(2 + 1) Dimensional Quasi Invariant Set And The Exact Solution Of Linear Heat Equation And High Dimensional Reaction-diffusion Equations

It is very important to obtain the exact solutions of partial different equa-tions,which widely appears in almost all the scientifc fields such as physics, chemistry,biology,ecomomy and so on.The invariant sets method is an effec-tive approach.In this paper,we use invariant sets to solve(2+1)-dimensional quasilinear heat equation with reaction term and higher order nonlinear reaction-diffusion equations.Firstly,we study(2+1)-dimensional quasilinear heat equation by constructing the functional invariant set E0={u:ux=vxf(t)F(u),uy=vyf(t)F(u)}. And we also consisder the generall invariant set E1={u:ux=vxf(t)F(u),uy=vy9(t)F(u)} in the subscase of v(x,y)=(x2+y2)/2,where,f(t)≠9(t).Secondly,we discuss higher-dimensional reaction-diffusion equations with scourse term ut=A1(u)uxx+A2(u)uyy+A3(u)uzz+B1(u)u2x+B2(u)u2y+B3(u)u2z+Q(u), and their solutions in terms of the invariant set S3={u:ux=F(u),uy=vyF(u),uz=uzF(u)}.Alternatively,we further extend this approach to(N+1)-dimensional equation. In other words,we study the equation by invariant set SN={u:ux=vxiF(u),i=1,2,…,N).We get some exact solutions of them, where v is a smooth func-tion of independent variables,F(u)is a smooth function to be determined, A(u),B(u),C(u),D(u),Q(u),Ai(u),Bi(u)(i=1,2,3)are the smooth func-tions of u.