An explicit analytic solution to the Thomas-Fermi equation.
type: Item
instance of: scholarly article
title: An explicit analytic solution to the Thomas-Fermi equation.
zbMATH Open document ID: 1034.34005
DOI: 10.1016/S0096-3003(02)00423-X
published in: Applied Mathematics and Computation
publication date: 2003-08-25T00:00:00Z
review text: The author considers the boundary value problem for the Thomas-Fermi equation \[ u'' (x) = \sqrt{u^{3} (x) / x}, x > 0, \quad u (0) = L, \quad u (+\infty ) = 0. \tag{1} \] With the homotopy analysis method the explicit analytic solution to problem (1) is given. This solution has the form \[ u (x) = \sum_{k=0}^{\infty } \sum_{n=1}^{4k+1} \alpha _{k, n} (1+x)^{-n},\tag{2} \] where the coefficients can be calculated successive with the help of recurrence formulae. The author gives the results of calculations, when, in place of the precise solution to (2), the corresponding \(m\)th-order approximation \(u(x) = \sum_{k=0}^{m } \sum_{n=1}^{4k+1} \alpha _{k, n} (1+x)^{-n}\) for \(m=40, 60\) is used.
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 1970655
zbMATH Keywords: explicit analytic solution
zbMATH Keywords: Thomas-Fermi equations
zbMATH Keywords: nonlinear ordinary differential equation
author: Shijun Liao
reviewed by: Vladimir L. Makarov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1970655