Asymptotic solution of neutron transport problems for small mean free paths

A method is presented for solving initial and boundary value problems for the energy dependent and one speed neutron transport equations. It consists in constructing an asymptotic expansion of the neutron density ψ(r, v, τ) with respect to a small parameter ε, which is the ratio of a typical mean free path of a neutron to a typical dimension of the domain under consideration. The density ψ is expressed as the sum of an interior part ψi, a boundary layer part ψb, and an initial layer part ψ0. Then ψi is sought as a power series in ε, while ψb decays exponentially with distance from a boundary or interface at a rate proportional to ε−1. Similarly ψ0 decays at a rate proportional to ε−1 with time after the initial time. For a near critical reactor, the leading term in ψi is determined by a diffusion equation. The leading term in ψb is determined by a half-space problem with a plane boundary. The initial and boundary conditions for the diffusion equation are obtained by requiring ψ0 and ψb to decay away from the initial instant and from the boundary, respectively. The results are illustrated by specializing them to the one speed case. The method may make it possible to treat more realistic and more complex problems than can be handled by other methods.