Robust higher-order numerical scheme for solving time-fractional singularly perturbed parabolic partial differential equations with large delay in time
This paper presents a robust higher-order numerical scheme for time-fractional singularly perturbed partial differential equations having large delay in time, where the time-fractional derivative term is taken in the Caputo sense with order α∈(0,1). The time-fractional singularly perturbed delay par...
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Elsevier
2025-07-01
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| Series: | International Journal of Thermofluids |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666202725002642 |
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| author | Habtamu Getachew Kumie Awoke Andargie Tiruneh Getachew Adamu Derese |
| author_facet | Habtamu Getachew Kumie Awoke Andargie Tiruneh Getachew Adamu Derese |
| author_sort | Habtamu Getachew Kumie |
| collection | DOAJ |
| description | This paper presents a robust higher-order numerical scheme for time-fractional singularly perturbed partial differential equations having large delay in time, where the time-fractional derivative term is taken in the Caputo sense with order α∈(0,1). The time-fractional singularly perturbed delay partial differential equation is the class of time-fractional delay partial differential equations in which the highest-order space derivative is multiplied by a very small positive parameter. The problem is discretized by the Alikhanov’s L2−1σ technique on a uniform mesh in the temporal direction and a hybrid finite difference scheme on a piecewise uniform mesh in the spatial direction. The hybrid scheme is composed of the mid-point upwind scheme in the outer region and central finite difference method in the boundary layer region. The uniform stability analysis and the bounds of the truncation error are performed. The convergence of the numerical scheme is proved in the maximum norm. It is shown that the proposed numerical scheme is parameter-uniformly convergent of order ON−γ+M−2ln2M, where N and M are the number of time and space mesh intervals, respectively and γ=3−α. To verify the applicability of the proposed numerical scheme, three examples are considered and the results agree with the theoretical concepts discussed. Moreover, comparisons are conducted with some of the methods available in literature and the present scheme outperforms from these methods. Finally, the numerical results verify that the proposed scheme is higher-order and ϵ-uniformly convergent. |
| format | Article |
| id | doaj-art-c16ff420888449e4b035124fefcc015e |
| institution | DOAJ |
| issn | 2666-2027 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | Elsevier |
| record_format | Article |
| series | International Journal of Thermofluids |
| spelling | doaj-art-c16ff420888449e4b035124fefcc015e2025-08-20T03:17:32ZengElsevierInternational Journal of Thermofluids2666-20272025-07-012810131710.1016/j.ijft.2025.101317Robust higher-order numerical scheme for solving time-fractional singularly perturbed parabolic partial differential equations with large delay in timeHabtamu Getachew Kumie0Awoke Andargie Tiruneh1Getachew Adamu Derese2Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia; Department of Mathematics, Debre Markos University, Debre Markos, Ethiopia; Corresponding author at: Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia.Department of Mathematics, Bahir Dar University, Bahir Dar, EthiopiaDepartment of Mathematics, Bahir Dar University, Bahir Dar, EthiopiaThis paper presents a robust higher-order numerical scheme for time-fractional singularly perturbed partial differential equations having large delay in time, where the time-fractional derivative term is taken in the Caputo sense with order α∈(0,1). The time-fractional singularly perturbed delay partial differential equation is the class of time-fractional delay partial differential equations in which the highest-order space derivative is multiplied by a very small positive parameter. The problem is discretized by the Alikhanov’s L2−1σ technique on a uniform mesh in the temporal direction and a hybrid finite difference scheme on a piecewise uniform mesh in the spatial direction. The hybrid scheme is composed of the mid-point upwind scheme in the outer region and central finite difference method in the boundary layer region. The uniform stability analysis and the bounds of the truncation error are performed. The convergence of the numerical scheme is proved in the maximum norm. It is shown that the proposed numerical scheme is parameter-uniformly convergent of order ON−γ+M−2ln2M, where N and M are the number of time and space mesh intervals, respectively and γ=3−α. To verify the applicability of the proposed numerical scheme, three examples are considered and the results agree with the theoretical concepts discussed. Moreover, comparisons are conducted with some of the methods available in literature and the present scheme outperforms from these methods. Finally, the numerical results verify that the proposed scheme is higher-order and ϵ-uniformly convergent.http://www.sciencedirect.com/science/article/pii/S2666202725002642Time-fractionalSingularly perturbedBoundary layerL2 – 1σ methodHybrid schemeParameter uniform |
| spellingShingle | Habtamu Getachew Kumie Awoke Andargie Tiruneh Getachew Adamu Derese Robust higher-order numerical scheme for solving time-fractional singularly perturbed parabolic partial differential equations with large delay in time International Journal of Thermofluids Time-fractional Singularly perturbed Boundary layer L2 – 1σ method Hybrid scheme Parameter uniform |
| title | Robust higher-order numerical scheme for solving time-fractional singularly perturbed parabolic partial differential equations with large delay in time |
| title_full | Robust higher-order numerical scheme for solving time-fractional singularly perturbed parabolic partial differential equations with large delay in time |
| title_fullStr | Robust higher-order numerical scheme for solving time-fractional singularly perturbed parabolic partial differential equations with large delay in time |
| title_full_unstemmed | Robust higher-order numerical scheme for solving time-fractional singularly perturbed parabolic partial differential equations with large delay in time |
| title_short | Robust higher-order numerical scheme for solving time-fractional singularly perturbed parabolic partial differential equations with large delay in time |
| title_sort | robust higher order numerical scheme for solving time fractional singularly perturbed parabolic partial differential equations with large delay in time |
| topic | Time-fractional Singularly perturbed Boundary layer L2 – 1σ method Hybrid scheme Parameter uniform |
| url | http://www.sciencedirect.com/science/article/pii/S2666202725002642 |
| work_keys_str_mv | AT habtamugetachewkumie robusthigherordernumericalschemeforsolvingtimefractionalsingularlyperturbedparabolicpartialdifferentialequationswithlargedelayintime AT awokeandargietiruneh robusthigherordernumericalschemeforsolvingtimefractionalsingularlyperturbedparabolicpartialdifferentialequationswithlargedelayintime AT getachewadamuderese robusthigherordernumericalschemeforsolvingtimefractionalsingularlyperturbedparabolicpartialdifferentialequationswithlargedelayintime |