A critical analysis of the conformable derivative
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866913228111478784 |
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| author | Abdelhakim, Ahmed A. Machado, José A. Tenreiro |
| author_facet | Abdelhakim, Ahmed A. Machado, José A. Tenreiro |
| contents | 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 $α$-derivative of $f$ at some point $x>0$, where $0<α<1$, is the pointwise product $x^{1-α}f^{\prime}(x)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1805_08133 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | A critical analysis of the conformable derivative Abdelhakim, Ahmed A. Machado, José A. Tenreiro Classical Analysis and ODEs 26A33, 34A08, 74D05 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 $α$-derivative of $f$ at some point $x>0$, where $0<α<1$, is the pointwise product $x^{1-α}f^{\prime}(x)$. 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. |
| title | A critical analysis of the conformable derivative |
| topic | Classical Analysis and ODEs 26A33, 34A08, 74D05 |
| url | https://arxiv.org/abs/1805.08133 |