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.
Highlights
Fractional differential equations are those equations in which an unknown function appears under a fractional order differentiation operator
A brief discussion is included to show that the fractional order derivatives and integrals incorporate a fading 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
Summary
Fractional differential equations are those equations in which an unknown function appears under a fractional (or non-integer) order differentiation operator. In the case of initial value problems of integer order calculus, the initial values correspond to the respective initial values of the state variables and this is the case for the initial value problems with the Caputo fractional derivatives This is not the case with initial value problems with the Riemann-Liouville fractional derivatives. Our method is based on an application of a symbolic operational calculus to derive an analogue of the variation of constants formula well known in the theory of integer order differential equations Such a variation of constants formula for the fractionalized initial value problem is shown to reduce to the well known formula of the integer order system when the fractional order. In the opinion of these authors this appears to be the first time that such a method has been proposed in the study of the stability of fractional (non-integer) order differential equations; we remark that a comparison principle and the positivity of certain Mittag-Leffler functions have been used by [18] in their stability considerations of fractional differential equations
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