Dynamically Consistent Approximate Rational Solutions to the Thomas-Fermi Equation

TL;DR

This paper introduces two rational approximate solutions to the Thomas-Fermi equation using dynamic consistency, compares their accuracy with numerical solutions, and derives new integral relations for the equation's solutions.

Contribution

The paper presents novel rational approximations to the Thomas-Fermi equation based on dynamic consistency principles, enhancing analytical understanding and solution accuracy.

Findings

The two rational solutions closely match numerical results.

Comparison shows the approximations are highly accurate.

New integral relations for solutions are derived.

Abstract

We construct two rational approximate solutions to the Thomas-Fermi (TF) nonlinear differential equation. These expressions follow from an application of the principle of dynamic consistency. In addition to examining differences in the predicted numerical values of the two approximate solutions, we compare these values with an accurate numerical solution obtained using a fourth-order Runge-Kutta method. We also present several new integral relations satisfied by the bounded solutions of the TF equation