Solution of nonlinear telegraph equation with discontinuities along the transmission line using meshless collocation method
In this paper, we present a novel hybrid numerical technique combining radial basis functions (RBFs) and the finite difference method (FDM) for solving the one-dimensional telegraph interface model (TIM). The presence of interfaces or discontinuities along the transmission line requires additional b...
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| Format: | Article |
| Language: | English |
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Elsevier
2025-06-01
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| Series: | Partial Differential Equations in Applied Mathematics |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2666818125001469 |
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| author | Muhammad Asif Tabassum Gul Muhammad Bilal Riaz Faisal Bilal |
| author_facet | Muhammad Asif Tabassum Gul Muhammad Bilal Riaz Faisal Bilal |
| author_sort | Muhammad Asif |
| collection | DOAJ |
| description | In this paper, we present a novel hybrid numerical technique combining radial basis functions (RBFs) and the finite difference method (FDM) for solving the one-dimensional telegraph interface model (TIM). The presence of interfaces or discontinuities along the transmission line requires additional boundary or interface conditions to accurately describe the voltage and current behaviors at these junctions. Our method incorporates these interface conditions, approximating spatial partial derivatives using RBFs and time derivatives using FDM. To validate our approach, we applied it to several benchmark linear and nonlinear TIMs. For linear models, the resulting algebraic system is solved using Gauss elimination, while for nonlinear models, we employed the quasi-Newton method to linearize the nonlinear terms. We evaluated the method’s performance by calculating the maximum absolute errors (MAEs) and root mean square errors (RMSEs) for different grid points (GPs). Additionally, we compared the true and estimate solutions through three dimensional (3D)graphs. The numerical results show that our technique provides simple implementation, quick convergence, and outstanding accuracy for both linear and nonlinear models. This hybrid approach proves to be a highly effective and reliable tool for solving telegraph interface problems. |
| format | Article |
| id | doaj-art-7f8ecd79a0f14ae5b22c1a0c2e2eedc8 |
| institution | OA Journals |
| issn | 2666-8181 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Partial Differential Equations in Applied Mathematics |
| spelling | doaj-art-7f8ecd79a0f14ae5b22c1a0c2e2eedc82025-08-20T01:49:32ZengElsevierPartial Differential Equations in Applied Mathematics2666-81812025-06-011410121910.1016/j.padiff.2025.101219Solution of nonlinear telegraph equation with discontinuities along the transmission line using meshless collocation methodMuhammad Asif0Tabassum Gul1Muhammad Bilal Riaz2Faisal Bilal3Department of Mathematics, University of Peshawar, Pakistan; Corresponding author.Department of Mathematics, University of Peshawar, PakistanIT4Innovations, VSB - Technical University of Ostrava, Ostrava, Czech Republic; Applied Science Research Center, Applied Science Private University, Amman, JordanDepartment of Mathematics, University of Peshawar, PakistanIn this paper, we present a novel hybrid numerical technique combining radial basis functions (RBFs) and the finite difference method (FDM) for solving the one-dimensional telegraph interface model (TIM). The presence of interfaces or discontinuities along the transmission line requires additional boundary or interface conditions to accurately describe the voltage and current behaviors at these junctions. Our method incorporates these interface conditions, approximating spatial partial derivatives using RBFs and time derivatives using FDM. To validate our approach, we applied it to several benchmark linear and nonlinear TIMs. For linear models, the resulting algebraic system is solved using Gauss elimination, while for nonlinear models, we employed the quasi-Newton method to linearize the nonlinear terms. We evaluated the method’s performance by calculating the maximum absolute errors (MAEs) and root mean square errors (RMSEs) for different grid points (GPs). Additionally, we compared the true and estimate solutions through three dimensional (3D)graphs. The numerical results show that our technique provides simple implementation, quick convergence, and outstanding accuracy for both linear and nonlinear models. This hybrid approach proves to be a highly effective and reliable tool for solving telegraph interface problems.http://www.sciencedirect.com/science/article/pii/S2666818125001469Hyperbolic telegraph equationInterface modelsMeshless collocation methodFinite difference method |
| spellingShingle | Muhammad Asif Tabassum Gul Muhammad Bilal Riaz Faisal Bilal Solution of nonlinear telegraph equation with discontinuities along the transmission line using meshless collocation method Partial Differential Equations in Applied Mathematics Hyperbolic telegraph equation Interface models Meshless collocation method Finite difference method |
| title | Solution of nonlinear telegraph equation with discontinuities along the transmission line using meshless collocation method |
| title_full | Solution of nonlinear telegraph equation with discontinuities along the transmission line using meshless collocation method |
| title_fullStr | Solution of nonlinear telegraph equation with discontinuities along the transmission line using meshless collocation method |
| title_full_unstemmed | Solution of nonlinear telegraph equation with discontinuities along the transmission line using meshless collocation method |
| title_short | Solution of nonlinear telegraph equation with discontinuities along the transmission line using meshless collocation method |
| title_sort | solution of nonlinear telegraph equation with discontinuities along the transmission line using meshless collocation method |
| topic | Hyperbolic telegraph equation Interface models Meshless collocation method Finite difference method |
| url | http://www.sciencedirect.com/science/article/pii/S2666818125001469 |
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