On symmetry-preserving difference scheme to a generalized Benjamin equation and third-order Burgers equation

Abstract In this paper, an exposition of a method is presented for discretizing a generalized Benjamin equation and third-order Burgers equation while preserving their Lie point symmetries. By using local conservation laws, the potential systems of original equation are obtained, which can be used to construct the invariant difference models and symmetry-preserving difference models of original equation, respectively. Furthermore, this method is very effective and can be applied to discrete high-order nonlinear evolution equations.

[1]  G. Quispel,et al.  Continuous symmetries of differential-difference equations: the Kac-van Moerbeke equation and Painlevé reduction , 1992 .

[2]  Luc Vinet,et al.  Lie group formalism for difference equations , 1997 .

[3]  E. Hopf.  The partial differential equation ut + uux = μxx , 1950 .

[4]  Anjan Biswas,et al.  Symmetry reduction, exact group-invariant solutions and conservation laws of the Benjamin-Bona-Mahoney equation , 2013, Appl. Math. Lett..

[5]  Xiaorui Hu,et al.  Nonlocal symmetries related to Bäcklund transformation and their applications , 2012, 1201.3409.

[6]  Changzheng Qu,et al.  Classification of coupled systems with two-component nonlinear diffusion equations by the invariant subspace method , 2009 .

[7]  V. Dorodnitsyn.  Transformation groups in net spaces , 1991 .

[8]  P. Olver.  Applications of lie groups to differential equations , 1986 .

[9]  Decio Levi,et al.  Non-classical symmetry reduction: example of the Boussinesq equation , 1989 .

[10]  N. Ibragimov.  Transformation groups applied to mathematical physics , 1984 .

[11]  Vladimir Dorodnitsyn,et al.  Lie Point Symmetry Preserving Discretizations for Variable Coefficient Korteweg–de Vries Equations , 2000 .

[12]  Willy Hereman,et al.  Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method , 1986 .

[13]  V. Dorodnitsyn,et al.  Symmetry-preserving difference schemes for some heat transfer equations , 1997, math/0402367.

[14]  Yifang Liu,et al.  Symmetry analysis of the option pricing model with dividend yield from financial markets , 2011, Appl. Math. Lett..

[15]  Clarkson?Kruskal Direct Similarity Approach for Differential-Difference Equations , 2005 .

[16]  Shou-Fu Tian,et al.  Lie symmetries and nonlocally related systems of the continuous and discrete dispersive long waves system by geometric approach , 2015 .

[17]  Vladimir Dorodnitsyn,et al.  Noether-type theorems for difference equations , 2001 .

[18]  D. Levi,et al.  Symmetries of the discrete Burgers equation , 1999 .

[19]  G. Bluman,et al.  Symmetries and differential equations , 1989 .

[20]  Roberto Floreanini,et al.  Lie symmetries of finite‐difference equations , 1995 .

[21]  P. Winternitz,et al.  Symmetries, Lagrangian Formalism and Integration of Second Order Ordinary Difference Equations , 2003 .

[22]  Abdul-Majid Wazwaz.  A study on the (2+1)-dimensional and the (2+1)-dimensional higher-order Burgers equations , 2012, Appl. Math. Lett..

[23]  S. Lou,et al.  Similarity reductions of the KP equation by a direct method , 1991 .

[24]  The Using of Conservation Laws in Symmetry-Preserving Difference Scheme , 2013 .

[25]  Christodoulos Sophocleous,et al.  Exact solutions of a remarkable fin equation , 2008, Appl. Math. Lett..

[26]  D. Levi,et al.  Symmetries and conditional symmetries of differential difference equations , 1993 .