On the group classification of variable-coefficient nonlinear diffusion–convection equations

We consider the variable coefficient diffusion–convection equation of the form f(x)ut=[g(x)D(u)ux]x+h(x)K(u)uxf(x)ut=[g(x)D(u)ux]x+h(x)K(u)ux which has considerable interest in mathematical physics, biology and chemistry. We present a complete group classification for this class of equations. Also we derive equivalence transformations between equations that admit Lie symmetries. Furthermore, we obtain mappings that connect variable and constant coefficient equations. Exact solutions of special forms of this equations are constructed using Lie symmetries and equivalence transformations.