Elementary Solutions of the Reduced Three-Dimensional Transport Equation

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Abstract
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.