Symmetry Of The Generalized Nonlinear Reaction Diffusion Equations

Due to the fact that nonlinear phenomena widely exist in natural and social world, and the research for them mostly can be reduced to solving the nonlin-ear equations, therefore the study of exact solutions to nonlinear equations has become a widespread concern. Among numerous methods, the ones relating to symmetry groups are most effective and universal to solve nonlinear partial dif-ferential equations. Based on Lie point symmetry method put forward by Sophis Lie, a Norway mathematician, many symmetry groups methods came out and applied to studying exact solutions and symmetry reduction of equations, these methods including conditional symmetry, Lie-Backlund symmetry, generalized conditional symmetry and so on.Because of most models'lacking Lie-Backlund symmetry, Zhdanov, Fokas and Liu proposed the generalized conditional symmetry, which constructs equa-tions'exact solutions of important physical values. So it was proved very efficient in the study of the exact solutions and symmetry reductions of partial differential equations.In this paper, the generalized conditional symmetry method is used to re-search the nonlinear diffusion equations ut=[A(u)u_xⁿ]x+P(x, u)ux+Q(x, u), in which the diffusion is power law, i. e. A(u)=um. We utilize the second-order generalized conditional symmetries such asη=uxx+H(u)u_x²+G(x, u)ux+ F(x, u) andη= uxx+H(u)u_x²+G(x, u)(ux)^2-n+F(x, u)(ux)^1-n to classify and reduce this equation. And according to the compatibility of the generalized con-ditional symmetry and the considered equations, exact solutions of some relative equations are given. These solutions will offer sound ground to the explanation of some physical phenomena.