Parameter uniform hybrid numerical method for time-dependent singularly perturbed parabolic differential equations with large delay
ABSTRACTIn this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the probl...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Taylor & Francis Group
2024-12-01
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| Series: | Applied Mathematics in Science and Engineering |
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| Online Access: | https://www.tandfonline.com/doi/10.1080/27690911.2024.2328254 |
| _version_ | 1797262796522520576 |
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| author | Zerihun Ibrahim Hassen Gemechis File Duressa |
| author_facet | Zerihun Ibrahim Hassen Gemechis File Duressa |
| author_sort | Zerihun Ibrahim Hassen |
| collection | DOAJ |
| description | ABSTRACTIn this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme's application is demonstrated through two test problems. |
| first_indexed | 2024-04-25T00:02:48Z |
| format | Article |
| id | doaj.art-83763f97bea245e19d0fca2fa72ffab7 |
| institution | Directory Open Access Journal |
| issn | 2769-0911 |
| language | English |
| last_indexed | 2024-04-25T00:02:48Z |
| publishDate | 2024-12-01 |
| publisher | Taylor & Francis Group |
| record_format | Article |
| series | Applied Mathematics in Science and Engineering |
| spelling | doaj.art-83763f97bea245e19d0fca2fa72ffab72024-03-14T06:34:24ZengTaylor & Francis GroupApplied Mathematics in Science and Engineering2769-09112024-12-0132110.1080/27690911.2024.2328254Parameter uniform hybrid numerical method for time-dependent singularly perturbed parabolic differential equations with large delayZerihun Ibrahim Hassen0Gemechis File Duressa1Department of Mathematics, Arba Minch University, Arba Minch, EthiopiaDepartment of Mathematics, Jimma University, Jimma, EthiopiaABSTRACTIn this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme's application is demonstrated through two test problems.https://www.tandfonline.com/doi/10.1080/27690911.2024.2328254Non-polynomial splineBakhvalov–Shishkin meshhybrid numerical schemeparametric convergentRichardson extrapolation65N12 |
| spellingShingle | Zerihun Ibrahim Hassen Gemechis File Duressa Parameter uniform hybrid numerical method for time-dependent singularly perturbed parabolic differential equations with large delay Applied Mathematics in Science and Engineering Non-polynomial spline Bakhvalov–Shishkin mesh hybrid numerical scheme parametric convergent Richardson extrapolation 65N12 |
| title | Parameter uniform hybrid numerical method for time-dependent singularly perturbed parabolic differential equations with large delay |
| title_full | Parameter uniform hybrid numerical method for time-dependent singularly perturbed parabolic differential equations with large delay |
| title_fullStr | Parameter uniform hybrid numerical method for time-dependent singularly perturbed parabolic differential equations with large delay |
| title_full_unstemmed | Parameter uniform hybrid numerical method for time-dependent singularly perturbed parabolic differential equations with large delay |
| title_short | Parameter uniform hybrid numerical method for time-dependent singularly perturbed parabolic differential equations with large delay |
| title_sort | parameter uniform hybrid numerical method for time dependent singularly perturbed parabolic differential equations with large delay |
| topic | Non-polynomial spline Bakhvalov–Shishkin mesh hybrid numerical scheme parametric convergent Richardson extrapolation 65N12 |
| url | https://www.tandfonline.com/doi/10.1080/27690911.2024.2328254 |
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