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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.

Linked References

  1. D. Bai and L. Zhang, “Numerical studies on a novel split-step quadratic B-spline finite element method for the coupled Schrödinger-KdV equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1263–1273, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. D. Bai and L. Zhang, “The finite element method for the coupled Schrödinger-KdV equations,” Physics Letters A, vol. 373, no. 26, pp. 2237–2244, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. I. Christie, D. F. Griffiths, A. R. Mitchell, and J. M. Sanz-Serna, “Product approximation for nonlinear problems in the finite element method,” IMA Journal of Numerical Analysis, vol. 1, no. 3, pp. 253–266, 1981. View at Publisher · View at Google Scholar · View at MathSciNet
  4. J. M. Sanz-Serna and I. Christie, “Petrov-Galerkin methods for nonlinear dispersive waves,” Journal of Computational Physics, vol. 39, no. 1, pp. 94–102, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. S. W. Schoombie, “Spline Petrov-Galerkin methods for the numerical solution of the Korteweg-de Vries equation,” IMA Journal of Numerical Analysis, vol. 2, no. 1, pp. 95–109, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. Wang, “Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 17–35, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. S. Ismail, “Numerical solution of a coupled Korteweg-de Vries equations by collocation method,” Numerical Methods for Partial Differential Equations, vol. 25, no. 2, pp. 275–291, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. S. Ismail, “A fourth-order explicit schemes for the coupled nonlinear Schrödinger equation,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 273–284, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. S. Ismail, “Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method,” Mathematics and Computers in Simulation, vol. 78, no. 4, pp. 532–547, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. S. Ismail and T. R. Taha, “A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation,” Mathematics and Computers in Simulation, vol. 74, no. 4-5, pp. 302–311, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M. S. Ismail and S. Z. Alamri, “Highly accurate finite difference method for coupled nonlinear Schrödinger equation,” International Journal of Computer Mathematics, vol. 81, no. 3, pp. 333–351, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. S. Ismail and T. R. Taha, “Numerical simulation of coupled nonlinear Schrödinger equation,” Mathematics and Computers in Simulation, vol. 56, no. 6, pp. 547–562, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. H. P. Bhatt and A. Q. M. Khaliq, “Higher order exponential time differencing scheme for system of coupled nonlinear Schrödinger equations,” Applied Mathematics and Computation, vol. 228, pp. 271–291, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  14. R. Abardeh, M. Ramezanpour, A. Doosthoseini, and E. Rezaie, “New method for solving coupled Schrödinger KdV equation,” Applied Mathematical Sciences, vol. 7, no. 86, pp. 4273–4280, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  15. D. Kaya and S. M. El-Sayed, “On the solution of the coupled Schrödinger-KdV equation by the decomposition method,” Physics Letters A, vol. 313, no. 1-2, pp. 82–88, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Labidi, G. Ebadi, E. Zerrad, and A. Biswas, “Analytical and numerical solutions of the Schrödinger-KdV equation,” Pramana—Journal of Physics, vol. 78, no. 1, pp. 59–90, 2012. View at Publisher · View at Google Scholar · View at Scopus
  17. K. Appert and J. Vaclavik, “Dynamics of coupled solitons,” Physics of Fluids, vol. 20, no. 11, pp. 1845–1849, 1977. View at Google Scholar · View at Scopus
  18. L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäuser, Boston, Mass, USA, 1997. View at Publisher · View at Google Scholar · View at MathSciNet