Shahid Beheshti University Computational Mathematics and Computer Modeling with Applications (CMCMA) 2783-4859 1 1 2022 06 01 A comparison between pre-Newton and post-Newton approaches for solving a physical singular second-order boundary problem in the semi-infinite interval 116 125 103730 10.52547/CMCMA.1.1.116 EN Amir Hosein Hadian Rasanan School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran Mehran Nikarya Department of Electrical Engineering and Information Technology, Iranian Research Organization for Science and Technology (IROST), Tehran, Iran Mohammad Mahdi Moayeri Department of Computer and Data Sciences, Shahid Beheshti University, Tehran, Iran. Arman Bahramnezhad Department of Computer and Data Sciences, Shahid Beheshti University, Tehran, Iran Journal Article 2022 10 29 In this paper, two numerical approaches based on the Newton iteration method with spectral algorithms are introduced to solve the Thomas-Fermi equation. That Thomas-Fermi equation is a nonlinear singular ordinary differential equation (ODE) with a boundary condition in infinite. In these schemes, the Newton method is combined with a spectral method where in one of those, by the Newton method we convert nonlinear ODE to a sequence of linear ODE and then, solve them using the spectral method. In another one, by the spectral method, the nonlinear ODE is converted to a system of nonlinear algebraic equations, then, this system is solved by the Newton method. In both approaches, the spectral method is based on the fractional order of rational Gegenbauer functions. Finally, the obtained results of the two introduced schemes are compared to each other in accuracy, runtime, and iteration number. Numerical experiments are presented showing that our methods are as accurate as the best results obtained until now. https://cmcma.sbu.ac.ir/article_103730_52acfa13c0dd4449bdf6b6929535fd51.pdf