Exact solutions of diffusion-convection equations

Abstract

(1) where f = f(x), g = g(x), h = h(x), A = A(u) and B = B(u) are arbitrary smooth functions of their variables, f(x)g(x)A(u) 6. Our aim is not to give a physical interpretation of the solution of diffusion equations (that is too huge and cannot be reached in the scope of a short paper), but to list the already known exact solutions of equations from the class under consideration. However, in some cases we give a short discussion of the nature of the listed solutions. The majority of the listed solutions have been obtained by means of different symmetry methods, such as reduction with respect to Lie and non-Lie symmetries, separation of variables, equivalence transformations, etc. Let us note that the constant coefficient diffusion equations ( = g = 1, B = 0) are well