Lie Symmetries And Invariant Solutions Of Higher Order Elliptic Equations

With the developing of science and technology, the research on partial differential equations has been deepened and expanded, involving life science,information science, geographic science, environmental science and many other areas. Therefore, it is of great scientific significance to study the properties of partial differential equations and its solving methods. At present, there are some methods to find the exact solutions of partial differential equations, including inverse scattering method, Darboux transforma-tion method, Backlund transformation method, Hirota bilinear method,function expan-sion method, Lie symmetry method and variable separation method.In this paper, we use the Lie symmetry method to study the high order linear elliptic equations, the results are as follows:1. We review the Lie symmetry method comprehensively;2. Discuss the elliptic equations in two cases and obtain its Lie algebra3. Calculate 1-dimensional optimal system of the Lie sub-algebra of the elliptic equation;4. Give the reductions and some exact solutions of the elliptic equation;5. The applications of Lie symmetry method on the initial boundary value problem of four order elliptic equations are given.These results have a good reference value for the further study of partial differential equations.