About an analytical application of the spherical symmetric diffusion equation (Q1938474)

A time-dependent solution of the diffusion approximation of the radiative transfer equation (see, e.g. [\textit{G. C.\ Pomraning}, The equations of radiation hydrodynamics. Pergamon Press (1973)]) in a hot (\(10^7\) K) plutonium sphere close to thermodynamic equilibrium is derived. Therefore the authors focus on solving the radiative transport problem analytically for high-temperature matter with normal particle densities (no numbers given) and sphere radii (5 cm, 30 cm), which is embedded in a vacuum. Absorption and diffusion coefficients depend only on the material properties. The radiation mean free paths are approximated within the frame of a hydrogen-like atomic model. The initial equation describing the radiative diffusion problem is separated into two nonlinear ordinary differential equations. The spatial and temporal boundary conditions are taken from the neutron transport theory introducing an additional extrapolation for solutions in a vacuum. While the time-dependent ordinary differential equation is solved analytically, the solution of the spatial-dependent equation is found on the basis of a (as mentioned) general valid successive iteration scheme [\textit{R. P.\ Kanwal}, Linear integral equations. Theory and technique.. New York-London: Academic Press (1971; Zbl 0219.45001)]. There the term in the differential equation which does not contain a derivative is first substituted by the black-body (zero-order) solution. After solving the obtained ordinary differential equation numerically, the result is used in the derivative-free term, etc.. It is shown that the numerical solution (after 7 iteration steps) is in rather good agreement with the second-order analytical solution. Nevertheless, as also mentioned by the authors, the convergence of the iteration scheme should be discussed in more detail in future.