Rational Approximation for Two-Point Boundary value problems

TL;DR

This paper introduces a novel method using Hankel determinants for solving nonlinear two-point boundary value problems, demonstrating rapid convergence and applicability to various physical equations.

Contribution

The paper presents a new Padé-Hankel based approach for efficiently solving nonlinear boundary value problems with demonstrated success on multiple physical models.

Findings

Rapid convergence of root sequences towards unknown parameters.

Successful application to vortex profile, renormalization group, Riccati, and Thomas-Fermi equations.

Identification of limitations where the method does not apply.

Abstract

We propose a method for the treatment of two--point boundary value problems given by nonlinear ordinary differential equations. The approach leads to sequences of roots of Hankel determinants that converge rapidly towards the unknown parameter of the problem. We treat several problems of physical interest: the field equation determining the vortex profile in a Ginzburg--Landau effective theory, the fixed--point equation for Wilson's exact renormalization group, a suitably modified Wegner--Houghton's fixed point equation in the local potential approximation, a Riccati equation, and the Thomas--Fermi equation. We consider two models where the approach does not apply in order to show the limitations of our Pad\'{e}--Hankel approach.