In this paper,a nonlinear wave equation with variable coefficients is studied,interestingly,this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressi...In this paper,a nonlinear wave equation with variable coefficients is studied,interestingly,this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities.With the aid of Lou’s direct method1,the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained.The corresponding numerical examples of exact solutions are presented by using different coefficients.Particularly,while the variable coefficients are taken as some special constants,the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients,which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials.Using the same method to solve the nonlinear wave equation,the validity and rationality of this method are verified.展开更多
基金This work is supported by the National Natural Science Foundation of China(Nos.11672069,11702059,11232003,11672062)The Ph.D.Programs Foundation of Ministry of Education of China(No.20130041110050)+3 种基金the Research Startup Project Plan for Liaoning Doctors(No.20141119)the Fundamental Research Funds for the Central Universities(20000101)the Natural Science Foundation of Liaoning Province(No.20170540199)111 Project(B08014).
文摘In this paper,a nonlinear wave equation with variable coefficients is studied,interestingly,this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities.With the aid of Lou’s direct method1,the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained.The corresponding numerical examples of exact solutions are presented by using different coefficients.Particularly,while the variable coefficients are taken as some special constants,the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients,which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials.Using the same method to solve the nonlinear wave equation,the validity and rationality of this method are verified.
