On two integro-differential equations arising in particle transport theory

Abstract

Two integro-differential equations arising in particle transport theory are solved explicitly using a technique involving difference equations. The physical problems to which these equations apply concern the energy-time and energy-space distributions of fast particles (neutrons, atoms, gamma -rays, etc.) as they slow down in a host medium. One of the equations involves the first-order derivative with respect to time or space and describes particles which scatter essentially in the forward direction. The other equation assumes a diffusive motion with almost isotropic scattering and hence involves a second-order space derivative. Solutions are obtained in heterogeneous media where the number density of scatterers varies continuously in space and also for a series of contiguous slabs in which the material properties remain constant but change discontinuously from slab to slab. The slowing-down density and energy deposition functions are discussed and evaluated explicitly in some special cases.