Analytic solutions of the nonlinear radiation diffusion equation with an instantaneous point source in non-homogeneous media

Analytical solutions to the nonlinear radiation diffusion equation with an instantaneous point source for a non-homogeneous medium with a power law spatial density profile are presented. The solutions are a generalization of the well-known solutions for a homogeneous medium. It is shown that the solutions take various qualitatively different forms according to the value of the spatial exponent. These different forms are studied in detail for linear and non-linear heat conduction. In addition, by inspecting the generalized solutions, we show that there exist values of the spatial exponent such that the conduction front has constant speed or even accelerates. Finally, various solution forms are compared in detail to numerical simulations, and a good agreement is achieved.