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http://dx.doi.org/10.5666/KMJ.2020.60.1.73

Fractional-Order Derivatives and Integrals: Introductory Overview and Recent Developments  

Srivastava, Hari Mohan (Department of Mathematics and Statistics, University of Victoria)
Publication Information
Kyungpook Mathematical Journal / v.60, no.1, 2020 , pp. 73-116 More about this Journal
Abstract
The subject of fractional calculus (that is, the calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past over four decades, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of mathematical, physical, engineering and statistical sciences. Various operators of fractional-order derivatives as well as fractional-order integrals do indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this survey-cum-expository article is to present a brief elementary and introductory overview of the theory of the integral and derivative operators of fractional calculus and their applications especially in developing solutions of certain interesting families of ordinary and partial fractional "differintegral" equations. This general talk will be presented as simply as possible keeping the likelihood of non-specialist audience in mind.
Keywords
fractional calculus; fractional-order integrals; fractional-order derivatives; differential equations; Integral equations; Cauchy-Goursat integral formula; differintegral equations; special functions; mathematical physics; Fuchsian and non-Fuchsian differential equations; fractional kinetic equations; Laplace and Sumudu transforms; Mittag-Leffler type functions; fractional integral operators;
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