Symmetry Reductions And Exact Solutions Of The Nonlinear Diffusion Equations With Variable Coefficients

Nonlinear diffusion equations, as an important class of parabolic equations arise from a variety of diffusion phenomena that widely exist in nature. These equations are mathematical models of physical problems appeared in many fields such as filtration, phase transition, biochemistry and dynamics of biological groups. In recent years, much attention has been paid on the study of nonlinear dif-fusion equations with variable coefficients, and much progress has been made. The purpose of this dissertation is to study the symmetry reductions, classifications and exact solutions of the nonlinear diffusion equations with variable coefficients by using the generalized conditional symmetry method, the sign-invariant method and the invariant-subspace method.In this dissertation, we focus on the following three respects.1. Using the generalized conditional symmetry method, choose the symmetry with characteristic n=[k(u)]xt, we consider symmetry reductions. classifications and functionally additive sepa-rable variable solutions of the nonlinear diffusion equation of variable coefficients with source term f(x)ut=(g(x)D(u)ux)x+q(x)Q(u).2. Taking the second order symmetry in the form η(x,u)=uxx+H(u)u2x+a(x)ux and the first order sign-invariant form J(x,u)=ut-A(x,u)ux2-B(x,u)ux-C(x,u) respectively,we apply the generalized conditional symmetry method and the sign-invariant method,to study the symmetry reductions,sign-invariants and exact solutions of the nonlinear difusion equation with variable coefficients f(x)ut=(g(x)D(u)ux)x+h(x)P(u)ux+q(x)Q(u).3.Via the invariant-subspace method associated with the generalized condi-tional symmetry method,we consider the nonlinear difusion equation with vari-able coefficients ut=(g(x)D(u)ux)x+h(x)P(u)ux+q(x)Q(u) where the nonlinear generalized conditional symmetry with characteristicη(x,u)=[k(u)]nx+a1(x)[k(u)](n-1)x+…+an(x)k(u)(n≤5,n∈Z+). By the transformation v=k(u),we find the corresponding new equation vt=9(x)A(v)vxx+g(x)B(v)v2x+[g’(x)A(v)+h(x)C(v)]vx+q(x)E(v) admit the linear generalized conditional symmetry with characteristic σ(x,v):vnx+a1(x)v(n-1)x+…+an(x)v(n≤5,n∈Z+). Combining the compatibility of σ=0with the corresponding equations,the symmetry reductions and classifications of the new equations are obtained.Exact solutions with the generalized separable variable in the form are constructed in the invariant-subspaces By the map v=k(u), these exact solutions can be transformed to the generalized functionally separable solutions of the original equations that admit the nonlinear generalized conditional symme-try with characteristic η.