Initial-value methods for the basic boundary-value problem and integral equation of radiative transfer

Abstract

We consider the problem of radiative transfer in a plane-parallel atmosphere which absorbs radiation and scatters it isotropically, there being a continuous distribution of primary internal sources of radiation. Our aim is to produce an effective numerical technique for calculating the internal intensity and the source functions. A major consideration is the computer's ability to solve initial-value problems for large systems of ordinary differential equations. We derive new differential-integral equations for the internal intensity at a fixed point and direction, and the source function at a fixed point, the new independent variable being the thickness of the atmosphere. Using standard quadrature techniques, we approximate these equations by systems of ordinary differential equations whose numerical treatment is straightforward. This work may be viewed as a new approach to the study of integral equations and linear unstable two-point boundary-value problems.