The b and h functions for integral equations with displacement kernels: A computational method and an application to radiative transfer

Abstract

Many problems in the theory of radiative transfer reduce to the solution of Fredholm integral equations with displacement kernels. Frequently, we are interested in the solutions of the Fredholm integral equations as well as certain functionals on the solution (reflection and transmission coefficients, etc.). Earlier it was shown that these functionals can be expressed algebraically in terms of the basic functions b and h. Normally, these functions are computed as solutions of an initial-value problem. Since they represent internal interactions due to isotropic illuminations, they are also solutions of a linear two-point boundary-value problem, which, unfortunately, s unstable. The purpose of this paper is to show that this unstable problem can be solved using a Gram-Schmidt orthogonalization scheme. This is demonstrated by making comparisons against earlier calculations using the initial-value method. In addition, the process is ideally suited to take advantage of multitasking on parallel processors.