Research Article Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions Raka Jovanovic, Sabre Kais, and Fahhad H. Alharbi Qatar Environment and Energy Research Institute (QEERI), Qatar Foundation, P.O. Box 5825, Doha, Qatar Correspondence should be addressed to Raka Jovanovic; rjovanovic@qf.org.qa Received 8 July 2014; Revised 11 September 2014; Accepted 16 September 2014; Published 12 November 2014 Academic Editor: Mehmet Sezer Copyright © 2014 Raka Jovanovic et al. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We present an eicient spectral methods solver for the homas-Fermi equation for neutral atoms in a semi-ininite domain. he ordinary diferential equation has been solved by applying a spectral method using an exponential basis set. One of the main advantages of this approach, when compared to other relevant applications of spectral methods, is that the underlying integrals can be solved analytically and numerical integration can be avoided. he nonlinear algebraic system of equations that is derived using this method is solved using a minimization approach. he presented method has shown robustness in the sense that it can ind high precision solution for a wide range of parameters that deine the basis set. In our test, we show that the new approach can achieve a very high rate of convergence using a small number of bases elements. We also present a comparison of recently published results for this problem using spectral methods based on several diferent basis sets. he comparison shows that our method is highly competitive and in many aspects outperforms the previous work. 1. Introduction In this paper we focus on solving the homas-Fermi (TF) equation, which is of great importance for a wide range of physical problems. Some of its applications are the deter- mination of efective nuclear charge in heavy atoms and in inding efective potentials for self-consistent calculations. he equation is a nonlinear ordinary diferential equation that is solved on a semi-ininite domain. he scaled TF equation [1] is given in  2   2 = 1 √   3/2 (1) subjected to the following boundary conditions (0)=1, lim →∞ ()=0. (2) Due to its importance this equation has been solved by many diferent methods like the perturbative approach [2], homotopy analysis method [3, 4], quasilinearization approaches [5], and Pad´ e approximations [6]. he complexity of solving this relatively simple looking equation is that it is singular at both endpoints. One of the complexities in applying spectral methods (SM) to the TF equation is the fact that it is deined on a semi- ininite domain. Signiicant research has been conducted on applying SM on ininite and semi-ininite domains [7– 15]. his has been achieved by implementing a wide range of approaches varying from using suitable basis sets and truncating the numerical window to forcing size scaling. Very good results for such problems have been achieved by using nonclassical orthogonal basis sets for systems [9], mapped orthogonal systems [16, 17], Laguerre functions [10], mapped Legendre functions [11], and mapped Fourier sine series [12]. Recently several approaches have been developed for solving the TF equation using pseudo spectral methods. he greatest efort in applying this type of approach has been done by using diferent versions of rational Chebyshev functions. In the work of Parand and Shahini [18] the original equation has been solved by using this basis set. In his article, Boyd [19] presents a spectral method based on the same basis set but applying it to the transformed version of the equation as given Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 168568, 8 pages http://dx.doi.org/10.1155/2014/168568