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

TL;DR

This paper compares two spectral-Newton numerical methods for solving the nonlinear singular Thomas-Fermi equation on an infinite interval, demonstrating comparable accuracy and efficiency.

Contribution

It introduces and compares two novel spectral-Newton approaches using fractional Gegenbauer functions for solving the Thomas-Fermi equation.

Findings

Both methods achieve high accuracy consistent with best existing results.

The methods show competitive runtime and iteration counts.

Numerical experiments validate the effectiveness of the proposed schemes.

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 boundary condition in infinite. In these schemes, the Newton method is combined with a spectral method where in one of those, by Newton method we convert nonlinear ODE to a sequence of linear ODE then, solve them using the spectral method. In another one, by the spectral method the nonlinear ODE be converted to system of nonlinear algebraic equations, then, this system is solved by Newton method. In both approaches, the spectral method is based on fractional order of rational Gegenbauer functions. Finally, the obtained results of 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 which obtained until now.