A critical analysis of the conformable derivative

TL;DR

This paper demonstrates that conformable fractional derivatives are equivalent to classical derivatives multiplied by a power function, questioning their significance in modeling fractional phenomena.

Contribution

It proves that conformable derivatives do not extend classical differentiability and shows their limitations in accurately modeling fractional physical systems.

Findings

Conformable derivatives are equivalent to classical derivatives times a power function.

Using conformable derivatives reduces fractional problems to integer-order differential equations.

The study highlights the inadequacy of conformable derivatives in representing true fractional phenomena.

Abstract

We prove that conformable ``fractional" differentiability of a function $f:[0,\infty[\,\longrightarrow \mathbb{R}$ is nothing else than the classical differentiability. More precisely, the conformable $\alpha$-derivative of $f$ at some point $x>0$, where $0<\alpha<1$, is the pointwise product $x^{1-\alpha}f^{\prime}(x)$. This proves the lack of significance of recent studies of the conformable derivatives. The results imply that interpreting fractional derivatives in the conformable sense alters fractional differential problems into differential problems with the usual integer-order derivatives that no longer describe the original fractional physical phenomena. A general fractional viscoelasticity model is analysed to illustrate this state of affairs. We also test the modelling efficiency of the conformable derivative using a fractional model of viscoelastic deformation of tight sandstone, and a fractional world population growth model.