Solution of the one-dimensional transport equation for diffusion length problem with higher order approximation of the Chebyshev polynomials of second kind

In this study, solution of the one-dimensional neutron transport equation has been investigated for the diffusion length problem. The diffusion lengths of one-speed neutrons in a homogeneous slab with isotropic scattering and without source have been computed numerically using higher order approximations of first the traditional Legendre polynomials (PN) method and then the Chebyshev polynomials of second kind (UN) method. For this purpose, higher order moments of equations of both methods have been obtained after deriving the one-dimensional transport equation. Then, so derived equations of moments have been solved together and linear differential equations with constant coefficients have been obtained for each order of approximations. Finally, diffusion lengths of the neutrons have been computed for various scattering parameters by taking the inverse of the smallest root of the characteristic equation corresponding to each resultant differential equation.