THE METHOD OF ULTRASPHERICAL HARMONICS IN PARTICLE-TRANSPORT

Traditional methods of solving the transport equation commonly employ an expansion in spherical harmonics. Other choices for the expansion functions have been investigated in the past, but have not been fully explored due to the algebraic complexity associated with both the simplification procedures and the addition formula, although there are good theoretical reasons for thinking that better approximate solutions will emerge. In this paper we obtain the first eight coupled differential equations for the transport equation with anisotropic terms in plane geometry using Chebyshev polynomials, and employing computer algebra as a means of simplification. The generalization to ultra-spherical (Gegenbauer) polynomials is discussed, and some numerical examples are given.