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Potential method applied to Boussinesq equation. (English) Zbl 05675347
Summary: A solution of the nonlinear Boussinesq equation is presented using the potential similarity transformation method. The equation is first written in a conserved form, a potential function is then assumed reducing it to a system of equations which is further solved through the group transformation method. New transformations are found.
MSC:
76Fluid mechanics
References:
[1]Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles a la surface au fond, J. math. Pures appl. Ser. 217, 55-108 (1872) · Zbl 04.0493.04 · doi:http://gallica.bnf.fr/ark:/12148/bpt6k164163.f00000063
[2]Xu, L.; Auston, D. H.; Hasegawa, A.: Propagation of electromagnetic solitary waves in dispersive nonlinear dielectrics, Phys. rev. A 45, 3184-3193 (1992)
[3]Karpman, V. I.: Nonlinear waves in dispersive media, (1975)
[4]Clarkson, P. A.; Kruskal, M. D.: New similarity solutions of the Boussinesq equation, J. math. Phys., 2201-2213 (1989) · Zbl 0698.35137 · doi:10.1063/1.528613
[5]Levi, D.; Winternitz, P.: Non classical symmetry reduction: example of the Boussinesq equation, J. phys. A 22, 2915-2924 (1989) · Zbl 0694.35159 · doi:10.1088/0305-4470/22/15/010
[6]Clarkson, P. A.; Mansfield, E. L.: Algorithms for the non classical method of symmetry reductions, SIAM J. Appl. math. 54 – 56, 1693-1719 (1994) · Zbl 0823.58036 · doi:10.1137/S0036139993251846
[7]Vil’danov, A. N.: Integrable boundary value problem for the Boussinesq equation, Theor. math. Phys. 141 – 142, 1494-1508 (2004) · Zbl 1178.35339 · doi:10.1023/B:TAMP.0000046559.46997.5c
[8]Yan, Z.: A similarity transformations and exact solutions for a family of higher dimensional generalized Boussinesq equations, Phys. lett. A 361, 223-230 (2007) · Zbl 1170.35089 · doi:10.1016/j.physleta.2006.07.047
[9]Wang, S.; Xue, H.: Global solution for a generalized Boussinesq equation, Appl. math. Comput. 204, 130-136 (2008) · Zbl 1161.35469 · doi:10.1016/j.amc.2008.06.059
[10]Wang, Yu-Zhu: Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation, Nonlinear anal. 70, 465-482 (2009) · Zbl 1161.35470 · doi:10.1016/j.na.2007.12.018
[11]Bruzón , M. S.; Gandarias, M. L.: Symmetries for a family of Boussinesq equations with nonlinear dispersion, Com. nonlinear sci. Numer. simul. 14, 3250-3257 (2009) · Zbl 1221.35326 · doi:10.1016/j.cnsns.2009.01.005
[12]Rashed, A. S.; Kassem, M. M.: Group analysis for natural convection from a vertical plate, J. comput. Appl. math. 222, 392-403 (2008) · Zbl 1148.76051 · doi:10.1016/j.cam.2007.11.010
[13]Kassem, M. M.: Group analysis of a non-Newtonian flow past a vertical plate subjected to a heat constant flux, Int. J. Appl. math. Mech. zamm. 88, 661-673 (2008)
[14]Kassem, M. M.; Rashed, A. S.: Group similarity transformation of a time dependent chemical convective process, Appl. math. Comput. 215, 1671-1684 (2009)
[15]Debnath, L.: Nonlinear partial differential equations for scientists and engineers, (1997) · Zbl 0892.35001