Differential approximations for transient radiative transfer through a participating medium exposed to collimated irradiation

First, we apply the modified differential approximation (MDA) suggested by Chandrasekhar to transient radiative transfer in a scattering planar medium exposed to collimated pulse irradiation. Next, a hybrid method of the P₁/₃ approximation suggested by Olson and the MDA is developed. The hybrid method may be referred to as the modified P₁/₃ approximation (MP₁/₃A) and is also applied to the same example. Comparisons of the results obtained by solving the MDA, the MP₁/₃A and the exact integral equation are made. The comparisons show that the temporal distribution of the transmissivity obtained by the MDA contains a small protuberance or an abrupt slope change, which decreases with the decrease of the scattering albedo. The results obtained by the MP₁/₃A are more accurate than those obtained by the MDA for most of the cases considered, because the MP₁/₃A corrects the propagation speed of the transmitted radiation.