Numerical solutions of the time-dependent thermalization problem in infinite 1/ν poisoned media as well as in finite media in the diffusion approximation have been obtained using an eigenfunction expansion of the neutron-density function in a discrete-energy representation. This eigenfunction method is compared with a method based on direct integration of the Boltzmann equation using a discrete-energy mesh for the scattering integral and a first-order Taylor series for the time integration. Both methods of calculation have given the same results where compared in the area of time-dependent and steady-state spectra. The Wigner-Wilkins Mass-1 and Nelkin scattering models have been used with particular emphasis on the computation of time-dependent, asymptotic, steady-state spectra and diffusion parameters and the determination of their sensitivity to the scattering kernel. It is found that time-dependent spectra are rather sensitive to the scattering kernel, particularly at times of the order of a few microseconds after the introduction of a neutron pulse in the case of hydrogenous moderators. The eigenvalues and eigenfunctions for both realistic scattering kernels show the characteristics predicted for simpler analytic models. Both discrete and continuum eigenvalues have been found with the eigenfunctions corresponding to the continuum eigenvalues exhibiting a characteristic singular behavior. An interpolation scheme to determine steady-state spectra in hydrogenous moderators is also presented. The method, which is based on interpolating in the reciprocal of the infinite-medium neutron lifetime, gives very good agreement with directly computed spectra in the range of 200 to 15 microseconds lifetime. A perturbation method based upon the infinite-medium eigenfunctions is used to compute diffusion parameters for the decay constant in water; this method, through terms in B4, yields the decay constant to better than 1% in comparison with the exact diffusion theory result for B2 = 1.0 cm-2.