Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations
A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which le...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/364360 |
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| _version_ | 1832558235717992448 |
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| author | A. H. Bhrawy M. A. Alghamdi |
| author_facet | A. H. Bhrawy M. A. Alghamdi |
| author_sort | A. H. Bhrawy |
| collection | DOAJ |
| description | A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results. |
| format | Article |
| id | doaj-art-3dc7401f36554b3a8d3297aec54f38cf |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-3dc7401f36554b3a8d3297aec54f38cf2025-02-03T01:32:50ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/364360364360Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral ApproximationsA. H. Bhrawy0M. A. Alghamdi1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi ArabiaA shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results.http://dx.doi.org/10.1155/2012/364360 |
| spellingShingle | A. H. Bhrawy M. A. Alghamdi Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations Abstract and Applied Analysis |
| title | Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations |
| title_full | Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations |
| title_fullStr | Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations |
| title_full_unstemmed | Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations |
| title_short | Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations |
| title_sort | numerical solutions of odd order linear and nonlinear initial value problems using a shifted jacobi spectral approximations |
| url | http://dx.doi.org/10.1155/2012/364360 |
| work_keys_str_mv | AT ahbhrawy numericalsolutionsofoddorderlinearandnonlinearinitialvalueproblemsusingashiftedjacobispectralapproximations AT maalghamdi numericalsolutionsofoddorderlinearandnonlinearinitialvalueproblemsusingashiftedjacobispectralapproximations |