Abstract
A b s t r a c t: The subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It does 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 paper* is to present a brief elementary and introductory approach to the theory of fractional calculus and its applications especially in developing solutions of certain interesting families of ordinary and partial fractional differintegral equations. Relevant connections of some of the results presented in this lecture with those obtained in many other earlier works on this subject will also be indicated.
Highlights
The concept of it fractional calculus that is, calculus of integrals and derivatives of any arbitrary real or complex order) seems to have stemmed from a question raised in the year 1695 by Marquis de l'Hôpital (1661-1704) to Gottfried Wilhelm Leibniz (1646-1716), which sought the meaning of Leibniz's notation dny dx n for the derivative of order n ∈ 0 := {0,1,2,...} when n = 1
This leads to superslow or intermediate processes that, in mathematical physics, we may refer to as fractional phenomena
Each of the following general results is capable of yielding particular solutions of many simpler families of linear ordinary fractional differintegral equations
Summary
To the theories of differential, integral, and integro-differential equations, and special functions of mathematical physics as well as their extensions and generalizations in one and more variables, some of the areas of present-day applications of fractional calculus include. The first work, devoted exclusively to the subject of fractional calculus, is the book by Oldham and Spanier [39]. Today there exist at least two international journals which are devoted almost entirely to the subject of fractional calculus: (i) Journal of Fractional Calculus and (ii) Fractional Calculus and Applied Analysis In this expository lecture, we aim at presenting an elementary and introductory overview of the theory of fractional calculus and of some of its applications
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