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Exact and numerical traveling wave solutions of Whitham-Broer-Kaup equations. (English) Zbl 1082.65580
Summary: We find the explicit and numerical traveling wave solutions of Whitham-Broer-Kaup (for short WBK) equations, which contain blow up solutions and periodic solutions, by using the decomposition method. By using this method, the solutions were calculated in the form of a convergent power series with easily computable components. The convergence of the method as applied to the Whitham-Broer-Kaup equations is illustrated numerically.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L75 Higher-order nonlinear hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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References:
[1] Whitham, G.B., Proc. R. soc. London ser. A, 299, 6, (1967)
[2] Broer, L.J., Appl. sci. res., 31, 377, (1975)
[3] Kaup, D.J., Prog. theor. phys., 54, 396, (1975)
[4] Kupershmidt, B.A., Comm. math. phys., 99, 51, (1985)
[5] Ablowitz, M.J.; Clarkson, P.A., Soliton, nonlinear evolution equations and inverse scattering, (1991), Cambridge University Press New York · Zbl 0762.35001
[6] Cox, D., Ideal, varieties and algorithms, (1991), Springer New York
[7] Whitham, G.B., Linear and nonlinear waves, (1974), John Wiley New York · Zbl 0373.76001
[8] Wang, M.L., Phys. lett. A, 216, 67, (1996)
[9] Wang, M.L., Phys. lett. A, 199, 169, (1995)
[10] Yan, Z.Y.; Zhang, H.Q., Phys. lett. A, 252, 291, (1999)
[11] Xie, F.; Yan, Z.; Zhang, H., Phys. lett. A, 285, 76, (2001)
[12] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Acad. Publ. Boston · Zbl 0802.65122
[13] Adomian, G., J. math. anal. appl., 135, 501, (1988)
[14] Abbaoui, K.; Cherruault, Y.; Seng, V., Math. comp. model., 2, 89, (1995)
[15] Cherruault, Y., Kybernetes, 18, 31, (1989)
[16] Rèpaci, A., Appl. math. lett., 3, 35, (1990)
[17] Cherruault, Y.; Adomian, G., Math. comput. model., 18, 103, (1993)
[18] Abbaoui, K.; Pujol, M.J.; Cherruault, Y.; Himoun, N.; Grimalt, P., Kybernetes, 30, 1183, (2001)
[19] Kaya, D., Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation, Appl. math. comp., 147, 69-78, (2004) · Zbl 1037.35069
[20] Kaya, D., Exact and numerical soliton solutions of some coupled KdV and mkdv equations, Appl. math. comp., 151, 775-787, (2004) · Zbl 1048.65096
[21] Kaya, D.; El-Sayed, S.M., Chaos, solitons & fractals, 17, 869, (2003)
[22] Kaya, D.; Aassila, M., Phys. lett. A, 299, 201, (2002)
[23] Kaya, D., Int. J. comp. math., 72, 531, (1999)
[24] Kaya, D., Int. J. comp. math., 75, 235, (2000)
[25] Wazwaz, A.M., Math. comput. simul., 56, 269, (2001)
[26] Kaya, D., Int. J. math. math. sci., 27, 675, (2001)
[27] Wazwaz, A.M., Partial differential equations: methods and applications, (2002), Balkema Publishers The Netherlands · Zbl 0997.35083
[28] Cherruault, Y., Modeles et methodes mathematiques pour LES sciences du vivant, (1998), Presses Universitaires de France Paris · Zbl 0929.92003
[29] Cherruault, Y., Optimisation methodes locales et globales, (1999), Presses Universitaires de France Paris · Zbl 0928.49001
[30] Cherruault, Y.; Adomian, G.; Abbaoui, K.; Rach, R., Ijbc, 38, 89, (1995)
[31] Abbaoui, K.; Cherruault, Y., Math. comput. model., 20, 60, (1994)
[32] Abbaoui, K.; Cherruault, Y., Math. comput. model., 28, 103, (1994)
[33] Seng, V.; Abbaoui, K.; Cherruault, Y., Math. comput. model., 24, 59, (1996)
[34] Abbaoui, K.; Cherruault, Y., Comp. math. appl., 29, 103, (1995)
[35] Ngarhasta, N.; Some, B.; Abbaoui, K.; Cherruault, Y., Kybernetes, 31, 61, (2002)
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