Browse > Article
http://dx.doi.org/10.14317/jami.2011.29.3_4.563

SYMMETRY REDUCTIONS, VARIABLE TRANSFORMATIONS AND EXACT SOLUTIONS TO THE SECOND-ORDER PDES  

Liu, Hanze (Department of Mathematics, Binzhou University)
Liu, Lei (The Center for Economic Research, Shandong University)
Publication Information
Journal of applied mathematics & informatics / v.29, no.3_4, 2011 , pp. 563-572 More about this Journal
Abstract
In this paper, the Lie symmetry analysis is performed on the three mixed second-order PDEs, which arise in fluid dynamics, nonlinear wave theory and plasma physics, etc. The symmetries and similarity reductions of the equations are obtained, and the exact solutions to the equations are investigated by the dynamical system and power series methods. Then, the exact solutions to the general types of PDEs are considered through a variable transformation. At last, the symmetry and integration method is employed for reducing the nonlinear ODEs.
Keywords
Lie symmetry analysis; Similarity reduction; Dynamical system method; Power series method; Exact solution;
Citations & Related Records
연도 인용수 순위
  • Reference
1 H.Liu, J.Li, L.Liu, Lie symmetries, optimal systems and exact solutions to the fifth-order KdV type equations, J. Math. Anal. Appl., 368 (2010), 551-558.   DOI   ScienceOn
2 V.A.Galaktionov, S.R.Svirshchevskii, Exact solutions and invariant subspaces of nonlinear partial differential equations in Mechanics and Physics, Chapman and Hall/CRC, 2006.
3 H.Liu, J.Li, Lie symmetry analysis and exact solutions for the extended mKdV equation, Acta Appl. Math., 109 (2010), 1107-1119.   DOI   ScienceOn
4 H.Liu, J.Li, Lie symmetries, conservation laws and exact solutions for two rod equations, Acta Appl. Math., 110 (2010), 573-587.   DOI   ScienceOn
5 Y.Stepanyants, On stationary solutions of the reduced Ostrovsky equation: periodic wave, compactons and compound solitons, Chaos, Solitons and Fractals, 28 (2006), 193-204.   DOI   ScienceOn
6 H.Liu, J.Li, Q.Zhang, Lie symmetry analysis and exact explicit solutions for general Burg- ers' equation, J. Comput. Appl. Math., 228 (2009), 1-9.   DOI   ScienceOn
7 H.Liu, J.Li, L.Liu, Lie group classifications and exact solutions for two variable-coefficient equations, Appl. Math. Comput., 215 (2009), 2927-2935.   DOI   ScienceOn
8 H.Liu, J.Li, L.Liu, Painleve analysis, Lie symmetries, and exact solutions for the time- dependent coefficients Gardner equations, Nonlinear Dyn., 59 (2010), 497-502.   DOI   ScienceOn
9 C.Qu, C.Zhu, Classification of coupled systems with two-component nonlinear diffusion equations by the invariant subspace method, J. Phys. A: Math. Theor., 42 (2009), 475201 (27pp).
10 H.Liu, J.Li, Lie symmetry analysis and exact solutions for the short pulse equation, Nonlinear Anal. TMA, 71 (2009), 2126-2133.   DOI   ScienceOn
11 P.J.Olver, Applications of Lie groups to differential equations, in: Grauate texts in Math- ematics, vol.107, Springer, New York, 1993.
12 G.W.Bluman, S.C.Anco, Symmetry and Integration Methods for Differential Equations, in: Applied Mathematical Sciences, vol.154, Springer-Verlag, New York, 2002.
13 J.Li, Exact explicit peakon and periodic cusp wave solutions for several nonlinear wave equations, J. Dyn. Diff. Equat., 20 (2008), 909-922.   DOI   ScienceOn
14 C.Qu, Q.Huang, Symmetry reductions and exact solutions of the affine heat equation, J. Math. Anal. Appl., 346 (2008), 521-530.   DOI   ScienceOn
15 E.Parkes, Explicit solutions of the reduced Ostrovsky equation, Chaos, Solitons and Fractals, 31 (2007), 602-610.   DOI   ScienceOn
16 A.Sakovich, S.Sakovich, Solitary wave solutions of the short pulse equation, J. Phys. A: Math. Gen., 39 (2006), L361-367.   DOI