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A collocation method with cubic B-splines for solving the MRLW equation. (English) Zbl 1133.65085

The modified regularized long wave equation is solved by a collocation method based on cubic B-splines. The time differencing is done by a leapfrog method. A von Neumann stability analysis is given for a linearlized form of the method. Note that L. R. T. Gardner, G. A. Gardner, F. A. Ayoub, and N. K. Amein [Arab. J. Sci. Eng. 22, 183–193 (1997; Zbl 0893.35113)] previously solved this problem using quintic B-splines.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
76B25 Solitary waves for incompressible inviscid fluids

Citations:

Zbl 0893.35113
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References:

[1] Alexander, M. E.; Morris, J. L., Galerkin methods applied to some model equations for nonlinear dispersive waves, J. Comput. Phys., 30, 428-451 (1979) · Zbl 0407.76014
[2] 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
[3] 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
[4] Gardner, L. R.T.; Gardner, G. A., Solitary wave of the regularized long wave equation, J. Comput. Phys., 91, 441-459 (1990) · Zbl 0717.65072
[5] Gardner, L. R.T.; Gardner, G. A.; Dag, I., A B-spline finite element method for the regularized long wave equation, Comm. Numer. Meth. Eng., 11, 59-68 (1995) · Zbl 0819.65125
[6] Gardner, L. R.T.; Gardner, G. A.; Dogan, A., A least squares finite element scheme for the RLW equation, Comm. Numer. Meth. Eng., 12, 795-804 (1996) · Zbl 0867.76040
[7] Gardner, L. R.T.; Gardner, G. A.; Ayoub, F. A.; Amein, N. K., Approximations of solitary waves of the MRLW equation by B-spline finite element, Arab. J. Sci. Eng., 22, 183-193 (1997) · Zbl 0893.35113
[8] Kaya, D.; El-Sayed, S. M., An application of the decomposition method for the generalized KdV and RLW equations, Chaos, Solitons and Fractals, 17, 869-877 (2003) · Zbl 1030.35139
[9] A.K. Khalifa, Theory and applications of the collocation method with splines for ordinary and partial differential equations, Ph.D. Thesis, Heriot-Watt University, 1979.; A.K. Khalifa, Theory and applications of the collocation method with splines for ordinary and partial differential equations, Ph.D. Thesis, Heriot-Watt University, 1979.
[10] Peregrine, D. H., Calculations of the development of an undular bore, J. Fluid Mech., 25, 2, 321-330 (1966)
[11] Raslan, K. R., A computational method for the regularized long wave (RLW) equation, Appl. Math. Comput., 176, 1101-1118 (2005) · Zbl 1082.65582
[12] Smith, G. D., Numerical Solution of Partial Differential Equation: Finite Difference Method (1978), Learendom Press: Learendom Press Oxford · Zbl 0389.65040
[13] Soliman, A. A.; Hussien, M. H., Collocation solution for RLW equation with septic splines, Appl. Math. Comput., 161, 623-636 (2005) · Zbl 1061.65102
[14] Soliman, A. A.; Raslan, K. R., Collocation method using quadratic B-spline for the RLW equation, Internat. J. Comput. Math., 78, 399-412 (2001) · Zbl 0990.65116
[15] Zhang, L., A finite difference scheme for generalized long wave equation, Appl. Math. Comput., 168, 962-972 (2005) · Zbl 1080.65079
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