An analytic solution to the Thomas–Fermi equation

Abstract

A perturbative procedure due to Bender et al. (here referred to as the BMPS procedure) [J. Math. Phys. 30, 1447 (1989)] and useful in solving difficult nonlinear problems, has been used here to solve the Thomas–Fermi (T–F) equation. The present work attempts to balance the ease of the ensuing analysis with the use of an analytic, zero-order function that already contains a good deal of the nonlinearity of the T–F equation. The initial slope of the T–F potential is computed with 0.35% error in a second-order application of the theory.

Comment on: “Series solution to the Thomas–Fermi equation” [Phys. Lett. A 365 (2007) 111]

“…However, one is surprised that with so many adjustable parameters the results are far from impressive, even at remarkable great perturbation orders [16,17]. For example the [30/30] Padé approximant of the HAM series yields u (0) with three exact digits [17], while the [1/1] Padé approximant of the δ expansion [28] provides slightly better results [29,30]. A more convenient expansion of the solution of the TF equation leads to many more accurate digits [10,11] with less terms.…”

An adaptive algorithm for the Thomas–Fermi equation

“…It plays an essential role to calculate accurately the initial slope. The perturbative approach [6][7][8] usually reduces the ThomasFermi model to a series of linear boundary value problems and produces a convergent analytic series solution. In [9], Fernández et al applied the powerseries expansions of the solution of the Thomas-Fermi equation for small and large values of coordinate and matched them.…”

On the rational second kind Chebyshev pseudospectral method for the solution of the Thomas–Fermi equation over an infinite interval

“…The Thomas-Fermi problem has been solved by different techniques. The authors of [26][27][28] used a perturbative approach to determine analytic solutions for the Thomas-Fermi equation. Bender et al [26] replaced the right-hand side of this equation by one which contains the parameter δ, i.e., y ′′ (…”

“…(1) into a set of linear equations with associated boundary conditions. Laurenzi [27] applied a perturbative method by combining it with an alternate choice of the nonlinear term of Eq. (1) to produce a rapidly converging analytic solution.…”