Exact Solutions And Conservation Laws Of Several Kinds Of Nonlinear Partial Differential Equations

The governing equations of many physical problems are represented as differential equations.For example,the population problem,the mixed problem and the falling problem are fundamental physical phenomena observed in differential form.When researchers systematically study the differential equations arose from important physical problems,the discovery of any explicit solution plays a key role in the study of their dynamic behavior.The study of symmetric analysis,exact solutions and conservation laws of nonlinear differential equations can assist in the scientific interpretation and engineering application of complex physical phenomena.In this paper,combined with the symbolic calculation software Maple,the exact solutions and conservation laws of the following three kinds of nonlinear partial differential equation models are investigated by using the theory of the Lie symmetry analysis.(1)The exact solution and conservation law of convective Cahn-Hilliard equation are studied.By constructing B?cklund transformation and using the Lie symmetry analysis,several sets of exact solutions of the equation in different forms are obtained under certain gauge conditions,and conservation laws are constructed by combining the symmetry of the equation.(2)The exact solutions and conservation laws of the time-fractional Gardner equation with time-dependent coefficients are studied.The Lie symmetry group of the equation is analyzed and divided into six cases according to the parameters of the equation,the symmetry reduction of different cases is realized,the power series solution is obtained accordingly,and the conservation law of the equation is constructed.(3)The exact solutions and conservation laws of the space-time fractional Boussinesq system are studied.Based on the Lie symmetry analysis of fractional partial differential equation,the infinitesimal generators and group-invariant solutions of the equation are obtained.Combined with the power series expansion method,the exact power series solution of the system is obtained,and the convergence of the solution is analyzed.The conservation law of the equation is constructed by using symmetry and nonlinear self-adjoint conditions.