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Abstract and Applied Analysis
Volume 2014, Article ID 705204, 8 pages
http://dx.doi.org/10.1155/2014/705204
Research Article

Petrov-Galerkin Method for the Coupled Schrödinger-KdV Equation

Department of Mathematics, College of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia

Received 27 March 2014; Accepted 24 May 2014; Published 15 June 2014

Academic Editor: Mohammad T. Darvishi

Copyright © 2014 M. S. Ismail 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.

Abstract

Petrov-Galerkin method is used to derive a numerical scheme for the coupled Schrödinger-KdV (SKdV) equations, where we have used the cubic B-splines as a test functions and a linear B-splines as a trial functions. Product approximation technique is used to deal with the nonlinear terms. An implicit midpoint rule and the Runge-Kutta method of fourth-order (RK4) are used to discretize in time. A block nonlinear pentadiagonal system is obtained. We solve this system by the fixed point method. The resulting scheme has a fourth-order accuracy in space direction and second-order in time direction in case of the implicit midpoint rule and it is unconditionally stable by von Neumann method. Using the RK4 method the scheme will be linear and fourth-order in time and space directions, and it is also conditionally stable. The exact soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness and the efficiency of the proposed schemes.