Exact Solutions And Conservation Laws Of Some Nonlinear Partial Differential Equations

Most of the physical and mathematical phenomena in nature can be described by nonlinear partial differential equation(systems),such as fluid mechanics,nonlinear dynamics,optical fiber and acoustics,condensation physics and so on.Therefore,the exact solution,symmetry and conservation laws of nonlinear partial differential equations(systems)play a very important role in the study of mathematical and physical models.At present,researchers at home and abroad have used many methods to solve the solutions of nonlinear partial differential equations.With the development of society,these methods are also widely used in other fields.In this paper,the Lie symmetry analysis method is used to explore the symmetry,exact solution,convergence and conservation laws of some nonlinear partial differential equations(systems).It is mainly divided into the following three parts:In the first part,a class of fourth-order nonlinear Parabolic equation by using the Lie group method,and the optimal system is obtained by using the symbolic calculation tool Maple.Thus,the similarity reduction is carried out under the optimal system,the fourth-order nonlinear partial differential equation is divided into ordinary differential equations,and the power series solution of the equation is obtained.Then the convergence of the solution is judged.Finally,the conservation law of the equation is constructed,and we proved that the equation is conserved.In the second part,the nonlinear time fractional order equation is studied,which is mainly based on the Lie group method.First,the optimal system of the equation is solved.On the basis of the optimal system,the equation is symmetrically reduced and transformed into a nonlinear ordinary differential equation,and the power series solution is obtained.The convergence of the solution is proved.Finally,the conservation law of the equation is constructed by using the new conservation law.In the third part,the modified time fractional Korteweg-de Vries(Kdv)equations are studied based on Lie group method,the optimal system is obtained by Lie symmetry analysis,the fractional equations are reduced to ordinary differential equations,and then the exact solution of the original equation is obtained by combining the power series method.Studying the properties of the solution is equivalent to studying the physical meaning of the nonlinear phenomenon.Finally,it is proved that the equation satisfies the conservation law.