A Successive Approximation Approach to Problems in Radiative Transfer with a Differential Formulation

Abstract

The radiation intensity in a gray participating medium is expressed in a differential form. The energy equation for radiative transfer becomes an infinite-order differential equation. Utilizing the method of weighted residuals and introducing some appropriate formulations for the intensity boundary conditions, a method of successive approximations is developed. The solution method is applied to a one-dimensional problem with linear-anisotropic scattering. This problem is chosen because of its practical importance and the availability of exact solutions. A first-order closed-form result, which has never been derived analytically before, is obtained and shown to have good accuracy. Successive higher-order approximate solutions are also presented. These solutions are easily attainable algebraically and converge quickly to the exact result. To illustrate the possible applicability of the solution method for multidimensional problems, the first-order solution to a simple two-dimensional problem is presented. Results show that based on the present approach, reasonably accurate approximate solutions can be generated with some simple mathematical developments.