The characteristic equation of radiative transfer theory

Abstract

The investigation of radiative transfer processes in turbid media of different configurations is often reduced to solving boundary-value problems for the integrodifferential radiative transfer equation (RTE) [1--12] or corresponding integral equation (IE) for source functions [5, 6, 8]. Finite-difference methods [9--13], iteration techniques [10, 14], doubling method [4, 7] and Monte Carlo method [15, 16] are widely used to solve the abovementioned equations. An important role in developing rigorous analytical and approximate (in particular, numerical) methods for solving radiative transfer boundary-value problems for the case of a plane-parallel turbid medium was played by an heuristic approach first advanced in [2, 17] based on using classical invariance principles (CIP). More abstract interpretation of these principles allows one to formulate the classical variant of the invariance embedding method [18] and to use it to reduce boundary-value problems of radiative transfer theory (RTT) and other sections of mathematical physics to solving Cauchy problems. These problems are more suitable for the use of numerical techniques as compared to a starting boundary-value problem. The various applications of this method were given, for example, in [5, 18--22]. Case's method [10, 23, 24], integral transformation technique [5, 6, 25--30] and Viener-Hopf's method [31] should be pointed out among methods that allow one to obtain rigorous and asymptotic solutions of RTT problems for the cases of plane-parallel, spherical and cylindrical symmetric media.