Abstract and Applied Analysis
Volume 2012 (2012), Article ID 418943, 16 pages
http://dx.doi.org/10.1155/2012/418943
Research Article

A Jacobi Dual-Petrov Galerkin-Jacobi Collocation Method for Solving Korteweg-de Vries Equations

Ali H. Bhrawy1,2 and M. M. Al-Shomrani1,3

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
3Faculty of Computer Science and Information Technology, Northern Border University, Saudi Arabia

Received 15 April 2012; Revised 22 June 2012; Accepted 29 June 2012

Academic Editor: Xiaodong Yan

Copyright © 2012 Ali H. Bhrawy and M. M. Al-Shomrani. 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.

Abstract

The present paper is devoted to the development of a new scheme to solve the initial-boundary value Korteweg-de Vries equation which models many physical phenomena such as surface water waves in a channel. The scheme consists of Jacobi dual-Petrov Galerkin-Jacobi collocation method in space combined with Crank-Nicholson-leap-frog method in time such that at each time step only a sparse banded linear algebraic system needs to be solved. Numerical results are presented to show that the proposed numerical method is accurate and efficient for Korteweg-de Vries equations and other third-order nonlinear equations.