On equiconvergence of ultraspherical polynomials solution of one-speed neutron transport equation

Ultraspherical polynomial approximation is used in slab criticality calculations for strongly anisotropic scattering. A unique and general formulation is developed for slab criticality condition whose sub-cases are spherical harmonics approximation, Chebyshev polynomial approximation of first and second kind. Since Legendre polynomials, Chebyshev Polynomials of first and second kinds are special cases of ultraspherical polynomials; our formulation inherently covers all these approximations and lets one to employ any other ultraspherical polynomial approximation in the solution of one-speed neutron transport equation. Our calculations showed that solution of one-speed neutron transport equation for various degrees of anisotropy and cross-section parameters is almost insensitive to the choice of ultraspherical polynomial with the present days’ computing capabilities. In other words, as much as high order ultraspherical polynomial approximation is used the solution converges to the same value for a specified problem regardless the type of the ultraspherical polynomial assigned in the solution as equiconvergence theorem of Jacobi polynomials states.