A comment on some new definitions of fractional derivative

TL;DR

This paper critically examines recent fractional derivative definitions by Caputo-Fabrizio and Atangana-Baleanu, situating them within the broader Prabhakar fractional integral framework and analyzing their implications in differential equations and viscoelasticity.

Contribution

It demonstrates that these new operators are special cases of Prabhakar integrals and clarifies their relation to classical fractional derivatives, challenging their novelty.

Findings

Caputo-Fabrizio operator is not truly fractional.

Atangana-Baleanu operator is fractional but related to classical derivatives.

These operators do not provide new insights into viscoelasticity theory.

Abstract

After reviewing the definition of two differential operators which have been recently introduced by Caputo and Fabrizio and, separately, by Atangana and Baleanu, we present an argument for which these two integro-differential operators can be understood as simple realizations of a much broader class of fractional operators, i.e. the theory of Prabhakar fractional integrals. Furthermore, we also provide a series expansion of the Prabhakar integral in terms of Riemann-Liouville integrals of variable order. Then, by using this last result we finally argue that the operator introduced by Caputo and Fabrizio cannot be regarded as fractional. Besides, we also observe that the one suggested by Atangana and Baleanu is indeed fractional, but it is ultimately related to the ordinary Riemann-Liouville and Caputo fractional operators. All these statements are then further supported by a precise analysis of differential equations involving the aforementioned operators. To further strengthen our narrative, we also show that these new operators do not add any new insight to the linear theory of viscoelasticity when employed in the constitutive equation of the Scott-Blair model.