Laguerre pseudospectral approximation to the Thomas-Fermi equation

We propose an iterative method to solve the nonlinear Thomas-Fermi equation based on Laguerre pseudospectral approximation. Different from the direct use of pseudospectral methods for solving the problem in the literature, we represent the solution of the Thomas-Fermi equation as sum of two parts due to its singularity at the origin. Both parts regard the initial slope as a parameter unknown to be determined. One "singular" part is a power-series expansion. The other "smooth" part satisfies a nonlinear two-point boundary value problem. We use a Laguerre pseudospectral method to discretize and solve it numerically by the Newton iteration. Moreover, an introduced new boundary condition is employed innovatively to construct an outer iteration to determine the initial slope. We show numerically this separation of solution as two parts is necessary for improvement of accuracy by direct Laguerre approximations of the Thomas-Fermi equation. Comparisons of our numerical results with those obtained by other methods show effectiveness and accuracy of our approach.