Certain Reaction Diffusion Separation Of Variables And Exact Solution Of The Equation (s)

It is well known that the method of separation of variables is an efficient way to solve linear partial differential equation(PDE). For the nonlinear PDE, there also have many method to obtain separation solutions, such as the Lie symmetry, differential Stackel matrices method, direct method, anstaze approach, geometric method, condition symmetry, generalized condition symmetry, formal variable separation approach and so on.In the chapter 2, we find the separation of variables solutions for nonlinear reaction diffusion equation (2.4) by utilizing the group foliation method. We introduce a automorphic systemLet G(x,u) = g(x)h(u), then (2.4) implies a polynomial which each term in the expression is a product of u-dependent function and x-dependent function. Our aim is to find all set of functions A(x),B(x),g(x),D(u),Q(u),and h(u) that the original equation has functional separable solutionsWe know that the model from the real world not only single evolution equations, but also many models are system. So there are many papers devoted to the investigation of existence and uniqueness problems, asymptotic behaviour of solutions, and so on. On the other hand, there are only a few papers devoted to the searcher for exact solutions of system of the form(3.6). In chapter 3, we utilize the sign-invariant theory to study the system (3.6) which introduced originally by Galaktionov et al. We consider the exact solutions for system (3.6), and we obtain some separable solutions of the corresponding system.