Symmetry Reduction And Exact Solutions Of Several Class Of Partial Differential Equations

This paper makes full use of Lie symmetry group theory to study several types of partial differential equations.A group invariant solution and conservation law for a class of biological chemo taxis model are considered,by the mathematical software Maple,the symmetric group and invariant solutions of the one-dimensional inverse mean curvature flow,the symmetric group and invariant solutions of mean curvature flow with a linear external forced field.The group invariant solution and conservation law for some equations are obtained in terms of the symmetric reduction method,by solving the symmetric group of the partial differential equation.In the first chapter,the one-dimensional inverse average curvature flow is transformed into a linear parabolic partial differential equation by using the support function of strictly convex closed plane curves.Using Lie point symmetry group theory,the equation is reduced symmetrically and group invariant solutions are discussed.In the second chapter,mean curvature flow with a linear external forced field is transformed into a class of degenerate nonlinear parabolic partial differential equation through the support function of strictly convex closed plane curves.The generators of the equation are obtained by applying the method of Lie symmetry group extension vector field and corresponding single parameter to find exact solution of the equation.In the third chapter,a biological chemo taxis model is derived for a class of biological chemo taxis phenomena.Three infinitesimal generators of the equation are obtained by applying the Lie symmetric group method.The corresponding invariant solutions are obtained by using the symmetric reduction method.Finally,using the method of conservation law proposed by Ibragimov,three conservation laws of infinitely small generators corresponding to the biological chemo taxis model are obtained.In summary,the main work of this paper is to use the Lie symmetry group theory method to solve several types of partial differential equations.Firstly,the evolution equation is obtained through the support function line of the strictly convex closed plane curve.And then the Lie point symmetry group theory method and the reduction method reduces the partial differential equation into an ordinary differential equation.And it selects the appropriate symmetric transformation to obtain the solution of the original equation and the conservation law of some equations.