Solutions of the radiation diffusion equation

It has been generally accepted that it is difficult to obtain analytic solutions of the radiation diffusion equation because of its strong nonlinearity. When attempting to obtain solutions for cases where a heat front is defined this belief may have arisen from attempts to impose boundary conditions inappropriately. For situations in which material properties can be represented as power laws of the temperature (with energy density going as Tβ and opacity as T−α) and material motion is neglected, the difficulty is removed by using the velocity of the heat front as the boundary condition, and working in the parameter U = T4+α−β. Using this prescription solutions are obtained which make no reference to a heated surface. A surface is then imposed so that the solutions can represent radiation waves flowing into initially cold material. Some features of the general solution are discussed before looking at a self-similar subset of the solutions and an approximate form these adhere to. It is shown that the approximations continue to accurately predict both the penetration depth of a radiation wave and the total amount of energy contained in the profile for a wide range of other solutions.