A novel exponentially fitted finite-difference method for time-fractional singularly perturbed convection–diffusion problems with variable coefficients
This study presents an exponentially fitted finite-difference scheme for addressing singularly perturbed convection–diffusion problems involving the time-fractional derivative. The Caputo fractional derivative defines the time-fractional derivative. Then, the implicit finite-difference method is use...
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| Format: | Article |
| Language: | English |
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Frontiers Media S.A.
2025-03-01
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| Series: | Frontiers in Applied Mathematics and Statistics |
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| Online Access: | https://www.frontiersin.org/articles/10.3389/fams.2025.1541766/full |
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| author | Worku Tilahun Aniley Gemechis File Duressa |
| author_facet | Worku Tilahun Aniley Gemechis File Duressa |
| author_sort | Worku Tilahun Aniley |
| collection | DOAJ |
| description | This study presents an exponentially fitted finite-difference scheme for addressing singularly perturbed convection–diffusion problems involving the time-fractional derivative. The Caputo fractional derivative defines the time-fractional derivative. Then, the implicit finite-difference method is used to discretize the temporal variable in a uniform mesh discretization. To manage the effect of the perturbation parameter on the solution profile, an exponentially fitted factor is introduced into the resulting system of ordinary differential equations. Finally, on a uniform spatial domain discretization, an exponentially fitted scheme is developed using the Numerov finite-difference approach. The ε-uniform of the proposed scheme is rigorously demonstrated, confirming that it is uniformly convergent with a convergence order of O((Δt)2−α+M−1). The validity of the proposed method is illustrated through model examples. The numerical results match the theoretical predictions and demonstrate that the proposed method is more accurate than some recent existing methods. |
| format | Article |
| id | doaj-art-a0e9f4dd12734f16a2479e076eef4d72 |
| institution | OA Journals |
| issn | 2297-4687 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | Frontiers Media S.A. |
| record_format | Article |
| series | Frontiers in Applied Mathematics and Statistics |
| spelling | doaj-art-a0e9f4dd12734f16a2479e076eef4d722025-08-20T02:17:53ZengFrontiers Media S.A.Frontiers in Applied Mathematics and Statistics2297-46872025-03-011110.3389/fams.2025.15417661541766A novel exponentially fitted finite-difference method for time-fractional singularly perturbed convection–diffusion problems with variable coefficientsWorku Tilahun AnileyGemechis File DuressaThis study presents an exponentially fitted finite-difference scheme for addressing singularly perturbed convection–diffusion problems involving the time-fractional derivative. The Caputo fractional derivative defines the time-fractional derivative. Then, the implicit finite-difference method is used to discretize the temporal variable in a uniform mesh discretization. To manage the effect of the perturbation parameter on the solution profile, an exponentially fitted factor is introduced into the resulting system of ordinary differential equations. Finally, on a uniform spatial domain discretization, an exponentially fitted scheme is developed using the Numerov finite-difference approach. The ε-uniform of the proposed scheme is rigorously demonstrated, confirming that it is uniformly convergent with a convergence order of O((Δt)2−α+M−1). The validity of the proposed method is illustrated through model examples. The numerical results match the theoretical predictions and demonstrate that the proposed method is more accurate than some recent existing methods.https://www.frontiersin.org/articles/10.3389/fams.2025.1541766/fullNumerov methodtime-fractionalconvection–diffusionCaputo fractional derivativefinite-differenceexponentially fitting factor |
| spellingShingle | Worku Tilahun Aniley Gemechis File Duressa A novel exponentially fitted finite-difference method for time-fractional singularly perturbed convection–diffusion problems with variable coefficients Frontiers in Applied Mathematics and Statistics Numerov method time-fractional convection–diffusion Caputo fractional derivative finite-difference exponentially fitting factor |
| title | A novel exponentially fitted finite-difference method for time-fractional singularly perturbed convection–diffusion problems with variable coefficients |
| title_full | A novel exponentially fitted finite-difference method for time-fractional singularly perturbed convection–diffusion problems with variable coefficients |
| title_fullStr | A novel exponentially fitted finite-difference method for time-fractional singularly perturbed convection–diffusion problems with variable coefficients |
| title_full_unstemmed | A novel exponentially fitted finite-difference method for time-fractional singularly perturbed convection–diffusion problems with variable coefficients |
| title_short | A novel exponentially fitted finite-difference method for time-fractional singularly perturbed convection–diffusion problems with variable coefficients |
| title_sort | novel exponentially fitted finite difference method for time fractional singularly perturbed convection diffusion problems with variable coefficients |
| topic | Numerov method time-fractional convection–diffusion Caputo fractional derivative finite-difference exponentially fitting factor |
| url | https://www.frontiersin.org/articles/10.3389/fams.2025.1541766/full |
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