Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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COLLOCATION METHOD USING QUARTIC B-SPLINE FOR NUMERICAL SOLUTION OF THE MODIFIED EQUAL WIDTH WAVE EQUATION

  • Islam, Siraj-Ul (Department of Basic Sciences, NWFP University of Engineering and Technology) ;
  • Haq, Fazal-I (Department of Maths, Stats and Computer Science, NWFP Agricultural University) ;
  • Tirmizi, Ikram A. (Faculty of Engineering Sciences, GIK Institute of Engineering Sciences)
  • Received : 2009.10.10
  • Accepted : 2009.12.23
  • Published : 2010.05.30
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Abstract

A Numerical scheme based on collocation method using quartic B-spline functions is designed for the numerical solution of one-dimensional modified equal width wave (MEW) wave equation. Using Von-Neumann approach the scheme is shown to be unconditionally stable. Performance of the method is validated through test problems including single wave, interaction of two waves and use of Maxwellian initial condition. Using error norms $L_2$ and $L_{\infty}$ and conservative properties of mass, momentum and energy, accuracy and efficiency of the suggested method is established through comparison with the existing numerical techniques.

Keywords

References

  1. K. O. Abdulloev and H. Bogolubsky and V. G. Makhankov, One more example of inelastic soliton interaction, Phys. Lett. A,56(1967), 427-428.
  2. A. Esen, A lumped Galerkin method for the numerical solution of the modified equal width wave equation using quadratic B-splines, Int. J. Comput. Math. 83(2006),no.5-6, 449-459. https://doi.org/10.1080/00207160600909918
  3. A. Esen and S. Kutluay, Solitary wave solutions of the modified equal width wave equation, Comm. Nonlinear Science and Numer. Simul. 13(2008), 1538-1546. https://doi.org/10.1016/j.cnsns.2006.09.018
  4. D. J. Evans and K. R. Raslan, Solitary waves for the generalized equal width (GEW) equation, Int. J. Comput. Math. 82(2005), no. 4, 445-455.
  5. L. R. T. Gardner and G. Gardner and T. Geyikli, The boundary forced MKdV equation, J. Comput. Phys. 11(1994), 5-12.
  6. L. Junfeng, He's variational iteration method for the modified equal width equation, Chaos, Solitons and Fractals 39(2009), 2102-2109. https://doi.org/10.1016/j.chaos.2007.06.104
  7. A. R. Mitchell and D. F. Griffiths, The finite difference equations in partial differential equations, John Wiley and Sons, New York, 1980.
  8. P. J. Morrison and J. D. Meiss and J. R. Carey, Scattering of RLW solitary waves, Physica D 11(1984), 324-336. https://doi.org/10.1016/0167-2789(84)90014-9
  9. P. M. Prenter, Splines and Variational Methods, Wiley, New York, 1975.
  10. K. R. Raslan, Collocation method using cubic B-spline for the generalised equal width equation, Int. J. Simul. Process Modell. 1(2006), no. 2, 37-44.
  11. B. Saka, Algorithms for numerical solution of the modified equal width wave equation using collocation method, Math. Comp. Modl. 45(2007), 1096-1117. https://doi.org/10.1016/j.mcm.2006.09.012
  12. A. M. Wazwaz, The tanh and sine-cosine methods for a reliable treatment of the modified equal width equation and its invariants, Comm. Nonlinear Science and Numer. Simul. 11(2006), 148-160. https://doi.org/10.1016/j.cnsns.2004.07.001
  13. S. I. Zaki, Solitary wave interactions for the modified equal width equation, Comput. Phys. Commun. 126(2000), 219-231. https://doi.org/10.1016/S0010-4655(99)00471-3
  14. I. Dag and B. Saka and D. Irk, Galerkin method for the numerical solution of the RLW equation using quintic B-splines, J. Comput. Appl. Maths 190(2006), no. 1-2. 532-547. https://doi.org/10.1016/j.cam.2005.04.026