×

Solitary waves for the MRLW equation. (English) Zbl 1179.65126

Summary: This work presents a computational comparison study of quadratic, cubic, quartic and quintic splines for solving the modified regularized long wave (MRLW) equation. Collocation schemes with quadratic and cubic splines are found to be unconditionally stable. The fourth-order Runge-Kutta method has been used to solve the collocation schemes when quartic and quintic B-splines are used. The three invariants of motion have been evaluated to determine the conservation properties of the suggested algorithms. Comparisons of results due to different schemes with the exact values shows the accuracy and efficiency of the proposed schemes. Results corresponding to higher order splines are more accurate than those corresponding to lower order splines.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kaya, D.; El-Sayed, S. M., An application of the decomposition method for the generalized KdV and RLW equations, Chaos Solitons Fractals, 17, 869-877 (2003) · Zbl 1030.35139
[2] Kaya, D., A numerical simulation of solitary wave solutions of the generalized regularized long wave equation, Appl. Math. Comput., 149, 833-841 (2004) · Zbl 1038.65101
[3] Raslan, K. R., A computational method for the regularized long wave (RLW) equation, Appl. Math. Comput., 176, 1101-1118 (2005) · Zbl 1082.65582
[4] Soliman, A. A.; Raslan, K. R., Collocation method using quadratic B-spline for the RLW equation, Int. J. Comput. Math., 78, 399-412 (2001) · Zbl 0990.65116
[5] Zaki, S. I., Solitary waves of splitted RLW equation, Comput. Phys. Commun., 138, 80-91 (2001) · Zbl 0984.65103
[6] Gardner, L. R.T.; Gardner, G. A.; Dogan, A., A least squares finite elements scheme for the RLW equation, Commun. Numer. Math. Eng., 12, 759-804 (1996) · Zbl 0867.76040
[7] Dag, I., Least squares quadratic B-splines finite element method for the regularized long wave equation, Comput. Methods Appl. Mech. Eng., 182, 205-215 (2000) · Zbl 0964.76042
[8] Dag, I.; Saka, B.; Irk, D., Application of cubic B-splines for numerical solution of the RLW equation, Appl. Math. Comput., 195, 373-389 (2004) · Zbl 1060.65110
[9] Soliman, A. A.; Hussien, M. H., Collocation solution for RLW equation with septic splines, Appl. Math. Comput., 161, 623-636 (2005) · Zbl 1061.65102
[10] Zhang, L., A finite difference scheme for generalized long wave equation, Appl. Math. Comput., 168, 962-972 (2005) · Zbl 1080.65079
[11] Khalifa, A. K.; Raslan, K. R.; Alzubaidi, H. M., A finite difference scheme for the MRLW and solitary wave interactions, Appl. Math. Comput., 189, 346-354 (2007) · Zbl 1123.65085
[12] Khalifa, A. K.; Raslan, K. R.; Alzubaidi, H. M., A collocation method with cubic B-splines for solving the MRLW equation, J. Comput. Appl. Math., 212, 406-418 (2008) · Zbl 1133.65085
[13] Gardner, L. R.T.; Gardner, G. A.; Ayoub, F. A.; Amein, N. K., Approximation of solitary waves of the MRLW equation by B-spline finite element, Arab. J. Sci. Eng., 22, 183-193 (1997) · Zbl 0893.35113
[14] Smith, G. D., Numerical Solution of Partial Differential Equations: Finite Difference Methods (1985), Oxford University Press · Zbl 0576.65089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.