An adaptive algorithm for the Thomas–Fermi equation

Abstract

A free boundary value problem is introduced to approximate the original Thomas-Fermi equation. The unknown truncated free boundary is determined iteratively. We transform the free boundary value problem to a nonlinear boundary value problem defined on [0,1]. We present an adaptive algorithm to solve the problem by means of the moving mesh finite element method. Comparison of our numerical results with those obtained by other approaches shows high accuracy of our method.Keywords Thomas-Fermi equation · Adaptive finite element method · Moving mesh method · Equidistribution · Free boundary value problem · Semi-infinite interval

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

“…The approximations of y ′ (0) computed by the present method and their relative errors are shown in Table 1. Obviously, this method is convergent by increasing the number of collocation points and obtaining suitable L. The comparison of the initial slope y ′ (0) calculated by the present work with values obtained by Liao [31], Khan [32], Yao [41] and Zhu et al [35] is given in Table 2, which shows that the present solution is highly accurate. Table 3 shows the approximations of y(x) obtained by the method proposed in this paper for N = 7 and L = 0.0958885, and those obtained by Khan [32] and Liao [42].…”

“…(12) and (15) on Eq. (22) we define Kobayashi result [40] y ′ (0) = −1.588071 Table 2 Comparison between methods in [31,32,41,35] and the present method for y ′ (0). As in a typical tau method and using Eq.…”

“…Yao [34] solved the Thomas-Fermi equation by an analytic technique named the homotopy analysis method with a more generalized set of basis function, and consequential auxiliary linear operator was introduced to provide a series solution. Zhu et al [35] approximated the original Thomas-Fermi equation by a nonlinear free boundary value problem. They then used an iterative method to solve the free boundary value problem.…”

An efficient numerical method for solving nonlinear Thomas-Fermi equation

“…Comparison of methods: Zaitsev et al [40] showed that the Adams-Bashforth and Runge-Kutta methods to solve this equation on unbounded domains are ill-conditioned, hence, researchers have used the methods of numerical and semi-analytical for solving the equation, and some researchers can calculate very good solutions. For example, authors of [55,57,58,59,60,61,64,68,70] used the analytical methods for solving the equation and Amore et al [68] were able to calculate the best solution using Pade-Hankel method, correct to 26 decimal places. Authors of [54,56,62,63,65,66,67] used the numerical methods for solving the equation and Parand & Delkhosh [73] were able to calculate the best solution using the combination of the quasilinearization method and the fractional order of rational Chebyshev collocation method, correct to 37 decimal places.…”

Numerical Solution of Falkner-Skan Equation by Iterative Transformation Method

“…The original problem has been formulated as a new free boundary value problem. Additionally, an adaptive algorithm for the Thomas-Fermi equation by means of the moving mesh finite element method has been developed [31].…”