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

Bruno Boisseau, P. Forgács, Hector Giacomini

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

A new (algebraic) approximation scheme to find global solutions of twopoint boundary value problems of ordinary differential equations (ODEs) is presented. The method is applicable for both linear and nonlinear (coupled) ODEs whose solutions are analytic near one of the boundary points. It is based on replacing the original ODEs by a sequence of auxiliary first-order polynomial ODEs with constant coefficients. The coefficients in the auxiliary ODEs are uniquely determined from the local behaviour of the solution in the neighbourhood of one of the boundary points. The problem of obtaining 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 ODEs arising in various problems in field theory. We treat in particular the static and rotationally symmetric global vortex, the skyrmion, the Abrikosov-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 ODEs 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.

Original languageEnglish
JournalJournal of Physics A: Mathematical and Theoretical
Volume40
Issue number9
DOIs
Publication statusPublished - Mar 2 2007

Fingerprint

Analytical Approximation
Approximation Scheme
Two-point Boundary Value Problem
Ordinary differential equations
boundary value problems
Boundary value problems
Ordinary differential equation
differential equations
approximation
Vortex
Vortex flow
Magnetic Monopoles
Anharmonic Oscillator
vortices
Ground State Energy
Monopole
Nonlinear Ordinary Differential Equations
Coefficient
magnetic monopoles
Energy Levels

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Modelling and Simulation
  • Statistics and Probability

Cite this

An analytical approximation scheme to two-point boundary value problems of ordinary differential equations. / Boisseau, Bruno; Forgács, P.; Giacomini, Hector.

In: Journal of Physics A: Mathematical and Theoretical, Vol. 40, No. 9, 02.03.2007.

Research output: Contribution to journalArticle

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