Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation
This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε, taking arbitrary value in the interval 0,1. For small values of ε, solution of th...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2021/6641236 |
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| author | Mesfin Mekuria Woldaregay Worku Tilahun Aniley Gemechis File Duressa |
| author_facet | Mesfin Mekuria Woldaregay Worku Tilahun Aniley Gemechis File Duressa |
| author_sort | Mesfin Mekuria Woldaregay |
| collection | DOAJ |
| description | This paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε, taking arbitrary value in the interval 0,1. For small values of ε, solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of ε. |
| format | Article |
| id | doaj-art-c9f9a86d2ab04b7384bd7011f04bc312 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-c9f9a86d2ab04b7384bd7011f04bc3122025-02-03T05:58:21ZengWileyAdvances in Mathematical Physics1687-91201687-91392021-01-01202110.1155/2021/66412366641236Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion EquationMesfin Mekuria Woldaregay0Worku Tilahun Aniley1Gemechis File Duressa2Department of Applied Mathematics, Adama Science and Technology University, Adama, EthiopiaDepartment of Mathematics, Jimma University, Jimma, EthiopiaDepartment of Mathematics, Jimma University, Jimma, EthiopiaThis paper deals with numerical treatment of singularly perturbed parabolic differential equations having large time delay. The highest order derivative term in the equation is multiplied by a perturbation parameter ε, taking arbitrary value in the interval 0,1. For small values of ε, solution of the problem exhibits an exponential boundary layer on the right side of the spatial domain. The properties and bounds of the solution and its derivatives are discussed. The considered singularly perturbed time delay problem is solved using the Crank-Nicolson method in temporal discretization and exponentially fitted operator finite difference method in spatial discretization. The stability of the scheme is investigated and analysed using comparison principle and solution bound. The uniform convergence of the scheme is discussed and proven. The formulated scheme converges uniformly with linear order of convergence. The theoretical analysis of the scheme is validated by considering numerical test examples for different values of ε.http://dx.doi.org/10.1155/2021/6641236 |
| spellingShingle | Mesfin Mekuria Woldaregay Worku Tilahun Aniley Gemechis File Duressa Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation Advances in Mathematical Physics |
| title | Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation |
| title_full | Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation |
| title_fullStr | Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation |
| title_full_unstemmed | Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation |
| title_short | Novel Numerical Scheme for Singularly Perturbed Time Delay Convection-Diffusion Equation |
| title_sort | novel numerical scheme for singularly perturbed time delay convection diffusion equation |
| url | http://dx.doi.org/10.1155/2021/6641236 |
| work_keys_str_mv | AT mesfinmekuriawoldaregay novelnumericalschemeforsingularlyperturbedtimedelayconvectiondiffusionequation AT workutilahunaniley novelnumericalschemeforsingularlyperturbedtimedelayconvectiondiffusionequation AT gemechisfileduressa novelnumericalschemeforsingularlyperturbedtimedelayconvectiondiffusionequation |