Positive solutions and subcriticality of energy dependent transport systems in slab geometry

Abstract

The energy dependent transport system in an anisotropic medium in slab geometry subjecting to internal source and incoming fluxes is investigated. The investigation is based on a corresponding integral equation of the boundary value problem from which a recursion formula for the determination of the solution is obtained. It is shown by using the notion of an upper solution that the convergence or divergence of the sequence of iterations depends solely on the existence or nonexistence of an upper solution. Through the construction of a suitable upper solution one can obtain an explicit estimate for the value of c, which represents the average number of secondary neutrons per collision, in terms of the (optical) slab length 2a so that the system is either critical or subcritical. It is shown in particular that if c<[1−E2(a)]−1, where E2(a) is the exponential integral of order two, then the integral equation has exactly one nonnegative solution for any nonnegative source and incoming fluxes. This result insures the subcriticality of the system as well as a constructive recursion formula for the determination of the solution. Estimates for a more general system and the energy independent system are also obtained.