Group Classifications, Symmetry Reductions and Exact Solutions to the Nonlinear Elastic Rod Equations

Abstract

In this paper, the Lie symmetry analysis are performed on the three nonlinear elastic rod (NER) equations. The complete group classifications of the generalized nonlinear elastic rod equations are obtained. The symmetry reductions and exact solutions to the equations are presented. Furthermore, by means of dynamical system and power series methods, the exact explicit solutions to the equations are investigated. It is shown that the combination of Lie symmetry analysis and dynamical system method is a feasible approach to deal with symmetry reductions and exact solutions to nonlinear PDEs.

Complete Group Classification and Exact Solutions to the Generalized Short Pulse Equation

“…It is known that the Lie symmetry analysis is a systematic and powerful method for tackling symmetries, CLs, and exact solutions to PDEs [1, 3–6, 8–14]. Furthermore, we find that the combination of Lie symmetry analysis and the dynamical system method is a feasible approach for dealing with exact solutions to PDEs [1, 12, 13]. Especially, we developed the generalized power series method for tackling exact solutions to the nonlinear evolution equations [3, 14].…”

Group Analysis and New Explicit Solutions of Simplified Modified Kawahara Equation with Variable Coefficients

“…However, the power series can be used to solve them. In view of this, we can find that the power series method [14,15,[20][21][22][23] is an effective tool of solving such ODEs. Moreover, from our model, we could find that these power series solutions are important for computations in numerical analysis and physical applications.…”

Lie Symmetry Analysis, Self-Adjointness and Conservation Law for a Type of Nonlinear Equation

“…[10] supplies the most general connection between Cls and pairs of symmetries and adjoint symmetries for non-self-adjoint systems. On the contribution of the Lie symmetry method, significant studies have been performed on the integrability of the nonlinear PDEs, group classification, optimal system, reduced solutions and conservation laws, such as [11][12][13][14][15][16][17][18][19][20][21][22][23], and references therein.…”