Computational Mathematics and Computer Modeling with Applications (CMCMA)

A comparison between pre-Newton and post-Newton approaches for solving a physical singular second-order boundary problem in the semi-infinite interval

https://doi.org/10.52547/CMCMA.1.1.116
Volume 1, Issue 1 - Serial Number 1
June 2022
Pages 116-125
Computational Mathematics and Computer Modeling with Applications (CMCMA)

Document Type : Regular paper

Authors

Amir Hosein Hadian Rasanan 1
Mehran Nikarya 2
Mohammad Mahdi Moayeri 3
Arman Bahramnezhad 4

1 School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

2 Department of Electrical Engineering and Information Technology, Iranian Research Organization for Science and Technology (IROST), Tehran, Iran

3 Department of Computer and Data Sciences, Shahid Beheshti University, Tehran, Iran.

4 Department of Computer and Data Sciences, Shahid Beheshti University, Tehran, Iran

Abstract
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.

Keywords

Pre-Newton method
Post-Newton method
Fractional order of rational Gegenbauer functions
Thomas-Fermi equation
Spectral method
  • Receive Date 29 October 2022
  • Revise Date 25 May 2023
  • Accept Date 24 July 2023

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