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Mathematical Problems in Engineering
Volume 2012, Article ID 896575, 13 pages
Research Article

An Extension of the Legendre-Galerkin Method for Solving Sixth-Order Differential Equations with Variable Polynomial Coefficients

1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt
3Department of Mathematics, Faculty of Science, Taibah University, Madinah 20012, Saudi Arabia

Received 26 April 2011; Accepted 12 December 2011

Academic Editor: Alexei Mailybaev

Copyright © 2012 A. H. Bhrawy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We extend the application of Legendre-Galerkin algorithms for sixth-order elliptic problems with constant coefficients to sixth-order elliptic equations with variable polynomial coefficients. The complexities of the algorithm are O(N) operations for a one-dimensional domain with () unknowns. An efficient and accurate direct solution for algorithms based on the Legendre-Galerkin approximations developed for the two-dimensional sixth-order elliptic equations with variable coefficients relies upon a tensor product process. The proposed Legendre-Galerkin method for solving variable coefficients problem is more efficient than pseudospectral method. Numerical examples are considered aiming to demonstrate the validity and applicability of the proposed techniques.