Lie Symmetry Methods And Their Applications In Differential Equations

This paper mainly studies the applications of Lie symmetry methods to solve differential equations. Lie symmetry offers a systemic way to reduce the order of the equation. Usually, we use standard method to solve the differential equations, sometimes it is too difficult, and so we make use of Lie symmetry to make it easy. It is very important to study group-invariant solutions and symmetries reduction for partial differential equations. It gives us a good tool to study PDE, It can reduce the amount of the calculation. And it helps us solve the equation effectively.In the third chapter, we first give introduction to the differential equations and the symmetry of the equation. The symmetries of the generalized KDV equation are obtained by parameter hypothesis. At last, we choose some simple symmetries to simplify the equation, and we obtain some group-invariant solutions of the generalized KDV equation.In the fourth chapter, we study the solutions about the generalized variable coefficient KDV equation by using the classical lie symmetry method. We use symmetries to reduce the equations and obtain new exact solutions. In this way, the PDE which is difficult to solve is convert to ODE which is easier to solve based on symmetry reduction. The fulfillment proves that this method is generally suitable for solving a major type nonlinear evolve equations.In the fifth chapter, the determining equations for a class of nonlinear partial differential equations with arbitrary order are considered. It shows that the determining equations for the nonclassical reduction can be obtained by requiring the compatibility between the original equation and the invariant surface condition. A simple PDE and BBM equation serve as examples to illustrating the feasibility of this method.