A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential Equations
This article proposes numerical algorithms for solving second-order and telegraph linear partial differential equations using a matrix approach that employs certain generalized Chebyshev polynomials as basis functions. This approach uses the operational matrix of derivatives of the generalized Cheby...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-12-01
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| Series: | Algorithms |
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| Online Access: | https://www.mdpi.com/1999-4893/18/1/2 |
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| author | Waleed Mohamed Abd-Elhameed Ramy M. Hafez Anna Napoli Ahmed Gamal Atta |
| author_facet | Waleed Mohamed Abd-Elhameed Ramy M. Hafez Anna Napoli Ahmed Gamal Atta |
| author_sort | Waleed Mohamed Abd-Elhameed |
| collection | DOAJ |
| description | This article proposes numerical algorithms for solving second-order and telegraph linear partial differential equations using a matrix approach that employs certain generalized Chebyshev polynomials as basis functions. This approach uses the operational matrix of derivatives of the generalized Chebyshev polynomials and applies the collocation method to convert the equations with their underlying conditions into algebraic systems of equations that can be numerically treated. The convergence and error bounds are examined deeply. Some numerical examples are shown to demonstrate the efficiency and applicability of the proposed algorithms. |
| format | Article |
| id | doaj-art-9a17cb99a5f046499d1deab2387bcdcb |
| institution | Kabale University |
| issn | 1999-4893 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj-art-9a17cb99a5f046499d1deab2387bcdcb2025-01-24T13:17:25ZengMDPI AGAlgorithms1999-48932024-12-01181210.3390/a18010002A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential EquationsWaleed Mohamed Abd-Elhameed0Ramy M. Hafez1Anna Napoli2Ahmed Gamal Atta3Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptDepartment of Mathematics, Faculty of Education, Matrouh University, Cairo 51511, EgyptDepartment of Mathematics and Computer Science, University of Calabria, 87036 Rende, ItalyDepartment of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, EgyptThis article proposes numerical algorithms for solving second-order and telegraph linear partial differential equations using a matrix approach that employs certain generalized Chebyshev polynomials as basis functions. This approach uses the operational matrix of derivatives of the generalized Chebyshev polynomials and applies the collocation method to convert the equations with their underlying conditions into algebraic systems of equations that can be numerically treated. The convergence and error bounds are examined deeply. Some numerical examples are shown to demonstrate the efficiency and applicability of the proposed algorithms.https://www.mdpi.com/1999-4893/18/1/2Chebyshev polynomialscollocation methodmatrix approachderivative formulasconvergence analysis |
| spellingShingle | Waleed Mohamed Abd-Elhameed Ramy M. Hafez Anna Napoli Ahmed Gamal Atta A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential Equations Algorithms Chebyshev polynomials collocation method matrix approach derivative formulas convergence analysis |
| title | A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential Equations |
| title_full | A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential Equations |
| title_fullStr | A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential Equations |
| title_full_unstemmed | A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential Equations |
| title_short | A New Generalized Chebyshev Matrix Algorithm for Solving Second-Order and Telegraph Partial Differential Equations |
| title_sort | new generalized chebyshev matrix algorithm for solving second order and telegraph partial differential equations |
| topic | Chebyshev polynomials collocation method matrix approach derivative formulas convergence analysis |
| url | https://www.mdpi.com/1999-4893/18/1/2 |
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