Parameter uniform finite difference formulation with oscillation free for solving singularly perturbed delay parabolic differential equation via exponential spline
Abstract Objective In this work, singularly perturbed time dependent delay parabolic convection-diffusion problem with Dirichlet boundary conditions is considered. The solution of this problem exhibits boundary layer at the right of special domain. In this layer the solution experiences steep gradie...
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2025-01-01
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| Online Access: | https://doi.org/10.1186/s13104-024-07005-1 |
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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 | Abstract Objective In this work, singularly perturbed time dependent delay parabolic convection-diffusion problem with Dirichlet boundary conditions is considered. The solution of this problem exhibits boundary layer at the right of special domain. In this layer the solution experiences steep gradients or oscillation so that traditional numerical methods may fail to provide smooth solutions. We developed oscillation free parameter uniform exponentially spline numerical method to solve the considered problem. Results In the temporal direction, the implicit Euler method is applied, and in the spatial direction, an exponential spline method with uniform mesh is applied. To handle the effect of perturbation parameter, an exponential fitting factor is introduced. For the developed numerical scheme, stability and uniform error estimates are examined. It is shown that the scheme is uniformly convergent of linear order in the maximum norm. Numerical examples are provided to illustrate the theoretical findings. |
| format | Article |
| id | doaj-art-d625adb9e611490f933b6d81aa954705 |
| institution | Kabale University |
| issn | 1756-0500 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | BMC |
| record_format | Article |
| series | BMC Research Notes |
| spelling | doaj-art-d625adb9e611490f933b6d81aa9547052025-01-26T12:13:17ZengBMCBMC Research Notes1756-05002025-01-0118111410.1186/s13104-024-07005-1Parameter uniform finite difference formulation with oscillation free for solving singularly perturbed delay parabolic differential equation via exponential splineZerihun Ibrahim Hassen0Gemechis File Duressa1Department of Mathematics, Arba Minch UniversityDepartment of Mathematics, Jimma UniversityAbstract Objective In this work, singularly perturbed time dependent delay parabolic convection-diffusion problem with Dirichlet boundary conditions is considered. The solution of this problem exhibits boundary layer at the right of special domain. In this layer the solution experiences steep gradients or oscillation so that traditional numerical methods may fail to provide smooth solutions. We developed oscillation free parameter uniform exponentially spline numerical method to solve the considered problem. Results In the temporal direction, the implicit Euler method is applied, and in the spatial direction, an exponential spline method with uniform mesh is applied. To handle the effect of perturbation parameter, an exponential fitting factor is introduced. For the developed numerical scheme, stability and uniform error estimates are examined. It is shown that the scheme is uniformly convergent of linear order in the maximum norm. Numerical examples are provided to illustrate the theoretical findings.https://doi.org/10.1186/s13104-024-07005-1Exponential splineOscillation-freeSingularly perturbed delay problemFitting factorConvection-diffusionUniform convergence |
| spellingShingle | Zerihun Ibrahim Hassen Gemechis File Duressa Parameter uniform finite difference formulation with oscillation free for solving singularly perturbed delay parabolic differential equation via exponential spline BMC Research Notes Exponential spline Oscillation-free Singularly perturbed delay problem Fitting factor Convection-diffusion Uniform convergence |
| title | Parameter uniform finite difference formulation with oscillation free for solving singularly perturbed delay parabolic differential equation via exponential spline |
| title_full | Parameter uniform finite difference formulation with oscillation free for solving singularly perturbed delay parabolic differential equation via exponential spline |
| title_fullStr | Parameter uniform finite difference formulation with oscillation free for solving singularly perturbed delay parabolic differential equation via exponential spline |
| title_full_unstemmed | Parameter uniform finite difference formulation with oscillation free for solving singularly perturbed delay parabolic differential equation via exponential spline |
| title_short | Parameter uniform finite difference formulation with oscillation free for solving singularly perturbed delay parabolic differential equation via exponential spline |
| title_sort | parameter uniform finite difference formulation with oscillation free for solving singularly perturbed delay parabolic differential equation via exponential spline |
| topic | Exponential spline Oscillation-free Singularly perturbed delay problem Fitting factor Convection-diffusion Uniform convergence |
| url | https://doi.org/10.1186/s13104-024-07005-1 |
| work_keys_str_mv | AT zerihunibrahimhassen parameteruniformfinitedifferenceformulationwithoscillationfreeforsolvingsingularlyperturbeddelayparabolicdifferentialequationviaexponentialspline AT gemechisfileduressa parameteruniformfinitedifferenceformulationwithoscillationfreeforsolvingsingularlyperturbeddelayparabolicdifferentialequationviaexponentialspline |