Fractionalization of a Class of Semi-Linear Differential Equations

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.