Spectral framework using modified shifted Chebyshev polynomials of the third-kind for numerical solutions of one- and two-dimensional hyperbolic telegraph equations
Abstract This investigation discusses a numerical approach to solving the hyperbolic telegraph equation in both one and two dimensions. Applying the Galerkin method is the basis of this approach. We use appropriate combinations of third-kind modified shifted Chebyshev polynomials (3KMSCPs) as basis...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2025-01-01
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| Series: | Boundary Value Problems |
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| Online Access: | https://doi.org/10.1186/s13661-024-01987-4 |
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| author | S. M. Sayed A. S. Mohamed E. M. Abo-Eldahab Y. H. Youssri |
| author_facet | S. M. Sayed A. S. Mohamed E. M. Abo-Eldahab Y. H. Youssri |
| author_sort | S. M. Sayed |
| collection | DOAJ |
| description | Abstract This investigation discusses a numerical approach to solving the hyperbolic telegraph equation in both one and two dimensions. Applying the Galerkin method is the basis of this approach. We use appropriate combinations of third-kind modified shifted Chebyshev polynomials (3KMSCPs) as basis functions to transform the governing partial differential equations into a collection of algebraic equations. Through spectral Galerkin techniques, we establish the convergence error to demonstrate that our algorithm is more effective and efficient. Five examples are examined to verify the effectiveness and resilience of the applied method by comparing errors and illustrating the results. Our results show that the current numerical solutions align closely with exact solutions. The current algorithm is simple to set up and is better suited to solving certain difficult partial differential equations. |
| format | Article |
| id | doaj-art-85d575aa7bbe45899ec4b1272dc19025 |
| institution | Kabale University |
| issn | 1687-2770 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj-art-85d575aa7bbe45899ec4b1272dc190252025-01-19T12:33:17ZengSpringerOpenBoundary Value Problems1687-27702025-01-012025113110.1186/s13661-024-01987-4Spectral framework using modified shifted Chebyshev polynomials of the third-kind for numerical solutions of one- and two-dimensional hyperbolic telegraph equationsS. M. Sayed0A. S. Mohamed1E. M. Abo-Eldahab2Y. H. Youssri3Department of Mathematics, Faculty of Science, Helwan UniversityDepartment of Mathematics, Faculty of Science, Helwan UniversityDepartment of Mathematics, Faculty of Science, Helwan UniversityDepartment of Mathematics, Faculty of Science, Cairo UniversityAbstract This investigation discusses a numerical approach to solving the hyperbolic telegraph equation in both one and two dimensions. Applying the Galerkin method is the basis of this approach. We use appropriate combinations of third-kind modified shifted Chebyshev polynomials (3KMSCPs) as basis functions to transform the governing partial differential equations into a collection of algebraic equations. Through spectral Galerkin techniques, we establish the convergence error to demonstrate that our algorithm is more effective and efficient. Five examples are examined to verify the effectiveness and resilience of the applied method by comparing errors and illustrating the results. Our results show that the current numerical solutions align closely with exact solutions. The current algorithm is simple to set up and is better suited to solving certain difficult partial differential equations.https://doi.org/10.1186/s13661-024-01987-4Modified shifted Chebyshev polynomialsSpectral methodsHyperbolic telegraph equationOperational matrix |
| spellingShingle | S. M. Sayed A. S. Mohamed E. M. Abo-Eldahab Y. H. Youssri Spectral framework using modified shifted Chebyshev polynomials of the third-kind for numerical solutions of one- and two-dimensional hyperbolic telegraph equations Boundary Value Problems Modified shifted Chebyshev polynomials Spectral methods Hyperbolic telegraph equation Operational matrix |
| title | Spectral framework using modified shifted Chebyshev polynomials of the third-kind for numerical solutions of one- and two-dimensional hyperbolic telegraph equations |
| title_full | Spectral framework using modified shifted Chebyshev polynomials of the third-kind for numerical solutions of one- and two-dimensional hyperbolic telegraph equations |
| title_fullStr | Spectral framework using modified shifted Chebyshev polynomials of the third-kind for numerical solutions of one- and two-dimensional hyperbolic telegraph equations |
| title_full_unstemmed | Spectral framework using modified shifted Chebyshev polynomials of the third-kind for numerical solutions of one- and two-dimensional hyperbolic telegraph equations |
| title_short | Spectral framework using modified shifted Chebyshev polynomials of the third-kind for numerical solutions of one- and two-dimensional hyperbolic telegraph equations |
| title_sort | spectral framework using modified shifted chebyshev polynomials of the third kind for numerical solutions of one and two dimensional hyperbolic telegraph equations |
| topic | Modified shifted Chebyshev polynomials Spectral methods Hyperbolic telegraph equation Operational matrix |
| url | https://doi.org/10.1186/s13661-024-01987-4 |
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