On the Numerical Integration of Singular Initial and Boundary Value Problems for Generalised Lane-Emden and Thomas-Fermi Equations

TL;DR

This paper introduces a geometric numerical method for solving singular initial and boundary value problems related to Lane-Emden and Thomas-Fermi equations, improving efficiency and robustness.

Contribution

It presents a novel geometric approach that transforms singular problems into unstable manifold computations, enabling effective solutions for complex differential equations.

Findings

Method successfully solves generalized Lane-Emden equations.

Approach effectively handles Thomas-Fermi equation.

Robust and efficient compared to traditional methods.

Abstract

We propose a geometric approach for the numerical integration of singular initial value problems for (systems of) quasi-linear differential equations. It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector field and thus into one which can be solved in an efficient and robust manner. Using the shooting method, our approach also works well for boundary value problems. As examples, we treat some (generalised) Lane-Emden equations and the Thomas-Fermi equation.