Why fractional derivatives with nonsingular kernels should not be used

TL;DR

This paper argues that fractional derivatives with nonsingular kernels, such as Caputo-Fabrizio and Atangana-Baleanu, are fundamentally flawed because they violate core principles of fractional calculus and do not provide new insights.

Contribution

The paper provides a rigorous mathematical critique showing these derivatives fail fundamental properties and are equivalent to existing derivatives, questioning their validity and usefulness.

Findings

Fail to satisfy the fundamental theorem of fractional calculus

Derivative at t=0 is always zero, imposing unnatural restrictions

These derivatives can be expressed in terms of integer and standard fractional derivatives

Abstract

In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time $t=0$ is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.