An accurate analytic solution to the Thomas-Fermi equation

TL;DR

This paper introduces improved base functions and an auxiliary operator in the homotopy analysis method to solve the Thomas-Fermi equation more efficiently and accurately, significantly reducing computational effort and increasing convergence.

Contribution

It presents a novel choice of base functions and auxiliary operator that enhance the homotopy analysis method for solving the Thomas-Fermi equation.

Findings

Convergence is doubled compared to previous methods.

Computational efforts are significantly reduced.

Higher accuracy achieved with Pade approximants.

Abstract

The explicit analytic solution of the Thomas Fermi equation thorough a new kind of analytic technique, namely the homotopy analysis method, was employed by Liao (Appl. Math. Comp. 144, (2003)). However, the base functions and the auxiliary linear differential operator chosen were such that the convergence to the exact solution was fairly slow. New base functions and auxiliary linear operator to form a better homotopy are the main concern of the present paper. It is known that proper choice of base functions and auxiliary operator is extremely significant in gaining the exact solution in order to reduce the computational cost. The proposed homotopy here not only greatly reduces the computational efforts by at least doubling the convergence of the homotopy series, but also enlarges the convergence region of the homotopy series as compared with that of Liao. Pade approximants to the obtained solutions increase the accuracy even to a higher degree. To support this, the explicit analytical expressions obtained using the proposed approach are compared with the numerically computed ones and those of Liao.