## International Journal of Differential Equations

### Numerical Solution of the Modified Equal Width Wave Equation

#### Abstract

Numerical solution of the modified equal width wave equation is obtained by using lumped Galerkin method based on cubic B-spline finite element method. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. Accuracy of the proposed method is discussed by computing the numerical conserved laws ${L}_{2}$ and ${L}_{\infty }$ error norms. The numerical results are found in good agreement with exact solution. A linear stability analysis of the scheme is also investigated.

#### Article information

Source
Int. J. Differ. Equ., Volume 2012 (2012), Article ID 587208, 15 pages.

Dates
Accepted: 30 September 2011
First available in Project Euclid: 24 January 2017

https://projecteuclid.org/euclid.ijde/1485226817

Digital Object Identifier
doi:10.1155/2012/587208

Mathematical Reviews number (MathSciNet)
MR2875240

Zentralblatt MATH identifier
1237.65110

#### Citation

Karakoç, Seydi Battal Gazi; Geyikli, Turabi. Numerical Solution of the Modified Equal Width Wave Equation. Int. J. Differ. Equ. 2012 (2012), Article ID 587208, 15 pages. doi:10.1155/2012/587208. https://projecteuclid.org/euclid.ijde/1485226817

#### References

• L. R. T. Gardner and G. A. Gardner, “Solitary waves of the regularised long-wave equation,” Journal of Computational Physics, vol. 91, no. 2, pp. 441–459, 1990.
• 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.
• P. J. Morrison, J. D. Meiss, and J. R. Cary, “Scattering of regularized-long-wave solitary waves,” Physica D. Nonlinear Phenomena, vol. 11, no. 3, pp. 324–336, 1984.
• Kh. O. Abdulloev, I. L. Bogolubsky, and V. G. Makhankov, “One more example of inelastic soliton interaction,” Physics Letters. A, vol. 56, no. 6, pp. 427–428, 1976.
• 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.
• T. Geyikli and S. Battal Gazi Karakoç, “Septic B-Spline Collocation Method for the Numerical Solution of the Modified Equal Width Wave Equation,” Applied Mathematics, vol. 2, no. 6, pp. 739–749, 2011.
• T. Geyikli and S. Battal Gazi Karakoç, “Petrov-Galerkin method with cubic Bsplines for solving the MEW equation,” Bulletin of the Belgian Mathematical Society. In press.
• A. Esen, “A numerical solution of the equal width wave equation by a lumped Galerkin method,” Applied Mathematics and Computation, vol. 168, no. 1, pp. 270–282, 2005.
• A. Esen, “A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic B-splines,” International Journal of Computer Mathematics, vol. 83, no. 5-6, pp. 449–459, 2006.
• B. Saka, “Algorithms for numerical solution of the modified equal width wave equation using collocation method,” Mathematical and Computer Modelling, vol. 45, no. 9-10, pp. 1096–1117, 2007.
• S. I. Zaki, “Solitary wave interactions for the modified equal width equation,” Computer Physics Communications, vol. 126, no. 3, pp. 219–231, 2000.
• S. I. Zaki, “Least-squares finite element scheme for the EW equation,” Computer Methods in Applied Mechanics and Engineering, vol. 189, no. 2, pp. 587–594, 2000.
• A.-M. Wazwaz, “The tanh and the sine-cosine methods for a reliable treatment of the modified equal width equation and its variants,” Communications in Nonlinear Science and Numerical Simulation, vol. 11, no. 2, pp. 148–160, 2006.
• B. Saka and Dağ, “Quartic B-spline collocation method to the numerical solutions of the Burgers' equation,” Chaos, Solitons and Fractals, vol. 32, no. 3, pp. 1125–1137, 2007.
• J. Lu, “He's variational iteration method for the modified equal width equation,” Chaos, Solitons and Fractals, vol. 39, no. 5, pp. 2102–2109, 2009.
• D. J. Evans and K. R. Raslan, “Solitary waves for the generalized equal width (GEW) equation,” International Journal of Computer Mathematics, vol. 82, no. 4, pp. 445–455, 2005.
• S. Hamdi, W. H. Enright, W. E. Schiesser, and J. J. Gottlieb, “Exact solutions of the generalized equal width wave equation,” in Proceedings of the International Conference on Computational Science and Its Application, vol. 2668, pp. 725–734, Springer, 2003.
• A. Esen and S. Kutluay, “Solitary wave solutions of the modified equal width wave equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1538–1546, 2008.
• P. M. Prenter, Splines and Variational Methods, John Wiley & Sons, New York, NY, USA, 1975.