Solutions of the steady, one-speed neutron transport equation for small mean free paths

The solution of the steady, one-speed neutron transport equation with isotropic scattering and small mean free path is obtained. The solution is asymptotic with respect to a small parameter ε, defined as the mean free path in terms of a unit length of the same order of magnitude as a typical dimension of the domain. The solution, the leading two terms of which are given, consists of a boundary layer solution plus an interior solution. The boundary layer solution decays exponentially with distance from the boundary, the decay rate being proportional to ε−1, and it shows the effects of boundary curvature and variations in the incoming flux along the boundary. The interior solution is a multiple of the source for subcritical domains, and depends on a diffusion equation for near critical domains. The boundary condition for the diffusion equation and an asymptotic criticality condition are derived.