An analytical approximation scheme to two point boundary value problems of ordinary differential equations

TL;DR

This paper introduces an algebraic approximation scheme for solving two point boundary value problems in ODEs, applicable to linear and nonlinear cases, and demonstrates its effectiveness on various physical models and problems.

Contribution

The paper presents a novel algebraic method for approximating global solutions of boundary value problems in ODEs, applicable to both linear and nonlinear equations, with demonstrated accuracy on physical models.

Findings

Successfully approximates connecting parameters for nonlinear ODEs

Accurately computes energies of skyrmions and monopoles

Effective for ODEs from field theory and renormalization group

Abstract

A new (algebraic) approximation scheme to find {\sl global} solutions of two point boundary value problems of ordinary differential equations (ODE's) is presented. The method is applicable for both linear and nonlinear (coupled) ODE's whose solutions are analytic near one of the boundary points. It is based on replacing the original ODE's by a sequence of auxiliary first order polynomial ODE's with constant coefficients. The coefficients in the auxiliary ODE's are uniquely determined from the local behaviour of the solution in the neighbourhood of one of the boundary points. To obtain the parameters of the global (connecting) solutions analytic at one of the boundary points, reduces to find the appropriate zeros of algebraic equations. The power of the method is illustrated by computing the approximate values of the ``connecting parameters'' for a number of nonlinear ODE's arising in various problems in field theory. We treat in particular the static and rotationally symmetric global vortex, the skyrmion, the Nielsen-Olesen vortex, as well as the 't Hooft-Polyakov magnetic monopole. The total energy of the skyrmion and of the monopole is also computed by the new method. We also consider some ODE's coming from the exact renormalization group. The ground state energy level of the anharmonic oscillator is also computed for arbitrary coupling strengths with good precision.