Thomas-Fermi Equation Solution by the Differential Analyzer

Abstract

A numerical solution of the Thomas-Fermi equation $\frac{d^2\phi }{d{x}^2}=\frac{{\phi }^{\frac{3}{2}}}{x^{\frac{1}{2}}}$, with the boundary conditions $\phi \mathrm{}\left(0\right)=1\mathrm{}$ and $\phi \left(\infty \right)=0\mathrm{}$, is presented, as obtained mechanically by means of the Differential Analyzer. This device for solving ordinary differential equations is described in a separate paper in the Journal of the Franklin Institute. The results are given for larger values of argument than have previously been reported, and at high values of argument precision has been improved. Over a part of the range, previously published results are checked, and the entire range is checked by an independent integration.