An integrodifferential equation for the two-dimensional reflection kernel

Abstract

Faber, Seth, and Wing [l] applied the method of doubling to the determination of the two-dimensional invariant imbedding operators (reflection, transmission, right-turn and left-turn) for a variety of discrete-directional models, including one in which motion of the underlying particles is allowed only in the two opposing directions along each of two mutually orthogonal axes (the four-compass-point model). In this work the application of the addition/doubling formulas was initiated at some “fundamental square,” for which the invariant-imbeclcling functions were determined on the physical basis of considering interactions only to first order in collisions (i.e., the possibility of an incident particle engendering two or more collisions in the fundamental square was neglected). The present work is intended as a continuation of that of [l], in that our