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Abstract and Applied Analysis
Volume 2014, Article ID 819367, 9 pages
http://dx.doi.org/10.1155/2014/819367
Research Article

A Numerical Solution for Hirota-Satsuma Coupled KdV Equation

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

Received 6 February 2014; Accepted 16 July 2014; Published 17 August 2014

Academic Editor: Fuding Xie

Copyright © 2014 M. S. Ismail and H. A. Ashi. 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

A Petrov-Galerkin method and product approximation technique are used to solve numerically the Hirota-Satsuma coupled Korteweg-de Vries equation, using cubic -splines as test functions and a linear -spline as trial functions. The implicit midpoint rule is used to advance the solution in time. Newton’s method is used to solve the block nonlinear pentadiagonal system we have obtained. The resulting schemes are of second order accuracy in both directions, space and time. The von Neumann stability analysis of the schemes shows that the two schemes are unconditionally stable. The single soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the schemes. The interaction of two solitons, three solitons, and birth of solitons is also discussed.