Solution methods for discrete-state Markovian initial value problems

Solution methods, both numerical and analytical, are considered for solving the Liouville master equation associated with discrete-state Markovian initial value problems. The numerical method, basically a moment (Galerkin) method, is very general and is validated and shown to converge rapidly by comparison with an earlier reported analytical result for the ensemble-averaged transmission of photons through a purely scattering statistical rod. An application of the numerical method to a simple problem in the extended kinetic theory of gases is given. It is also shown that for a certain restricted class of problems, the master equation can be solved analytically using standard Laplace transform techniques. This solution generalizes the analytical solution for the photon transmission problem to a wider class of statistical problems.