Time Reparametrization, Not Fractional Calculus: A Reassessment of the Conformable Derivative
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915696479305728 |
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| author | Ghazouani, Aziz El Amir, Fouad Ibrahim Abdou Mohamed, Khoulane Elomari, M'hamed |
| author_facet | Ghazouani, Aziz El Amir, Fouad Ibrahim Abdou Mohamed, Khoulane Elomari, M'hamed |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_22366 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Time Reparametrization, Not Fractional Calculus: A Reassessment of the Conformable Derivative Ghazouani, Aziz El Amir, Fouad Ibrahim Abdou Mohamed, Khoulane Elomari, M'hamed Analysis of PDEs Numerical Analysis Mathematical Physics Dynamical Systems 34A08, 35R11, 26A33, 37D45, 65L05 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. |
| title | Time Reparametrization, Not Fractional Calculus: A Reassessment of the Conformable Derivative |
| topic | Analysis of PDEs Numerical Analysis Mathematical Physics Dynamical Systems 34A08, 35R11, 26A33, 37D45, 65L05 |
| url | https://arxiv.org/abs/2512.22366 |