Analytic Error Estimates for Integral-Equation Eigenvalue Problems in Neutron Physics

The technique usually employed to estimate errors in approximation schemes for neutron physics problems is simply to compare the results with higher order approximations or purely numerical results, or with available experimental measurements. In this paper, an analytic error-estimating technique is developed for deriving error bounds for approximate eigenvalues, which depends only on the proximity of the exact and approximate eigenvalues and not on higher order approximations. An integral equation formulation is employed in developing the error estimating method, but the form of the integral equation kernel is not restricted, so that broad classes of integral equations may be treated. By means of the Green's function, differential-equation eigenvalue problems may also be handled. To illustrate the error estimating method, the space decay constant eigenvalue problem of neutron thermalization theory is discussed. Error bounds are developed for the space decay constant eigenvalues in both the Wilkins heavy-gas differential equation and Wigner-Wilkins integral-equation scattering models. The results obtained indicate that rigorous error estimates can be obtained with little computational effort.