ELEMENTARY SOLUTIONS OF REDUCED 3-DIMENSIONAL TRANSPORT EQUATION

By treating one of the space dimensions exactly and approximating the other two by the exp (-iB · r) assumption, which is suggested by asymptotic transport theory, it is possible to reduce the three-dimensional transport equation to an equation that is of one-dimensional form and that still contains details of the complete three-dimensional angular distribution. In this paper we develop the method of elementary solutions for the reduced transport equation in the case of time-independent, monoenergetic neutron transport in homogeneous media with isotropic scattering. The spectrum of the transport operator consists of a pair of discrete points if B2 is sufficiently small and a continuum which occupies a two-dimensional region in the complex spectral plane. The eigenfunctions possess full-range and half-range orthogonality and completeness properties, which are proved via the solution of two-dimensional integral equations using the theory of boundary-value problems for generalized analytic functions. As applications we solve the Green's function for an infinite homogeneous prism and the albedo operator for a semi-infinite homogeneous prism. Also discussed are possible generalizations of the method to more complicated forms of the reduced transport equation.