Formulation of a Ritz-Galerkin type procedure for the approximate solution of the neutron transport equation

• Published in: Journal of mathematical analysis and applications Vol. 50; no. 1; pp. 42 - 65
• Main Authors: Kaper, Hans G, Leaf, Gary K, Lindeman, Arthur J
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.1975
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/0022-247X(75)90037-2

The one-group neutron transport equation is commonly given as an integrodifferential equation for the neutron density ψ( x, ω) over a domain G × S in the five-dimensional phase space E 3 × S(¦ ω ¦ = 1) . In this paper we show how, by decomposing the domain of the transport operator into a complementary pair of manifolds by means of a projection operator, any transport problem can be formulated, on either manifold, in terms of a symmetric positive definite operator. We use Friedrichs' method to extend the operator to a selfadjoint operator and look for a generalized solution by minimizing a certain functional over the appropriate Hilbert space. A Ritz-Galerkin type approximation procedure is formulated, and an estimate for the difference between the exact and approximate solution is given. The procedure is illustrated for a special choice of finite dimensional subspace.