The Study Of Invariant Solutions Of Some Nonlinear Partial Differential Equations Based On Symmetry Group Methods

With the development of modern physics and mathematics,the nonlinear phenomena and nonlinear problems in physics have attracted a great deal of attention.Many nonlinear problems can be attributed to the study of nonlinear PDE.And it is very necessary to solve the solutions of nonlinear PDE.There are many methods to solve the PDE.Lie symmetry method is a universal and effective method,which is the basis of studying invariant solutions of nonlinear PDE.In this thesis,a class of PDEs and two kinds of PDE are studied based on classical Lie symmetry method,conditional symmetry method and approximate symmetry method.Symmetry classifications,one-dimensional optimal system,similarity reduction and invariant solutions of PDEs are obtained.In the first chapter,the methods of classical Lie symmetry,conditional symmetry and approximate symmetry are introduced and background,significance of this study are given.In the second chapter,the symmetry classifications of nonlinear PDEs with two arbitrary functions are obtained by using classical Lie symmetry method and two cases of these are further studied.We construct one-dimensional optimal system of Lie symmetry group and obtain similarity reduction and invariant solutions of PDEs.In addition,a special case of PDEs is studied by conditional symmetry method.And similarity reduction and some invariant solutions of PDEs are obtained on basis of conditional symmetry.In the third chapter,we study a class of nonlinear filtration equation vt= k?v_x?v_xx.Two cases of the equation are further studied respectively,when k?v_x?= e^xand k?v_x?=?v_x?n,and one-dimensional optimal system is constructed.Then we obtain similarity reduction and invariant solutions on basis of optimal system.In addition,this two cases are studied by conditional symmetry method.Similarity reduction and some invariant solutions are obtained on basis of conditional symmetry.In the fourth chapter,the perturbed Boussinesq equation is studied by approxi-mate symmetry method,which is proposed by Baikov,Gazizov and Ibragimov.We compute one-dimensional optimal system and analysis invariant solutions.And some approximate invariants are obtained finally.In the fifth chapter,we summary the research contents of this paper and look forward to further research.