Ann. nucl. Energy, Vol. 18, No. 3, pp. 147 154, 1991 0306-4549/91 $3.00+0.00 Printed in Great Britain. All rights reserved Copyright ~ 1991 Pergamon Press plc A FAMILY OF TRANSPORT EQUATIONS IN NEUTRON TRANSPORT THEORY ANIL K. PRINJA Chemical and Nuclear Engineering Depactment University of New Mexico Albuquerque, New Mexico 87131 United States of America (Received 13 October 1990) Abstract -- The backward formulation of problems in neutron transport is considered. Boundary conditions are developed from the adjoint system which in turn is rigorously obtained from a variational principle. The variational principle is derived by constraining the flux with the forward transport equation and aff'diated boundary/initial conditions, avoiding physically motivated Importance arguments. Extremising the constrained functional yields both the forward and adjoint equations as well as boundary/'mitial conditions as the Euler Lagrange system. The backward formulation provides a means of developing exact equations for the scalar flux and emergent current as well as internal distributions such as the slowing down density. I. INTRODUCTION The role of the variational principle in providing accurate estimates of flux-weighted integral quantities and as a Lagrangian which yields the transport equation as the correspond- ing Euler-Lagrange system (Davison,1957; Becker,1964; Kaplan,1969; Bell and Glasstone, 1970; S tacey, 1973) is well established in transport theory. Variational principles have also had a profound impact on the development of approximate solution methods, such as the finite element method in reactor physics applications (Kaplan and Davis,1967; Kang and Hansen,1973; Martin and Duderstadt,1977; Henry,1975). Success in this area has been particularly noteworthy for self-adjoint problems where powerful theorems can be rigorously proved and error estimates obtained (Duderstadt and Martin, 1979). However most problems of practical interest are non self-adjoint (eg described by the energy dependent transport equation for non-thermal neutrons) for which similar general results are not available and one generally resorts to Galerkin methods. Another complicating feature in non-self adjoint cases is the need for trial functions for the adjoint flux. These are not as easily constructed as for the forward flux for which physically motivated approximation schemes such as diffusion theory provide acceptable trial functions. Such reasoning does not carry over readily to the adjoint system. In this paper we demonstrate a related and useful role for the variational principle, namely as a link between forward and backward transport descriptions. The backward formulation has not received widespread attention in neutron transport but has formed the approach of choice in the field of atomic collisions in solids since the early work of Lindhard et al. (1963). The true backward transport equation differs from the forward 147