Spectral Method for Solving the Nonlinear Thomas-Fermi Equation Based on Exponential Functions

Abstract

We present an efficient spectral methods solver for the Thomas-Fermi equation for neutral atoms in a semi-infinite domain. The ordinary differential 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. The nonlinear algebraic system of equations that is derived using this method is solved using a minimization approach. The presented method has shown robustness in the sense that it can find high precision solution for a wide range of parameters that define 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 different basis sets. The comparison shows that our method is highly competitive and in many aspects outperforms the previous work.

New Numerical Solution For Solving Nonlinear Singular Thomas-Fermi Differential Equation

“…Table 3 shows the obtained values of y ′ (0) for various values of L and m = 45, and the absolute error with Parand et al [86]. Table 4 shows comparison of the obtained values of y(t) between the present method, Parand and Shahini [59], Jovanovic [80], and Liao [100] for various values of t. Table 5 shows the obtained values of y ′ (t) by the present method for various values of t. Fig. 2 shows the graphs of residual error Res(t) of the Eq.…”

“…For examples, recently, authors of [61,63,66,70,71,72,76,82,85] have used analytical methods to solve this equation, and the best solution for y ′ (0) was calculated by Amore et al [82] by using Pade-Hankel method, correct to 26 decimal places. Authors of [59,62,74,75,79,80,81] have used numerical methods to solve this equation, and the best solution for y ′ (0) was calculated by Parand et al [86] by using an iterative method based on the fractional order of rational Euler functions, correct to 27 decimal places. In these numerical methods, there is a numerical parameter that is selected by the authors.…”

An efficient numerical method for solving nonlinear Thomas-Fermi equation

“…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. In numerical methods, there is usually a numerical arbitrary parameter which selected by authors.…”

A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

“…Jovanovic et al in 2014 [54] solved the Thomas-Fermi equation by applying a spectral method using an exponential basis set in a semi-infinite domain. The goal of the spectral method approach is to find the values of coefficients a i that best satisfy the following equation:…”