A critical analysis of the conformable derivative

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 may no longer properly 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 on various fractional models. We find that, compared with the classical fractional derivative, the conformable framework results in a substantially larger error.