A numerical solution of the equal width wave equation using a fully implicit finite difference method

Abstract

In this paper, a fully implicit finite difference scheme for the numerical solution of the equal width wave (EW) equation is proposed. Since the EW equation is nonlinear the scheme leads to a system of nonlinear equations. At each time-step Newton’s method is used to solve this nonlinear system. The results and comparisons with analytical and other numerical values clearly show that results obtained using the fully implicit finite difference scheme are precise and reliable

Keywords

• Equal width wave equation; Finite differences; Solitary waves; Undular bore; Nonlinear partial differential equations

References

1. Ali A. H. A., Spectral method for solving the equal width equation based on Chebyshev polynomials, Nonlinear Dyn., 51, 59-70 (2008).
2. Archilla B. G., A spectral method for the equal width equation, J. Comput. Phys., 125, 395-402 (1996).
3. Arora R., Siddiqui Md. J., Singh V. P., Solutions of Inviscid Burgers’ and Equal Width Wave Equations by RDTM, Int. J. Appl. Phys. Math., 2, 212-214 (2012).
4. Dag I., Saka B., A cubic B-spline collocation method for the EW equation, Math. Comput. Appl., 9, 381-392 (2004).
5. Dogan A., Application of Galerkin’s method to equal width wave equation, Appl. Math. Comput., 160, 65-76 (2005).
6. Esen A. , A numerical solution of the equal width wave equation by a lumped Galerkin method, Appl. Math. Comput., 168, 270-282 (2005).
7. Esen A., Kutluay S., A linerized implicit finite-difference method for solving the equal width wave equation, Int. J. Comput. Math., 83, 319-330 (2006).
8. Gardner L. R. T., Gardner G. A., Solitary waves of the equal width wave equation, J. Comput. Phys., 101, 218-223 (1992).

9. Gardner L. R. T., Gardner G. A., Ayoub F. A., Amein N. K., Simulations of the EW undular bore, Commun. Num. Meth. Eng., 13, 583-592 (1997).
10. Haq F., Shah I. A., Ahmad S., Septic B-Spline Collocation method for numerical solution of the Equal Width Wave (EW) equation, Life Sci. J., 10, 253-260 (2013).
11. Khalifa A. K., Ali A. H., Raslan K. R., Numerical study for the equal width wave (EWE) equation, Mem. Fac. Sci. Kochi Uni., 20, 47-55 (1999).
12. Morrison P. J., Meiss J. D., Carey J. R., Scattering of regularized-long-wave solitary waves, Physica D, 11, 324-336 (1984).
13. Ramos J. I., Explicit finite difference methods for the EW and RLW equations, Appl. Math. Comput., 179, 622-638 (2006).
14. Raslan K. R., A computational method for the equal width equation, Int. J. Comput. Math., 81, 63-72 (2004).
15. Raslan K. R., Collocation method using quartic B-spline for the equal width (EW) equation, Appl. Math. Comput., 168, 795-805 (2005).
16. Saka B., A finite element method for equal width equation, Appl. Math. and Comput., 175, 730-747 (2006).
17. Saka B., Dag I., Dereli Y., Korkmaz A., Three different methods for numerical solution of the EW equation, Eng. Anal. Bound. Elem., 32, 556-566 (2008).
18. Zaki S. I., A least-squares finite element scheme for the EW equation, Comput. Meth. Appl. Mech. Engrg., 189, 587-594 (2000).
19. Zaki S. I., Solitary waves induced by the boundary forced EW equation, Comput. Meth. Appl. Mech. Engrg., 190, 4881-4887 (2001).