TITLE:
Fractionalization of a Class of Semi-Linear Differential Equations
AUTHORS:
Issic K. C. Leung, K. Gopalsamy
KEYWORDS:
Fractional Integral, Caputo Fractional Derivative, Fading Memory, Mittag-Leffler Functions, Complete Monotonicity, Fractionalization, Variation of Constants Formula, Fourier Integral Theorem, Mittag-Leffler Stability
JOURNAL NAME:
Applied Mathematics,
Vol.8 No.11,
November
30,
2017
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.