Jacobi polynomials approximation to the one-speed neutron transport equation

Jacobi polynomials, the largest family of classical orthogonal polynomial sequence, are used to obtain eigenspectrum of one-speed neutron transport equation for strongly anisotropic scattering. Previously detailed Ultraspherical or Gegenbauer polynomials approximation including spherical harmonics, Chebyshev polynomials approximation of first and second kinds is a special case of Jacobi polynomials approximation in which the two variables of the Jacobi polynomials are equal. Eigenspectrum calculations using the so-called Jacobi polynomials approximation have demonstrated that plane symmetrical systems prefer Ultraspherical polynomials approximation since their well-known peculiarity or symmetry property meets the requirement of symmetry in angular flux with respect to spatial coordinates for every order of approximation. In the more general case when two variables of Jacobi polynomial are unequal, symmetry in eigenvalues disappear in the low-order approximations. However, it has been shown that symmetrical eigenvalue pairs are reached asymptotically as the order of approximation is increased for unequal variables too. Jacobi polynomials approximation is further applied to homogeneous slab criticality problem with strongly anisotropic scattering and reflected boundaries. Additionally, very useful analytical recursive relations for the calculation of various types of integrals involving Jacobi polynomials are derived which are needed in the eigenspectrum and criticality calculations.