On finding solutions of kinetic problems

Abstract

RECENTLY a number of papers has been published concerned with the investigation of iterative methods far the numerical solution of kinetic problems, which are related to finding solutions of the neutron transport equation; as an example we may mention [1–5]. It is obvious that we cannot investigate the convergence of these methods sufficiently fully, determine the region of their application, the value of each algorithm put forward [2, 4] and find inexpensive algorithms, without knowing sufficiently well the properties of the exact solutions for the transport equation. The mathematical basis of many standard one-group kinetic problems was first given in [6–8]. In [9] a complete system of general functions, which are solutions of a one-group homogeneous kinetic equation was found for a plane layer. Analytic solutions of various problems are given in [10]. To clarify the nature of the approximation of solutions of various problems to exact solutions, we find for a number of problems the solutions of the kinetic equation written in self-conjugate form [8]. In Section 2 we extend the results of [9] to the q-dimensional case and in Section 4 to difference equations. In Section 3 we find a fundamental solution for an re-group problem, and in Section 5 give a solution of an m-group periodic problem. In Section 6 one of the variants of the KP-method [2–5] is given for solving m-group problems. For a periodic problem we find the value of the KP-method and show that in practice the KP-method cannot be improved.