Time Reparametrization, Not Fractional Calculus: A Reassessment of the Conformable Derivative

TL;DR

This paper critically reassesses the conformable derivative, showing it is a time reparametrization tool rather than a fractional operator, and emphasizes the importance of classical derivatives for modeling memory effects.

Contribution

The paper clarifies that the conformable derivative is equivalent to a classical derivative under a nonlinear time change, challenging its classification as a fractional operator.

Findings

Conformable derivative can be transformed into classical derivatives via time reparametrization.

Numerical simulations show no genuine nonlocal fractional effects with conformable derivatives.

Classical and true fractional derivatives better model memory-dependent phenomena.

Abstract

The conformable derivative has been promoted in numerous publications as a new fractional derivative operator. This article provides a critical reassessment of this claim. We demonstrate that the conformable derivative is not a fractional operator but a useful computational tool for systems with power-law time scaling, equivalent to classical differentiation under a nonlinear time reparametrization. Several results presented in the literature as novel fractional contributions can be reinterpreted within a classical framework. We show that problems formulated using the conformable derivative can be transformed into classical formulations via a change of variable. The solution is derived classically and then transformed back, this reformulation highlights the absence of genuinely nonlocal fractional effects. We provide a theoretical analysis, numerical simulations comparing conformable, classical, and truly fractional (Caputo) models, and discuss the reasons why this misconception persists. Our results suggest that classical derivatives, as well as established fractional derivatives, offer a more faithful framework for modeling memory-dependent phenomena.