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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| Published in | Abstract and Applied Analysis Vol. 2013; no. 2013; pp. 237 - 246-322 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Cairo, Egypt
Hindawi Limiteds
01.01.2013
Hindawi Puplishing Corporation Hindawi Publishing Corporation John Wiley & Sons, Inc Wiley |
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| Abstract | 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. |
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| AbstractList | 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. |
| Audience | Academic |
| Author | A. H. Bhrawy M. A. Alghamdi |
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| Copyright | Copyright © 2013 A. H. Bhrawy and M. A. Alghamdi. COPYRIGHT 2013 John Wiley & Sons, Inc. Copyright 2013 Hindawi Publishing Corporation |
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| SubjectTerms | Algorithms Approximation Approximation theory Boundary value problems Differential equations Functions (mathematics) Galerkin methods Initial conditions Mathematical analysis Mathematical research Numerical analysis Polynomials Spectra |
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| Title | A New Legendre Spectral Galerkin and Pseudo-Spectral Approximations for Fractional Initial Value Problems |
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