R. L. Bowden

Solution of the Initial-Value Transport Problem for Monoenergetic Neutrons in Slab Geometry

The initial-value transport problem of monoenergetic neutrons migrating in a thin slab is solved by applying the normal-mode expansion method of Case to the results of Lehner and Wing. Fredholm integral equations are derived for the expansion coefficients. In addition, exact expressions for the eigenvalues of the problem are derived and the results of calculations are presented. The solution is shown to have properties expected from elementary diffusion theory.

Multigroup neutron transport. I. Full range

A functional analytic approach to the N‐group, isotropic scattering, particle transport problem is presented. A full‐range eigenfunction expansion is found in a particularly compact way, and the stage is set for the determination of the half‐range expansion, which is discussed in a companion paper. The method is an extension of the work of Larsen and Habetler for the one‐group case.

on the solutions to a class of nonlinear integral equations arising in transport theory

Existence and uniqueness for the solutions to a class of nonlinear equations arising in transport theory are investigated in terms of a real parameter α which can take on positive and negative values. On the basis of contraction mapping and positivity properties of the relevant nonlinear operator, iteration schemes are proposed, and their convergence, either pointwise or in norm, is studied.

On the solution of nonlinear matrix integral equations in transport theory

The coupled nonlinear matrix integral equations for the matrices X (z) and Y (z) which factor the dispersion matrix Λ (z) of multigroup transport theory are studied in a Banach space X. By utilizing fixed‐point theorems we are able to show that iterative solutions converge uniquely to the ’’physical solution’’ in a certain sphere of X. Both isotropic and anisotropic scattering are considered.

Solution of the Transport Equation with Anisotropic Scattering in Slab Geometry

Some systematics which exist between eigenfunctions and adjoint singular integral equations arising in the solution of the transport equation in slab geometry are illustrated. The transport equation is shown to obey a singular integral equation and its relationship to the eigenfunction expansion‐method solution is shown. A new method for solving for the expansion coefficients in the eigenfunction expansion method is illustrated by solving Milnes problem. The role adjoint singular integral equations play in finding appropriate weight functions for use in orthogonality relations between the eigenfunctions of the transport equation is briefly discussed.

A note on the iteration of the Chandrasekhar nonlinear H‐equation

An iteration scheme to solve the Chandrasekhar H equation in the form H (μ) ={1−μ F 10[Ψ (s) H (s)]/(s +μ) ds}−1 is shown to converge monotonically and uniformly.

Resolvent integration techniques for generalized transport equations

A generalized class of ’’transport type’’ equations is studied, including most of the known exactly solvable models; in particular, the transport operator K is a scalar type spectral operator. A spectral resolution for K is obtained by contour integration techniques applied to bounded functions of K. Explicit formulas are developed for the solutions of full and half range problems. The theory is applied to anisotropic neutron transport, yielding results which are proved to be equivalent to those of Mika.

Solution of the Initial‐Value Neutron‐Transport Problem for a Slab with Infinite Reflectors

The initial‐value transport problem of monoenergetic neutrons migrating in a thin slab surrounded by infinitely thick reflectors is solved for isotropic scattering by using the normal‐mode expansion technique of Case. The results obtained indicate that the reflector may give rise to a branch‐cut integral term typical of a semi‐infinite medium while the central slab may contribute a summation over discrete residue terms. Exact expressions are obtained for these discrete time eigenvalues, and sample numerical results are presented showing the behavior of real time eigenvalues as a function of the material properties of the slab and reflector. In the limit of purely absorbing reflectors or a bare slab, the present solution has the properties which have been previously reported by others who used the approach of Lehner and Wing.

Analysis of the dispersion function for anisotropic longitudinal plasma waves

An analysis of the zeros of the dispersion function for longitudinal plasma waves is made. In particular, the plasma equilibrium distribution function is assumed to have two relative maxima and is not necessarily an even function. The results of this analysis are used to obtain the Wiener–Hopf factorization of the dispersion function. A brief analysis of the coupled nonlinear integral equations for the Wiener–Hopf factors is also presented.

Conservative neutron transport theory

A functional analytic development of the Case full‐range and half‐range expansions for the neutron transport equation for a conservative medium is presented. A technique suggested by Larsen is used to overcome the difficulties presented by the noninvertibility of the transport operator K−1 on its range. The method applied has considerable advantages over other approaches and is applicable to a class of abstract integro–differential equations.