Improved Solution of the Integral Transport Equation Across a Plane Boundary

Abstract

The method of singular derivatives is used to solve the mono-energetic integral transport equation in the vicinity of a plane boundary, by expanding the flux density in functions appropriate to this region. These are the "polylog" functions - integer powers of combinations of x and log x, x being the spatial variable measured in mean-free-paths. To obtain convergence of an infinite series, the formalism is transformed into a series of orthogonal (shifted Gegenbauer) polynomials of arbitrary order, and the problem then reduces to the solution of K simultaneous linear algebraic equations, where K is related to the truncation order of the polylogs. The method is useful for contiguous regions of arbitrary thickness, but breaks down far from the boundary, and for large values of K. Some numerical results are reported.