to the fact that it extends the newlineconcept of the derivatives and integrals to cater to the arbitrary orders. This report aims first newlineto expose the concepts, applicable definitions of Fractional Calculus Operators, Special newlineFunctions, Generalized Fractional Calculus and second to demonstrate how these functions newlineare used for solving several Fractional differential equations along with their applications in newlinediverse areas. The main objective of this thesis is to examine and investigate the generalized newlineFractional Operators and the composition of Special Functions with them. newline