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

TL;DR

This paper derives and analyzes analytic solutions for the nonlinear radiation diffusion equation with a point source in non-homogeneous media, revealing diverse behaviors depending on spatial density profiles and confirming results with numerical simulations.

Contribution

It generalizes known solutions for homogeneous media to non-homogeneous cases with power law density profiles, exploring different solution behaviors and front dynamics.

Findings

Solutions vary qualitatively with the spatial exponent.

Conduction fronts can be constant speed or accelerate depending on parameters.

Analytic solutions agree well with numerical simulations.

Abstract

Analytic 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 the conduction front has constant speed or even accelerates. Finally, the various solution forms are compared in detail to numerical simulations, and a good agreement is achieved.