Rational Chebyshev series for the Thomas–Fermi function: Endpoint singularities and spectral methods

Abstract

We solve the Thomas–Fermi problem for neutral atoms, uyy−(1/y)u3/2=0 on y∈[0,∞] with u(0)=1 and u(∞)=0, using rational Chebyshev functions TLn(y;L) to illustrate some themes in solving differential equations on a semi-infinite interval. L is a user-choosable numerical parameter. The Thomas–Fermi equation is singular at the origin, giving a TL convergence rate of only fourth order, but this can be removed by the change of variables, z=y with v(z)=u(y(z)). The function v(z) decays as z→∞ with a term in z−3, which is consistent with a geometric rate of convergence. However, the asymptotic series has additional terms with irrational fractional powers beginning with z−4.544. In spite of the faster spatial decay, the irrational powers degrade the convergence rate to slightly larger than tenth order. This vividly illustrates the subtle connection between the spatial decay of u(x) and the decay-with-degree of its rational Chebyshev series. The TL coefficients an(L) are hostages to a tug-of-war between a singularity on the negative real axis, which gives a geometric rate of convergence that slows with increasingL, and the slow inverse power decay for large z, which gives quasi-tenth order convergence with a proportionality constant that decreasesinversely as a power of L. For L=2, we can approximate uy(0) (=vzz(0)) to 1 part in a million with a truncation N of only 20. L=64 and N=600 gives uy(0)=−1.5880710226113753127186845, correct to 25 decimal places.