A shifted Legendre spectral method for fractional-order multi-point boundary value problems
In this article, a shifted Legendre tau method is introduced to get a direct solution technique for solving multi-order fractional differential equations (FDEs) with constant coefficients subject to multi-point boundary conditions. The fractional derivative is described in the Caputo sense. Also, th...
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| Published in | Advances in difference equations Vol. 2012; no. 1; pp. 1 - 19 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
09.02.2012
Springer Nature B.V BioMed Central Ltd |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1687-1847 1687-1839 1687-1847 |
| DOI | 10.1186/1687-1847-2012-8 |
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| Abstract | In this article, a shifted Legendre tau method is introduced to get a direct solution technique for solving multi-order fractional differential equations (FDEs) with constant coefficients subject to multi-point boundary conditions. The fractional derivative is described in the Caputo sense. Also, this article reports a systematic quadrature tau method for numerically solving multi-point boundary value problems of fractional-order with variable coefficients. Here the approximation is based on shifted Legendre polynomials and the quadrature rule is treated on shifted Legendre Gauss-Lobatto points. We also present a Gauss-Lobatto shifted Legendre collocation method for solving nonlinear multi-order FDEs with multi-point boundary conditions. The main characteristic behind this approach is that it reduces such problem to those of solving a system of algebraic equations. Thus we can find directly the spectral solution of the proposed problem. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. |
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| AbstractList | In this article, a shifted Legendre tau method is introduced to get a direct solution technique for solving multi-order fractional differential equations (FDEs) with constant coefficients subject to multi-point boundary conditions. The fractional derivative is described in the Caputo sense. Also, this article reports a systematic quadrature tau method for numerically solving multi-point boundary value problems of fractional-order with variable coefficients. Here the approximation is based on shifted Legendre polynomials and the quadrature rule is treated on shifted Legendre Gauss-Lobatto points. We also present a Gauss-Lobatto shifted Legendre collocation method for solving nonlinear multi-order FDEs with multi-point boundary conditions. The main characteristic behind this approach is that it reduces such problem to those of solving a system of algebraic equations. Thus we can find directly the spectral solution of the proposed problem. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. In this article, a shifted Legendre tau method is introduced to get a direct solution technique for solving multi-order fractional differential equations (FDEs) with constant coefficients subject to multi-point boundary conditions. The fractional derivative is described in the Caputo sense. Also, this article reports a systematic quadrature tau method for numerically solving multi-point boundary value problems of fractional-order with variable coefficients. Here the approximation is based on shifted Legendre polynomials and the quadrature rule is treated on shifted Legendre Gauss-Lobatto points. We also present a Gauss-Lobatto shifted Legendre collocation method for solving nonlinear multi-order FDEs with multi-point boundary conditions. The main characteristic behind this approach is that it reduces such problem to those of solving a system of algebraic equations. Thus we can find directly the spectral solution of the proposed problem. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms.[PUBLICATION ABSTRACT] |
| ArticleNumber | 8 |
| Author | Bhrawy, Ali H Al-Shomrani, Mohammed M |
| Author_xml | – sequence: 1 givenname: Ali H surname: Bhrawy fullname: Bhrawy, Ali H email: alibhrawy@yahoo.co.uk organization: Department of Mathematics, Faculty of Science, King Abdulaziz University, Department of Mathematics, Faculty of Science, Beni-Suef University – sequence: 2 givenname: Mohammed M surname: Al-Shomrani fullname: Al-Shomrani, Mohammed M organization: Department of Mathematics, Faculty of Science, King Abdulaziz University, Faculty Of Computer Science and Information Technology, Northern Border University |
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| Copyright | Bhrawy and Al-Shomrani; licensee Springer. 2012. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Springer International Publishing AG 2012 |
| Copyright_xml | – notice: Bhrawy and Al-Shomrani; licensee Springer. 2012. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. – notice: Springer International Publishing AG 2012 |
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| Keywords | collocation method direct method shifted Legendre polynomials multi-term FDEs tau method Gauss-Lobatto quadrature multi-point boundary conditions |
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| SubjectTerms | Analysis Approximation Boundary conditions Boundary value problems Derivatives Difference and Functional Equations Functional Analysis Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Ordinary Differential Equations Partial Differential Equations Quadratures Spectra |
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| Title | A shifted Legendre spectral method for fractional-order multi-point boundary value problems |
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