Fractionalization of a class of semi-linear differential equations

Kui Chiu Issic LEUNG, K. GOPALSAMY

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Abstract

The dynamics of a fractionalized semi-linear scalar differential equation is considered with a Caputo fractional derivative. By using a symbolic operational method, a fractional order initial value problem is converted into an equivalent Volterra integral equation of second kind. A brief discussion is included to show that the fractional order derivatives and integrals incorporate a fading memory (also known as long memory) and that the order of the fractional derivative can be considered to be an index of memory. A variation of constants formula is established for the fractionalized version and it is shown by using the Fourier integral theorem that this formula reduces to that of the integer order differential equation as the fractional order approaches an integer. The global existence of a unique solution and the global Mittag-Leffler stability of an equilibrium are established by exploiting the complete monotonicity of one and two parameter Mittag-Leffler functions. The method and the analysis employed in this article can be used for the study of more general systems of fractional order differential equations. Copyright © 2017 by authors and Scientific Research Publishing Inc.
Original languageEnglish
Pages (from-to)1715-1744
JournalApplied Mathematics
Volume8
Issue number11
DOIs
Publication statusPublished - Nov 2017

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Semilinear Differential Equations
Fractional Order
Complete Monotonicity
Variation of Constants Formula
Fading Memory
Fractional Order Differential Equations
Differential equation
Caputo Fractional Derivative
Mittag-Leffler Function
Fourier Integral
Integer
Long Memory
Volterra Integral Equations
Fractional Derivative
Semilinear
Unique Solution
Global Existence
Initial Value Problem
Two Parameters
Scalar

Citation

Leung, I. K. C., & Gopalsamy, K. (2017). Fractionalization of a class of semi-linear differential equations. Applied Mathematics, 8(11), 1715-1744. doi: 10.4236/am.2017.8111

Keywords

  • Fractional integral
  • Caputo fractional derivative
  • Fading memory
  • Mittag-leffler functions
  • Complete monotonicity
  • Fractionalization
  • Variation of constants formula
  • Fourier integral theorem
  • Mittag-leffler stability