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New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials

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Abstract

A family of symmetric generalized Jacobi polynomials (GJPs) is introduced and used for solving high even-order linear elliptic differential equations in one variable by the Galerkin method. Some efficient and accurate algorithms are developed and implemented for solving such equations. The use of GJPs leads to a simplified analysis, more precise error estimates and very efficient numerical algorithms. The methods lead to linear systems with specially structured matrices that can be efficiently inverted. Two numerical examples are presented aiming to demonstrate the accuracy and the efficiency of the proposed algorithms.

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Acknowledgments

The authors are very grateful to the anonymous referee for carefully reading the paper and for his comments and suggestions which have greatly improved the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

    E. H. Doha & W. M. Abd-Elhameed

  2. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

    W. M. Abd-Elhameed & A. H. Bhrawy

  3. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

    A. H. Bhrawy

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  1. E. H. Doha
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  2. W. M. Abd-Elhameed
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  3. A. H. Bhrawy
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Correspondence to W. M. Abd-Elhameed.

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Doha, E.H., Abd-Elhameed, W.M. & Bhrawy, A.H. New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials. Collect. Math. 64, 373–394 (2013). https://doi.org/10.1007/s13348-012-0067-y

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  • DOI: https://doi.org/10.1007/s13348-012-0067-y

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