Fitted Tension Spline Scheme for a Singularly Perturbed Parabolic Problem With Time Delay
A fitted tension spline numerical scheme for a singularly perturbed parabolic problem (SPPP) with time delay is proposed. The presence of a small parameter ε as a multiple of the diffusion term leads to the suddenly changing behaviors of the solution in the boundary layer region. This results in a c...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2024-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2024/9458277 |
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| author | Sisay Ketema Tesfaye Gemechis File Duressa Tekle Gemechu Dinka Mesfin Mekuria Woldaregay |
| author_facet | Sisay Ketema Tesfaye Gemechis File Duressa Tekle Gemechu Dinka Mesfin Mekuria Woldaregay |
| author_sort | Sisay Ketema Tesfaye |
| collection | DOAJ |
| description | A fitted tension spline numerical scheme for a singularly perturbed parabolic problem (SPPP) with time delay is proposed. The presence of a small parameter ε as a multiple of the diffusion term leads to the suddenly changing behaviors of the solution in the boundary layer region. This results in a challenging duty to solve the problem analytically. Classical numerical methods cause spurious nonphysical oscillations unless an unacceptable number of mesh points is considered, which requires a large computational cost. To overcome this drawback, a numerical method comprising the backward Euler scheme in the time direction and the fitted spline scheme in the space direction on uniform meshes is proposed. To establish the stability and uniform convergence of the proposed method, an extensive amount of analysis is carried out. Three numerical examples are considered to validate the efficiency and applicability of the proposed scheme. It is proved that the proposed scheme is uniformly convergent of order one in both space and time. Further, the boundary layer behaviors of the solutions are given graphically. |
| format | Article |
| id | doaj-art-7de2bf05e60f4e99be3fdd212d28edea |
| institution | Kabale University |
| issn | 1687-0042 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-7de2bf05e60f4e99be3fdd212d28edea2025-02-03T11:37:23ZengWileyJournal of Applied Mathematics1687-00422024-01-01202410.1155/2024/9458277Fitted Tension Spline Scheme for a Singularly Perturbed Parabolic Problem With Time DelaySisay Ketema Tesfaye0Gemechis File Duressa1Tekle Gemechu Dinka2Mesfin Mekuria Woldaregay3Department of Applied MathematicsDepartment of MathematicsDepartment of Applied MathematicsDepartment of Applied MathematicsA fitted tension spline numerical scheme for a singularly perturbed parabolic problem (SPPP) with time delay is proposed. The presence of a small parameter ε as a multiple of the diffusion term leads to the suddenly changing behaviors of the solution in the boundary layer region. This results in a challenging duty to solve the problem analytically. Classical numerical methods cause spurious nonphysical oscillations unless an unacceptable number of mesh points is considered, which requires a large computational cost. To overcome this drawback, a numerical method comprising the backward Euler scheme in the time direction and the fitted spline scheme in the space direction on uniform meshes is proposed. To establish the stability and uniform convergence of the proposed method, an extensive amount of analysis is carried out. Three numerical examples are considered to validate the efficiency and applicability of the proposed scheme. It is proved that the proposed scheme is uniformly convergent of order one in both space and time. Further, the boundary layer behaviors of the solutions are given graphically.http://dx.doi.org/10.1155/2024/9458277 |
| spellingShingle | Sisay Ketema Tesfaye Gemechis File Duressa Tekle Gemechu Dinka Mesfin Mekuria Woldaregay Fitted Tension Spline Scheme for a Singularly Perturbed Parabolic Problem With Time Delay Journal of Applied Mathematics |
| title | Fitted Tension Spline Scheme for a Singularly Perturbed Parabolic Problem With Time Delay |
| title_full | Fitted Tension Spline Scheme for a Singularly Perturbed Parabolic Problem With Time Delay |
| title_fullStr | Fitted Tension Spline Scheme for a Singularly Perturbed Parabolic Problem With Time Delay |
| title_full_unstemmed | Fitted Tension Spline Scheme for a Singularly Perturbed Parabolic Problem With Time Delay |
| title_short | Fitted Tension Spline Scheme for a Singularly Perturbed Parabolic Problem With Time Delay |
| title_sort | fitted tension spline scheme for a singularly perturbed parabolic problem with time delay |
| url | http://dx.doi.org/10.1155/2024/9458277 |
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