Lie Point Symmetries Of Differential Difference Equations

Partial diferential equations are important mathematical tools for describing nat-ural phenomena. Finding the solutions of partial diferential equations and researchingthe properties of the solutions are very important and forefront research topics. Lie sym-metries group theory is an efective and useful tool for obtaining the analytical solutionsof diferential equations. The basic idea of Lie symmetries group theory is finding thesymmetries group of given equations. The symmetries group of diferential equation is atransformation group that transformed one solution of diferential equation to another.How to find the symmetries of diferential-diference equations has been paid more atten-tion in recent years.In this paper, we will extend the prolonging method of Lie symmetries of diferentialequations to the diference equations. By using of the continuous theory we will extend theprolonging method to diferential-diference equations. At the same time we will introducean efective method for finding determining equations—-commuting flows. Finally, wewill use a series of theory to get the Lie symmetries of some diference equations anddiferential-diference equations.