Advanced Fitted Mesh Finite Difference Strategies for Solving ‘n’ Two-Parameter Singularly Perturbed Convection–Diffusion System
This paper proposes a robust finite difference method on a fitted Shishkin mesh to solve a system of <i>n</i> singularly perturbed convection–reaction–diffusion differential equations with two small parameters. Defined on the interval <inline-formula><math xmlns="http://www...
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MDPI AG
2025-02-01
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| Online Access: | https://www.mdpi.com/2075-1680/14/3/171 |
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| author | Jenolin Arthur Joseph Paramasivam Mathiyazhagan George E. Chatzarakis S. L. Panetsos |
| author_facet | Jenolin Arthur Joseph Paramasivam Mathiyazhagan George E. Chatzarakis S. L. Panetsos |
| author_sort | Jenolin Arthur |
| collection | DOAJ |
| description | This paper proposes a robust finite difference method on a fitted Shishkin mesh to solve a system of <i>n</i> singularly perturbed convection–reaction–diffusion differential equations with two small parameters. Defined on the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>, this system exhibits boundary layers due to the presence of small parameters, making accurate numerical approximations challenging. The method employs a piecewise uniform Shishkin mesh that adapts to layer regions and efficiently captures the solution’s behavior. The scheme is proven to be uniformly convergent with respect to the perturbation parameters, achieving nearly first-order accuracy. Comprehensive numerical experiments validate the theoretical results, illustrating the method’s robustness and efficiency in handling parameter-sensitive boundary layers. |
| format | Article |
| id | doaj-art-b886a74be65946ff9aab35ea4efcb2d0 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-b886a74be65946ff9aab35ea4efcb2d02025-08-20T03:43:30ZengMDPI AGAxioms2075-16802025-02-0114317110.3390/axioms14030171Advanced Fitted Mesh Finite Difference Strategies for Solving ‘n’ Two-Parameter Singularly Perturbed Convection–Diffusion SystemJenolin Arthur0Joseph Paramasivam Mathiyazhagan1George E. Chatzarakis2S. L. Panetsos3PG & Research Department of Mathematics, Bishop Heber College (Affiliated to Bharathidasan University), Trichy 620 017, Tamil Nadu, IndiaPG & Research Department of Mathematics, Bishop Heber College (Affiliated to Bharathidasan University), Trichy 620 017, Tamil Nadu, IndiaDepartment of Electrical and Electronic Engineering Educators, School of Pedagogical & Technological Education (ASPETE), 15122 Marousi, GreeceDepartment of Electrical and Electronic Engineering Educators, School of Pedagogical & Technological Education (ASPETE), 15122 Marousi, GreeceThis paper proposes a robust finite difference method on a fitted Shishkin mesh to solve a system of <i>n</i> singularly perturbed convection–reaction–diffusion differential equations with two small parameters. Defined on the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>, this system exhibits boundary layers due to the presence of small parameters, making accurate numerical approximations challenging. The method employs a piecewise uniform Shishkin mesh that adapts to layer regions and efficiently captures the solution’s behavior. The scheme is proven to be uniformly convergent with respect to the perturbation parameters, achieving nearly first-order accuracy. Comprehensive numerical experiments validate the theoretical results, illustrating the method’s robustness and efficiency in handling parameter-sensitive boundary layers.https://www.mdpi.com/2075-1680/14/3/171singularly perturbed differential equationsnumerical methodsconvection–diffusion equationsShishkin meshesboundary layeruniform convergence |
| spellingShingle | Jenolin Arthur Joseph Paramasivam Mathiyazhagan George E. Chatzarakis S. L. Panetsos Advanced Fitted Mesh Finite Difference Strategies for Solving ‘n’ Two-Parameter Singularly Perturbed Convection–Diffusion System Axioms singularly perturbed differential equations numerical methods convection–diffusion equations Shishkin meshes boundary layer uniform convergence |
| title | Advanced Fitted Mesh Finite Difference Strategies for Solving ‘n’ Two-Parameter Singularly Perturbed Convection–Diffusion System |
| title_full | Advanced Fitted Mesh Finite Difference Strategies for Solving ‘n’ Two-Parameter Singularly Perturbed Convection–Diffusion System |
| title_fullStr | Advanced Fitted Mesh Finite Difference Strategies for Solving ‘n’ Two-Parameter Singularly Perturbed Convection–Diffusion System |
| title_full_unstemmed | Advanced Fitted Mesh Finite Difference Strategies for Solving ‘n’ Two-Parameter Singularly Perturbed Convection–Diffusion System |
| title_short | Advanced Fitted Mesh Finite Difference Strategies for Solving ‘n’ Two-Parameter Singularly Perturbed Convection–Diffusion System |
| title_sort | advanced fitted mesh finite difference strategies for solving n two parameter singularly perturbed convection diffusion system |
| topic | singularly perturbed differential equations numerical methods convection–diffusion equations Shishkin meshes boundary layer uniform convergence |
| url | https://www.mdpi.com/2075-1680/14/3/171 |
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