Mathematical study of the solutions of a diffusion equation with exponential diffusion coefficient: explicit self-similar solutions to some boundary value problems

Abstract

Explicit self-similar solutions are systematically obtained for a nonlinear diffusion equation with a diffusion coefficient that depends exponentially on the transported magnitude (D(u)=K1ealpha u), for some boundary conditions associated with the parameter lambda of the transformation group. Group elements under which the Hamiltonian of this physical system remains invariant enable one to find new magnitudes which remain invariant under transformation for each specific lambda value in turn. This allows physical properties of the system to be established. The sign of alpha limits the possibility of obtaining self-similar solutions in any kind of diffusion process for those group elements belonging to the open interval (0, 1/2); if alpha >0 then only self-similar solutions for absorption processes will be found and if alpha <0, only the cession processes will possess self-similar solutions. The value lambda =0 leads to an analytic solution to the problem with free boundaries (the Stefan problem).