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

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.

5 pages, 3 tables, Latex

References in corpus (4)

• Exact Renormalization Group Equations. An Introductory Review
• Twisted Superconducting Semilocal Strings
• Precise variational tunneling rates for anharmonic oscillator with g<0
• Numerical and asymptotic analysis of the 't Hooft-Polyakov magnetic monopole

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• Conformal mappings versus other power series methods for solving ordinary differential equations: illustration on anharmonic oscillators
• Accurate eigenvalues of bounded oscillators
• New Approach to Bounded Quantum--Mechanical Models
• High precision solutions to quantized vortices within Gross-Pitaevskii equation