Exact Solutions of Diffusion–Convection Equations

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=0. 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 (f = g = 1, B = 0) are well investigated and some of exact solutions given below were summarized before in [26,48]. Our paper is organized as follows. First of all we adduce solutions of the linear heat equation obtained by means of various symmetry methods. In Section 3 the linearizable Burgers, Fujita–Storm and Focas–Yortsos equations are considered. Lie reduction of constant coefficient nonlinear diffusion equation (hB = 0, f = g = 1) is performed in Section 4. Solutions of constant coefficient diffusion equations with exponential nonlinearity are adduced in Section 5. Solutions of constant coefficient diffusion equations with power nonlinearity are presented in Section 6. The important particular case of such equations, namely, the fast diffusion equation, is studied in more detail in Section 8. Diffusion equations with other nonlinearities are briefly discussed in Section 9. The next considered case (Section 10) covers the nonlinear constant coefficient diffusion–convection equations (f = g = h = 1). In Section 11 we adduce a brief analysis of known solutions of n-dimensional radially symmetric nonlinear diffusion equations. In Section 12 exact solutions of some variable coefficient diffusion–convection equations are collected. At last, in Sections 13 and 14 we present a detailed analysis of interesting variable coefficient equations having distinguished invariance properties. In the Appendix A we adduce the complete results of group classification of equations (1) with respect to the extended group Ĝ∼ of equivalence transformations (22). Below, if it is not indicated separately, α, εi, λ, a, b, c, ci are arbitrary constants, ε = ±1. For convenience we use double numeration T.N of classification cases and local equivalence transformations, where T denotes the number of table and N does the number of case (or transformation, or solution) in table T. The notion “equation T.N” is used for the equation of form (1) where the parameter-functions f , g, h, A, B take values from the corresponding case.