An Iterative Technique for Obtaining Solutions of a Thomas–Fermi Equation

Abstract

A convergent iteration scheme is given for the boundary value problem $y'' = x^{{{ - 1} /2}} y^{{3 /2}} $, $y(0) = 1$, $ - y(b) + by'(b) = 0$. An iteration scheme based on eigenpairs of Hilbert–Schmidt operators obtained from a Green’s function representation for solutions of a linearization of the eigenvalue problem $u'' = \lambda x^{{{ - 1} /2}} u^{{3 / 2}} $, $ - \alpha u(0) + u'(0) = 0$, $ - u(1) + u'(1) = 0$ is shown to converge to a solution. The solution of the Thomas–Fermi equation for $b = \lambda ^{{2 / 3}} $ is then given by $y(x) = u(\lambda ^{{{ - 2} / 3}} x)$. Since the linear operators involved are Hilbert–Schmidt, the iteration lends itself to implementation through Galerkin methods.