Fractional Derivatives: The Perspective of System Theory

Abstract

This paper addresses the present day problem of multiple proposals for operators under the umbrella of “fractional derivatives”. Several papers demonstrated that various of those “novel” definitions are incorrect. Here the classical system theory is applied to develop a unified framework to clarify this important topic in Fractional Calculus.

Fractional Model for a Class of Diffusion-Reaction Equation Represented by the Fractional-Order Derivative

“…In this section, we recall another type of fractional derivative represented by series and developed in many works in the literature [20,28]. We particularly focus on the Grunwald-Letinov derivative [20,21] proposed to study some fractional phenomena. There exist many discussions related to the fractional derivatives operators, but all of them develop have advantages and inconveniences.…”

“…There exist many discussions related to the fractional derivatives operators, but all of them develop have advantages and inconveniences. There exist many criteria for the classification of the existing derivatives, those which respect Leibniz rule as presented in [20,21] and those which do not satisfies this property. The commutativity property is considered and discussed in the work of Atangana et al [29].…”

“…Definition 1. [18][19][20][21] We consider the function g : R −→ R, the Grunwald-Letnikov fractional derivative of order α of the function h is given by the equation…”

“…To be more precise, in this paper, we will study the fractional diffusion equation described by the Grunwald-Letnikov derivative. For history, there exist many discussions related to the good definition of the fractional derivatives, and many papers were addressed in these directions, see [20,21]. Nowadays the most used fractional derivatives in fractional calculus are: Caputo-Liouville derivative [19], Liouville-Riemann derivative [19], conformable derivative, Grunwald-Letnikov derivative, Atangana-Baleanu derivative [22], Caputo-Fabrizio derivative [23].…”

Criteria of limit‐point case for conformable fractional Sturm‐Liouville operators

“…We mention that this new kind derivative is of local type, since it satisfies the classical Leibnitz rule. It can be regarded as a weighted first derivative (see Ortigueira and Machado). Although under this definition and with sufficiently smooth coefficients, the corresponding operator may changed to a differential operator with power weight.…”