Nonclassical Symmetry Methods And The Applications In Partial Differential Equations

This paper mainly studies the applications of nonclassical symmetry methods to solve partial differential equations.Compared with the classical Lie method,it adds an invariant surface condition in the formation.Consequently,it makes calculation easy and obtains more group-invariant solutions.In the second chapter,we first introduce fundamental knowledge on differential equations and symmetry.It contains definition of vector fields,invariant group of algebraic equation,extension and generators of invariant group and so on.In the third chapter,we first give introduction to the differential equation and the symmetry of the differential equation.Next,we obtain symmetries of Rosenau-Hyman(RH) equation by using indeterminate coefficient method.In the fourth chapter,we study the nonclassical symmetries and group-invariant solutions of Boussinesq-Burgerse equation.In many papers,nonclassical symmetry method is used to solute one-dimensional partial differential equations.However,we improve the method and use it to solute two-dimensional partial differential equation in this paper.Consequently,nonclasical symmetries and group-invariant solutions of Boussinesq-Burgerse equation are obtained.In the fifth chapter,nonclasical symmetries and compatibility are studied. Likewise,compatibility is often found in one-dimensional partial differential equations, but we extend compatibility to two-dimensional partial differential equation and obtain nonclassical symmetries of two-dimensional heat equation.Finally,compatibiliity is successfully use to obtain the determining equations of two-dimensional heat equation.