A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems
We extend the application of the Galerkin method for treating the multiterm fractional differential equations (FDEs) subject to initial conditions. A new shifted Legendre-Galerkin basis is constructed which satisfies exactly the homogeneous initial conditions by expanding the unknown variable using...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/306746 |
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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 | We extend the application of the Galerkin method for treating the multiterm fractional differential equations (FDEs) subject to initial conditions. A new shifted Legendre-Galerkin basis is constructed which satisfies exactly the homogeneous initial conditions by expanding the unknown variable using a new polynomial basis of functions which is built upon the shifted Legendre polynomials. A new spectral collocation approximation based on the Gauss-Lobatto quadrature nodes of shifted Legendre polynomials is investigated for solving the nonlinear multiterm FDEs. The main advantage of this approximation is that the solution is expanding by a truncated series of Legendre-Galerkin basis functions. Illustrative examples are presented to ensure the high accuracy and effectiveness of the proposed algorithms are discussed. |
| format | Article |
| id | doaj-art-47de2278a42a4a2892dbde5f38768368 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-47de2278a42a4a2892dbde5f387683682025-02-03T05:46:24ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/306746306746A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value ProblemsA. 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 ArabiaWe extend the application of the Galerkin method for treating the multiterm fractional differential equations (FDEs) subject to initial conditions. A new shifted Legendre-Galerkin basis is constructed which satisfies exactly the homogeneous initial conditions by expanding the unknown variable using a new polynomial basis of functions which is built upon the shifted Legendre polynomials. A new spectral collocation approximation based on the Gauss-Lobatto quadrature nodes of shifted Legendre polynomials is investigated for solving the nonlinear multiterm FDEs. The main advantage of this approximation is that the solution is expanding by a truncated series of Legendre-Galerkin basis functions. Illustrative examples are presented to ensure the high accuracy and effectiveness of the proposed algorithms are discussed.http://dx.doi.org/10.1155/2013/306746 |
| spellingShingle | A. H. Bhrawy M. A. Alghamdi A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems Abstract and Applied Analysis |
| title | A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems |
| title_full | A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems |
| title_fullStr | A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems |
| title_full_unstemmed | A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems |
| title_short | A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems |
| title_sort | new legendre spectral galerkin and pseudo spectral approximations for fractional initial value problems |
| url | http://dx.doi.org/10.1155/2013/306746 |
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