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Algorithms for numerical solution of the modified equal width wave equation using collocation method. (English) Zbl 1121.65107

Summary: Quintic B-spline collocation algorithms for numerical solution of the modified equal width wave (MEW) equation are proposed. The algorithms are based on Crank-Nicolson formulation for time integration and quintic B-spline functions for space integration. Quintic B-spline collocation method over the finite intervals is also applied to the time split MEW equation and space split MEW equation. Results for the three algorithms are compared by studying the propagation of the solitary wave, interaction of the solitary waves, wave generation and birth of solitons.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L75 Higher-order nonlinear hyperbolic equations
35Q51 Soliton equations
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[1] Prenter, P.M., Splines and variational methods, (1975), J. Wiley New York · Zbl 0344.65044
[2] Morrison, P.J.; Meiss, J.D.; Carey, J.R., Scattering of RLW solitary waves, Physica D, 11, 324-336, (1984) · Zbl 0599.76028
[3] G.A. Gardner, L.R.T. Gardner, Modelling solitons of the Korteweg – de Vries equation with quintic splines, University of Wales, Bangor, UK, Maths Preprint series, No: 90.30, 1990 · Zbl 0717.65072
[4] Gardner, L.R.T.; Gardner, G.A.; Daǧ, İ., A B-spline finite element method for the regularized long wave equation, Commun. numer. meth. eng., 11, 59-68, (1995) · Zbl 0819.65125
[5] Zaki, S.I., Solitary wave interactions for the modified equal width equation, Comput. phys. comm., 126, 219-231, (2000) · Zbl 0951.65098
[6] Daǧ, İ., Least squares quadratic B-spline finite element method for the regularized long wave equation, Comput. methods appl. mech. engrg., 182, 205-215, (2000) · Zbl 0964.76042
[7] Zaki, S.I., A quintic B-spline finite elements scheme for the KdVB equation, Comput. methods appl. mech. engrg., 188, 121-134, (2000) · Zbl 0957.65088
[8] Hamdi, S.; Enright, W.H.; Schiesser, W.E.; Gottlieb, J.J., Exact solutions of the generalized equal width wave equation, (), 725-734
[9] Dağ, İ.; Doğan, A.; Saka, B., B-spline collocation methods for numerical solutions of RLW equation, Int. J. comput. math., 80, 743-757, (2003) · Zbl 1047.65088
[10] Saka, B.; Dağ, İ.; Doğan, A., A Galerkin method for the numerical solution of the RLW equation using quadratic B-splines, Int. J. comput. math., 81, 6, 727-739, (2004) · Zbl 1060.65109
[11] Saka, B.; Dağ, İ., A collocation method for the numerical solution of the RLW equation using cubic B-spline basis, Arab. J. sci. eng., 30, 39-50, (2005)
[12] Evans, D.J.; Raslan, K.R., Solitary waves for the generalized equal width (GEW) equation, Int. J. comput. math., 82, 4, 445-455, (2005) · Zbl 1064.65114
[13] Ramadan, M.A.; El-Danaf, T.S.; Alaal, F., A numerical solution of the burgers’ equation using septic B-splines, Chaos solitons fractals, 26, 795-804, (2005) · Zbl 1075.65127
[14] Dağ, İ.; Saka, B.; Irk, D., Galerkin method for the numerical solution of the RLW equation using quintic B-splines, J. comput. appl. math., 190, 532-547, (2006) · Zbl 1086.65094
[15] Wazwaz, A.M., The tanh and sine – cosine methods for a reliable treatment of the modified equal width equation and its variants, Commun. nonlinear sci. numer. simul., 11, 148-160, (2006) · Zbl 1078.35108
[16] Saka, B., A finite element method for equal width equation, Appl. math. comput., 175, 730-747, (2006) · Zbl 1088.65084
[17] B. Saka, İ. Dağ, Quartic B-spline collocation methods to the numerical solutions of the Burgers’ equation, Chaos Solitons Fractals (in press)
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