Nystrom-product integration for wiener-hopf equations with applications to radiative transfer

We discuss the numerical solution of the Wiener-Hopf integral equation u(x)-∫0∞k(x-t)u(t)dt=f(x), where k(x) has logarithmic singularity at x = 0, and decays exponentially as |x|→∞. An approximate solution u_n is defined by introducing a mesh with n subintervals on [0, ∞], and then approximating the truncated integral operator ∫0β(n)k(x-t)u(t)dt (where β(n) is some mesh point) by using a composite m-point quadrature rule. Because of the weak singularity in K, product integration is employed near t = x. We show that, under certain conditions which require the mesh and β(n) to be carefully chosen, {norm of matrix}u -u_n{norm of matrix}∞ = O(1/n^P) where m ≤p <m + 1 and p depends on the order of the quadrature rule. These results are illustrated by the numerical solution of a Wiener-Hopf equation from classical radiative transfer.