### 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 language | English |
---|---|

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 40 |

Issue number | 9 |

DOIs | |

Publication status | Published - Mar 2 2007 |

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### 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.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 40, no. 9. https://doi.org/10.1088/1751-8113/40/9/F01

}

TY - JOUR

T1 - An analytical approximation scheme to two-point boundary value problems of ordinary differential equations

AU - Boisseau, Bruno

AU - Forgács, P.

AU - Giacomini, Hector

PY - 2007/3/2

Y1 - 2007/3/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=47749103819&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=47749103819&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/40/9/F01

DO - 10.1088/1751-8113/40/9/F01

M3 - Article

VL - 40

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 9

ER -