Table of Contents
International Journal of Computational Mathematics
Volume 2014, Article ID 178024, 12 pages
http://dx.doi.org/10.1155/2014/178024
Research Article

Solving the Generalized Regularized Long Wave Equation Using a Distributed Approximating Functional Method

Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 002, South Africa

Received 23 March 2014; Revised 5 July 2014; Accepted 21 July 2014; Published 12 August 2014

Academic Editor: Tobias Preusser

Copyright © 2014 Edson Pindza and Eben Maré. 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. P. D. Ariel, “On a second parameter in the solution of the flow near a rotating disk by homotopy analysis method,” Communications in Numerical Analysis, vol. 2012, Article ID 00111, 13 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  2. H. Panahipour, “Application of extended tanh method to generalized Burgers-type equations,” Communications in Numerical Analysis, vol. 2012, Article ID 00058, pp. 1–14, 2012. View at Google Scholar · View at MathSciNet
  3. L. R. T. Gardner, G. A. Gardner, and T. Geyikli, “The boundary forced MKdV equation,” Journal of Computational Physics, vol. 113, no. 1, pp. 5–12, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. D. H. Peregrine, “Calculations of the development of an undular bore,” Journal of Fluid Mechanics, vol. 25, pp. 321–330, 1966. View at Publisher · View at Google Scholar
  5. D. H. Peregrine, “Long waves on a beach,” Journal of Fluid Mechanics, vol. 27, pp. 815–827, 1967. View at Google Scholar
  6. J. C. Lewis and J. A. Tjon, “Resonant production of solitons in the RLW equation,” Physics Letters. A, vol. 73, no. 4, pp. 275–279, 1979. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. T. B. Benjamin, “Internal waves of permanent form in fluids of great depth,” Journal of Fluid Mechanics, vol. 29, pp. 559–592, 1967. View at Google Scholar
  8. J. L. Bona, W. G. Pritchard, and L. R. Scott, “A comparison of solutions of two model equations for long waves,” Lectures in Applied Mathematics, vol. 20, pp. 235–267, 1983. View at Google Scholar
  9. D. Bhardwaj and R. Shankar, “Computational method for regularized long wave equation,” Computers & Mathematics with Applications, vol. 40, no. 12, pp. 1397–1404, 2000. View at Publisher · View at Google Scholar · View at Scopus
  10. B. Y. Guo and W. M. Cao, “The Fourier pseudospectral method with a restrain operator for the RLW equation,” Journal of Computational Physics, vol. 74, no. 1, pp. 110–126, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. B. Saka and I. Dag, “A collocation method for the numerical solution of the RLW equation using cubic B-spline basis,” The Arabian Journal for Science and Engineering, vol. 30, no. 1, pp. 39–50, 2005. View at Google Scholar · View at MathSciNet · View at Scopus
  12. J. I. Ramos, “Explicit finite difference metho ds for the EWand RLW equations,” Applied Mathematics and Computation, vol. 179, no. 2, pp. 622–638, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. L. R. T. Gardner and G. A. Gardner, “Solitary waves of the equal width wave equation,” Journal of Computational Physics, vol. 101, no. 1, pp. 218–223, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. B. Saka, “A finite element method for equal width equation,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 730–747, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. A. H. A. Ali, “Spectral method for solving the equal width equation based on Chebyshev polynomials,” Nonlinear Dynamics, vol. 51, no. 1-2, pp. 59–70, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. S. I. Zaki, “A least-squares finite element scheme for the EW equation,” Computer Methods in Applied Mechanics and Engineering, vol. 189, no. 2, pp. 587–594, 2000. View at Publisher · View at Google Scholar · View at Scopus
  17. K. R. Raslan, “A computational method for the equal width equation,” International Journal of Computer Mathematics, vol. 81, no. 1, pp. 63–72, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. R. Mokhtari and M. Mohammadi, “Numerical solution of GRLW equation using Sinc-collocation method,” Computer Physics Communications, vol. 181, no. 7, pp. 1266–1274, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. D. K. Hoffman, N. Nayar, O. A. Sharafeddin, and D. J. Kouri, “Analytic banded approximation for the discretized free propagator,” The Journal of Physical Chemistry, vol. 95, no. 21, pp. 8299–8305, 1991. View at Publisher · View at Google Scholar · View at Scopus
  20. D. K. Hoffman and D. J. Kouri, “Distributed approximating function theory: A general, fully quantal approach to wave propagation,” Journal of Physical Chemistry, vol. 96, no. 3, pp. 1179–1184, 1992. View at Publisher · View at Google Scholar · View at Scopus
  21. G. W. Wei, D. S. Zhang, D. J. Kouri, and D. K. Hoffman, “Distributed approximating functional approach to the Fokker-Planck equation: time propagation,” Journal of Chemical Physics, vol. 107, no. 8, pp. 3239–3246, 1997. View at Publisher · View at Google Scholar · View at Scopus
  22. D. S. Zhang, G. W. Wei, D. J. Kouri, and D. K. Hoffman, “Distributed approximating functional approach to the Fokker-Planck equation: eigenfunction expansion,” The Journal of Chemical Physics, vol. 106, no. 12, pp. 5216–5224, 1997. View at Publisher · View at Google Scholar · View at Scopus
  23. G. W. Wei, “A new algorithm for solving some mechanical problems,” Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 15–17, pp. 2017–2030, 2001. View at Publisher · View at Google Scholar · View at Scopus
  24. G. W. Wei, “Discrete singular convolution for the sine-Gordon equation,” Physica D. Nonlinear Phenomena, vol. 137, no. 3-4, pp. 247–259, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. G. W. Wei, D. S. Zhang, D. J. Kouri, and D. K. Hoffman, “Lagrange distributed approximating functionals,” Physical Review Letters, vol. 79, no. 5, pp. 775–778, 1997. View at Publisher · View at Google Scholar · View at Scopus
  26. B. Fornberg, A Practical Guide to Pseudos pectral Methods, vol. 1 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  27. S. Yu, S. Zhao, and G. W. Wei, “Local spectral time splitting method for first- and second-order partial differential equations,” Journal of Computational Physics, vol. 206, no. 2, pp. 727–780, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. J. Shen, T. Tang, and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, Berlin, Germany, 2011.
  29. L. N. Trefethen, “Is Gauss quadrature better than Clenshaw-Curtis?” SIAM Review, vol. 50, no. 1, pp. 67–87, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. A. K. Khalifa, K. R. Raslan, and H. M. Alzubaidi, “A finite difference scheme for the MRLW and solitary wave interactions,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 346–354, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. F. Stenger, Numerical Methods Based on Sinc and Analitic Functions, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar · View at MathSciNet