Comment on: "Stokes' first problem for heated flat plate with Atangana--Baleanu fractional derivative" [Chaos Solitons Fractals 117 (2018) 68]

TL;DR

This paper critiques a previous study on Stokes' first problem with Atangana--Baleanu fractional derivatives, highlighting a fundamental mathematical error in handling the Dirac delta function and discussing related issues in fractional rheological models.

Contribution

It identifies and explains a key mathematical mistake in a prior work, providing corrections and clarifications on the proper treatment of the Dirac delta in fractional differential equations.

Findings

The previous paper incorrectly treats the Dirac delta as zero.

Correct handling of the Dirac delta affects the solution accuracy.

Errors in fractional rheological modeling are also discussed.

Abstract

In the sense of distributions, the derivative of the Heaviside unit step function $H(t)$ is a generalized Dirac-$\delta$ distribution. If the velocity $V(t)$ of a flat plate is impulsive, as $V(t)=H(t)$ (i.e., it is suddenly set into motion with unit velocity at $t=0^+$), then its acceleration is $V'(t)=\delta(t)$. The Dirac-$\delta$ distribution has no point values. However, when the Dirac-$\delta$ is the forcing term of an ODE (in $t$), it contributes to the solution. The recently published paper [Chaos Solitons Fractals 117 (2018) 68] incorrectly treats the Dirac-$\delta$ function as being identically 0. This Comment analyzes the source of this error, and provides guidance on how to correct it (based on the established literature). The mathematical error identified is in addition to some issues about rheological models with fractional derivatives, which are also noted. That is to say, whether or not an "Atangana--Baleanu fractional derivative" is used in [Chaos Solitons Fractals 117 (2018) 68], the solution to Stokes' first problem provided therein is not correct.