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The Scientific World Journal
Volume 2014, Article ID 309264, 11 pages
Research Article

Two Legendre-Dual-Petrov-Galerkin Algorithms for Solving the Integrated Forms of High Odd-Order Boundary Value Problems

1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
2Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
3Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt

Received 25 October 2013; Accepted 20 November 2013; Published 27 January 2014

Academic Editors: S. Amat, M. Inc, and S. A. Tersian

Copyright © 2014 Waleed M. Abd-Elhameed 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.


Two numerical algorithms based on dual-Petrov-Galerkin method are developed for solving the integrated forms of high odd-order boundary value problems (BVPs) governed by homogeneous and nonhomogeneous boundary conditions. Two different choices of trial functions and test functions which satisfy the underlying boundary conditions of the differential equations and the dual boundary conditions are used for this purpose. These choices lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost. The various matrix systems resulting from these discretizations are carefully investigated, especially their complexities and their condition numbers. Numerical results are given to illustrate the efficiency of the proposed algorithms, and some comparisons with some other methods are made.