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

Aziz El Ghazouani, Fouad Ibrahim Abdou Amir, Khoulane Mohamed, M'hamed Elomari

TL;DR

This work analyzes the conformable derivative, arguing it is not a true fractional operator but a nonlinear time reparametrization of classical differentiation. By establishing that $D^{\\alpha}_{\\psi} f(t)=\\psi_{\\alpha}(t) f'(t)$ and introducing the time transform $\\tau=\\phi_{\\alpha}(t)$, the authors show that conformable models are mathematically equivalent to classical ones, yielding memory-free dynamics with altered time scales. Through ODE, PDE, evolution, and dynamical-systems examples, they demonstrate that conformable results can be recovered by a coordinate change, while Caputo/Riemann-Liouville derivatives produce genuine nonlocal memory effects. The findings urge researchers to treat conformable calculus as a computational reparametrization tool, not as a source of memory modeling, and to rely on established fractional calculus for memory-dependent phenomena. This has practical implications for modeling, analysis, and terminology in fractional-dynamics research.

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.

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

TL;DR

This work analyzes the conformable derivative, arguing it is not a true fractional operator but a nonlinear time reparametrization of classical differentiation. By establishing that and introducing the time transform , the authors show that conformable models are mathematically equivalent to classical ones, yielding memory-free dynamics with altered time scales. Through ODE, PDE, evolution, and dynamical-systems examples, they demonstrate that conformable results can be recovered by a coordinate change, while Caputo/Riemann-Liouville derivatives produce genuine nonlocal memory effects. The findings urge researchers to treat conformable calculus as a computational reparametrization tool, not as a source of memory modeling, and to rely on established fractional calculus for memory-dependent phenomena. This has practical implications for modeling, analysis, and terminology in fractional-dynamics research.

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.
Paper Structure (11 sections, 9 theorems, 67 equations, 6 figures)

This paper contains 11 sections, 9 theorems, 67 equations, 6 figures.

Table of Contents

  1. Introduction
  2. On the Interpretation of the Conformable Derivative
  3. Applications and Reformulations
  4. Linear Ordinary Differential Equations
  5. Partial Differential Equations
  6. Evolution Problems
  7. Dynamical Systems
  8. Effectiveness in Modeling Real-World Problems
  9. The misconception of Physical Relevance of Conformable Derivatives
  10. The Lorenz Attractor and the Conformable Derivative
  11. The Fractional Lorenz Attractor and the Caputo Derivative

Key Result

Theorem 1

Let $a \in [0, \infty)$. Then $f$ is $\psi_{\alpha}$-differentiable at $a$ if and only if the function $g(\tau) = f(\phi_{\alpha}^{-1}(\tau))$ is differentiable at $\tau_a = \phi_{\alpha}(a)$. Moreover, the derivatives are related by: where $D^{\alpha}_{\psi}$ denotes the $\psi_\alpha$ derivative operator.

Figures (6)

  • Figure 1: Phase space trajectories of the Lorenz system. (a) The classical Lorenz attractor. (b) The attractor under the conformable derivative formulation with $\alpha = 0.9$, exhibiting a temporally distorted but topologically equivalent structure.
  • Figure 2: Time series of the Lorenz system variables $x(t)$, $y(t)$, and $z(t)$. The conformable system (red) replicates the dynamics of the classical system (blue) on a rescaled time axis, confirming the equivalence under time reparametrization.
  • Figure 3: Dynamic characteristics of the conformable Lorenz system. (a) The norm of the system velocity $\|x^{\prime}(t)\|$, showing initial acceleration. (b) The nonlinear time transformation $\tau = t^\alpha/\alpha$ for $\alpha = 0.9$, which is the source of the observed kinematic distortion.
  • Figure 4: Two-dimensional phase space projections $(x,y)$, $(x,z)$, and $(y,z)$ of the Lorenz attractor. The geometric structure is preserved under the conformable derivative, demonstrating that the transformation affects only the temporal, not the spatial, evolution.
  • Figure 5: Comparative analysis of Lorenz attractors for $\alpha = 0.9$. (a) Classical derivative (reference). (b) Conformable derivative: a kinematically distorted version of (a). (c) Caputo fractional derivative: a structurally altered attractor exhibiting genuine memory effects and fractal roughness.
  • ...and 1 more figures

Theorems & Definitions (24)

  • Definition 1
  • Example 1
  • Theorem 1
  • Proof 1
  • Corollary 1
  • Remark 1
  • Example 2
  • Theorem 2
  • Proof 2
  • Proposition 1
  • ...and 14 more