Similarity temperature profiles for some nonlinear reaction - diffusion equations

A relevant tool in the study of the closed - form solutions of reaction - diffusion equations is the concept of phase - plane. The aim of this paper is to apply this approach to some simplified models of microwave heating problems. We investigate two models: one is power law dependence and the other is exponential dependence. In both cases, we assume a heat source term with spatial polynomial decay but increasing with temperature. The spatial polynomial decay is known to have been applied earlier under some physically reasonable assumptions. In particular, the solutions obtained permit us to compare the contribution of the heat source term and geometry. The results of this analysis are in keeping with what others have observed with nonlinear diffusion; they also apply to numerous equations which have hitherto not been studied.