A robust numerical scheme for singularly perturbed differential equations with spatio-temporal delays
In this article, we proposed and analyzed a numerical scheme for singularly perturbed differential equations with both spatial and temporal delays. The presence of the perturbation parameter exhibits strong boundary layers, and the large negative shift gives rise to a strong interior layer in the so...
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| Format: | Article |
| Language: | English |
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Frontiers Media S.A.
2023-03-01
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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.2023.1125347/full |
| _version_ | 1797861051776106496 |
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| author | Ababi Hailu Ejere Gemechis File Duressa Mesfin Mekuria Woldaregay Tekle Gemechu Dinka |
| author_facet | Ababi Hailu Ejere Gemechis File Duressa Mesfin Mekuria Woldaregay Tekle Gemechu Dinka |
| author_sort | Ababi Hailu Ejere |
| collection | DOAJ |
| description | In this article, we proposed and analyzed a numerical scheme for singularly perturbed differential equations with both spatial and temporal delays. The presence of the perturbation parameter exhibits strong boundary layers, and the large negative shift gives rise to a strong interior layer in the solution. The abruptly changing behaviors of the solution in the layers make it difficult to solve the problem analytically. Standard numerical methods do not give satisfactory results, unless a large mesh number is considered, which needs a massive computational cost. We treated such problem by proposing a numerical scheme using the implicit Euler method in the temporal variable and the nonstandard finite difference method in the spatial variable on uniform meshes. The stability and uniform convergence of the proposed scheme have been investigated and proved. To demonstrate the theoretical results, numerical experiments are carried out. From the theoretical and numerical results, we observed that the method is uniformly convergent of order one in time and of order two in space. |
| first_indexed | 2024-04-09T21:56:55Z |
| format | Article |
| id | doaj.art-ebcd1d1c4e9e4ab28a6a6408cb371dd3 |
| institution | Directory Open Access Journal |
| issn | 2297-4687 |
| language | English |
| last_indexed | 2024-04-09T21:56:55Z |
| publishDate | 2023-03-01 |
| publisher | Frontiers Media S.A. |
| record_format | Article |
| series | Frontiers in Applied Mathematics and Statistics |
| spelling | doaj.art-ebcd1d1c4e9e4ab28a6a6408cb371dd32023-03-24T05:30:53ZengFrontiers Media S.A.Frontiers in Applied Mathematics and Statistics2297-46872023-03-01910.3389/fams.2023.11253471125347A robust numerical scheme for singularly perturbed differential equations with spatio-temporal delaysAbabi Hailu Ejere0Gemechis File Duressa1Mesfin Mekuria Woldaregay2Tekle Gemechu Dinka3Department of Applied Mathematics, Adama Science and Technology University, Adama, EthiopiaDepartment of Mathematics, Jimma University, Jimma, EthiopiaDepartment of Applied Mathematics, Adama Science and Technology University, Adama, EthiopiaDepartment of Applied Mathematics, Adama Science and Technology University, Adama, EthiopiaIn this article, we proposed and analyzed a numerical scheme for singularly perturbed differential equations with both spatial and temporal delays. The presence of the perturbation parameter exhibits strong boundary layers, and the large negative shift gives rise to a strong interior layer in the solution. The abruptly changing behaviors of the solution in the layers make it difficult to solve the problem analytically. Standard numerical methods do not give satisfactory results, unless a large mesh number is considered, which needs a massive computational cost. We treated such problem by proposing a numerical scheme using the implicit Euler method in the temporal variable and the nonstandard finite difference method in the spatial variable on uniform meshes. The stability and uniform convergence of the proposed scheme have been investigated and proved. To demonstrate the theoretical results, numerical experiments are carried out. From the theoretical and numerical results, we observed that the method is uniformly convergent of order one in time and of order two in space.https://www.frontiersin.org/articles/10.3389/fams.2023.1125347/fullsingularly perturbed problemspatio-temporal delaysnonstandard finite differenceimplicit Euler methoduniform convergence |
| spellingShingle | Ababi Hailu Ejere Gemechis File Duressa Mesfin Mekuria Woldaregay Tekle Gemechu Dinka A robust numerical scheme for singularly perturbed differential equations with spatio-temporal delays Frontiers in Applied Mathematics and Statistics singularly perturbed problem spatio-temporal delays nonstandard finite difference implicit Euler method uniform convergence |
| title | A robust numerical scheme for singularly perturbed differential equations with spatio-temporal delays |
| title_full | A robust numerical scheme for singularly perturbed differential equations with spatio-temporal delays |
| title_fullStr | A robust numerical scheme for singularly perturbed differential equations with spatio-temporal delays |
| title_full_unstemmed | A robust numerical scheme for singularly perturbed differential equations with spatio-temporal delays |
| title_short | A robust numerical scheme for singularly perturbed differential equations with spatio-temporal delays |
| title_sort | robust numerical scheme for singularly perturbed differential equations with spatio temporal delays |
| topic | singularly perturbed problem spatio-temporal delays nonstandard finite difference implicit Euler method uniform convergence |
| url | https://www.frontiersin.org/articles/10.3389/fams.2023.1125347/full |
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