×

Potential method applied to Boussinesq equation. (English) Zbl 1423.35302

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:

35Q35 PDEs in connection with fluid mechanics
35A22 Transform methods (e.g., integral transforms) applied to PDEs
PDFBibTeX XMLCite
Full Text: DOI

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) · JFM 04.0493.04
[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), Pergamon: Pergamon New York
[4] Clarkson, P. A.; Kruskal, M. D., New similarity solutions of the Boussinesq equation, J. Math. Phys., 2201-2213 (1989) · Zbl 0698.35137
[5] Levi, D.; Winternitz, P., Non classical symmetry reduction: example of the Boussinesq equation, J. Phys. A, 22, 2915-2924 (1989) · Zbl 0694.35159
[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
[7] Vil’danov, A. N., Integrable boundary value problem for the Boussinesq equation, Theor. Math. Phys., 141-142, 1494-1508 (2004) · Zbl 1178.35339
[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
[9] Wang, S.; Xue, H., Global solution for a generalized Boussinesq equation, Appl. Math. Comput., 204, 130-136 (2008) · Zbl 1161.35469
[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
[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
[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
[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) · Zbl 1158.76036
[14] Kassem, M. M.; Rashed, A. S., Group similarity transformation of a time dependent chemical convective process, Appl. Math. Comput., 215, 1671-1684 (2009) · Zbl 1179.80048
[15] Debnath, L., Nonlinear Partial Differential Equations for Scientists and Engineers (1997), Birkhauser: Birkhauser Boston, p. 343 · Zbl 0892.35001
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.