Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
type: Item
instance of: scholarly article
title: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
zbMATH Open document ID: 0924.34008
author: Igor Podlubny
published in: Mathematics in Science and Engineering
publication date: 1999-09-01T00:00:00Z
review text: The subject of fractional calculus (that is, derivatives and integrals of any real or complex order) has gained importance during the past three decades due mainly to its demonstrated applications in such diverse fields of science and engineering as (for example) fluid flow, rheology, dynamical processes in self-similar and porous structures, diffusive transport akin to diffusion, electrical networks, probability and statistics, control theory of dynamical systems, viscoelasticity, electrochemistry of corrosion, and chemical physics. Indeed it provides several potentially useful tools for solving differential and integral equations, and other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The concept of fractional calculus seems to have stemmed from a letter from Marquis de l'Hôpital (1661-1704) to G. W. Leibniz (1646-1716), dated 30 September 1695, which sought the meaning of Leibniz's (currently popular) notation \({d^ny\over dx^n}\) for the derivative of order \(n\) when \(n={1\over 2}\). Fractional integrals of an arbitrary (real or complex) order are a generalization of the ordinary integral of order \(n(n\in\mathbb{N})\). Indeed, if we define the linear integral operators \({\mathcal I}\) and \({\mathcal K}\) by \[ ({\mathcal I}f)(x):=\int^x_0f(t)dt\tag{1} \] and \[ ( {\mathcal K}f)(x):=\int^\infty_xf(t)dt,\tag{2} \] then it is easily seen by iteration that \[ ({\mathcal I}^nf)(x)= {1\over(n-1)!} \int^x_0(x-t)^{n-1}f(t)dt \quad(n\in \mathbb{N}) \tag{3} \] and \[ ({\mathcal K}^nf)(x)= {1\over(n-1)!} \int^\infty_x (t-x)^{n-1}f(t)dt \quad (n\in\mathbb{N}). \tag{4} \] With a view to interpolating \((n-1)!\) between the positive integer values of \(n\), one can set \[ (n-1)!= \Gamma(n) \tag{5} \] in terms of the familiar Gamma function. Thus, in general, equations (3) and (4) would lead eventually to the Riemann-Liouville operator \({\mathcal R}^\mu\) and the Weyl operator \({\…
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 34Axx
Mathematics Subject Classification ID: 34-01
zbMATH DE Number: 1329842
zbMATH Keywords: fractional differential equations
zbMATH Keywords: fractional derivatives
zbMATH Keywords: existence
zbMATH Keywords: fractional calculus
zbMATH Keywords: Riemann-Liouville operator
zbMATH Keywords: Weyl operator
zbMATH Keywords: fractional integral
zbMATH Keywords: uniqueness
reviewed by: H. M. Srivastava
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1329842