Exact solutions of nonlinear diffusion-convection equations

We consider the class of nonlinear diffusion-convection equations which contain arbitrary functions of the dependent variable. We perform a thorough symmetry analysis of the general equation in one, two and three spatial dimensions. We identify all special forms of the two arbitrary functions which admit special symmetry properties and for these cases, attempt to reduce the governing equation to an differential equation. We show that reduction of the governing equation an ordinary differential equation is possible in many cases. We seek solutions the reduced equations and hence are able to construct time-dependent similarity solutions to the governing equation. We extend a previously derived method for reducing a power law case of our governing equation through our knowledge of the symmetries of the class of equations. As a result, we construct an infinite family of time-dependent solutions satisfying nonsingular initial conditions for special cases of the governing equation in both and three dimensions. Finally, we develop an inverse method by exploiting a linearisable form of our governing equation in one spatial dimension. The method is used to derive two solutions with distinct variable flux boundary conditions for an unsaturated soil