Abstract
We investigate the symmetries of the generalised KdV Equation by using the theory of Lie classical method. The similarities obtained are utilized to reduce the order of nonlinear partial differential equation. Some solutions of reduced differential equation are presented.
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References
Hong, H., Lee, H.: Korteweg-de Vries equation of ion acoustic surface waves. Phys. Plasmas 6, 3422–3424 (1999)
Satsuma, J., Kaup, D.J.: A B\(\ddot{a}\)cklund transformation for a higher order Korteweg-de Vries equation. J. Phys. Soc. Jpn. 43, 692–726 (1977)
Zhang, S.: Application of exp-function method to a KdV equation with variable coefficients. Phys. Lett. A 365, 448–453 (2007)
Singh, K., Gupta, R.K.: On symmetries and invariant solutions of a coupled KdV system with variable coefficients. Int. J. Math. Math. Sci. 23, 3711–3725 (2005)
Gupta, R.K., Singh, K.: Symmetry analysis and some exact solutions of cylindrically symmetric null fields in general relativity. Commun. Nonlinear Sci. Numer. Simul. 16, 4189–4196 (2011)
Abd-el-Malek, M.B., Helal, M.M.: Group method solutions of the generalized forms of Burgers, Burgers-KdV and KdV equations with time-dependent variable coefficients. Acta Mech. 221, 281–296 (2011)
Yan, Z.: A new auto-B\(\ddot{a}\)cklund transformation and its applications in finding explicit exact solutions for the general KdV equation. Chin. J. Phys. 40, 113–120 (2002)
Lie, S.: Vorlesungen uber. Differentialgliechurger mit Bekannten Infinite-simalen Transformationen, Teuber, Leipzig (1891)
Lie, S.: On integration of a class of linear partial differential equations by means of definite integrals. Arch. Math. 6, 328–368 (1881)
Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)
Arrigo, D.J., Beckham, J.R.: Nonclassical symmetries of evolutionary partial differential equations and compatibility. J. Math. Anal. Appl. 289, 55–65 (2004)
Kudryashov, N.A.: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fractals 24, 1217–1231 (2005)
Wang, M.L., Zhang, J.L., Li, X.Z.: The \(\frac{G^{\prime }}{G}\)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372, 417–423 (2008)
Bansal, A., Gupta, R.K.: Modified \(\frac{G^{\prime }}{G}\)-expansion method for finding exact wave solutions of the coupled KleinG-ordon-Schr\(\ddot{o}\)dinger equation. Math. Methods Appl. Sci. 35, 1175–1187 (2012)
Gupta, R.K., Bansal, A.: Similarity reductions and exact solutions of generalized Bretherton equation with time-dependent coefficients. Nonlinear Dyn. 71, 1–12 (2013)
Bluman, G.W., Anco, S.C.: Symmetries and Differential Equations. Springer, New York (1989)
Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Applied Mathematical Sciences, vol. 81. Springer, Berlin (1989)
Bruz\(\acute{o}\)n, M.S., Gandarias, M.L., Camacho, J.C.: Classical and nonclassical symmetries for a Kuramoto Sivashinsky equation with dispersive effects. Math. Methods Appl. Sci. 30, 2091–2100 (2007)
Gandarias, M.L., Bruz\(\acute{o}\)n, M.S.: Solutions through nonclassical potential symmetries for a generalized inhomogeneous nonlinear diffusion equation. Math. Methods Appl. Sci. 31, 753–767 (2008)
Kudryashov, N.A., Loguinova, N.B.: Extended simplest equation method for nonlinear differential equations. Appl. Math. Comput. 205, 396–402 (2008)
Molati, M., Ramollo, M.P.: Symmetry classification of the Gardner equation with time-dependent coefficients arising in stratified fluids. Commun. Nonlinear Sci. Numer. Simul. 17, 1542–1548 (2012)
Wazwaz, A.M., Triki, H.: Soliton solutions for a generalized KdV and BBM equations with time-dependent coefficients. Commun. Nonlinear Sci. Numer. Simul. 16, 1122–1126 (2011)
Li, W., Zhao, Y.-M.: Exact solutions for a generalized KdV equation with time-dependent coefficients and K(m, n) equation. Appl. Math. Sci. 6, 2203–2217 (2012)
Wazwaz, A.M.: The tanh method for compact and non compact solutions for variants of the KdV-Burger equations. Phys. D: Nonlinear Phenom. 213, 147–151 (2006)
Parkes, E.J., Duffy, B.R.: An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations. Comput. Phys. Commun. 98, 288–300 (1996)
El-Wakil, S.A., Abdou, M.A.: New exact travelling wave solutions using modified extended tanh-function method. Chaos Solitons Fractals 31, 840–852 (2007)
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Kumar, R., Bansal, A., Gupta, R.K. (2016). Some Solutions of Generalised Variable Coefficients KdV Equation by Classical Lie Approach. In: Cushing, J., Saleem, M., Srivastava, H., Khan, M., Merajuddin, M. (eds) Applied Analysis in Biological and Physical Sciences. Springer Proceedings in Mathematics & Statistics, vol 186. Springer, New Delhi. https://doi.org/10.1007/978-81-322-3640-5_19
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DOI: https://doi.org/10.1007/978-81-322-3640-5_19
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