Numerical approximation to ODEs using the error functional

Authors:
L. Bayón, J. M. Grau, M. M. Ruiz and P. M. Suárez

Journal:
Proc. Amer. Math. Soc. **140** (2012), 4295-4308

MSC (2010):
Primary 65L05, 47J30

Published electronically:
April 24, 2012

MathSciNet review:
2957220

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we present a new method for solving systems of ordinary nonlinear differential equations with initial conditions. The method is based on the transformation of the problem to an optimal control problem. We then solve it with a technique based on the use of an integral form of the Euler equation combined with the shooting method and the cyclic coordinate descent method. Our method substantially improves a previous approach that uses iterative dynamic programming to solve the associated optimal control problem. We consider the error functional instead of the classical global error, the error functional obtained by our method being lower than that obtained by classical methods. The method presented in this paper allows us to solve a wide range of *n*th order ordinary nonlinear differential equations with initial conditions.

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Additional Information

**L. Bayón**

Affiliation:
Department of Mathematics, University of Oviedo, E.P.I. Campus of Viesques, Gijón, 33203, Spain

Email:
bayon@uniovi.es

**J. M. Grau**

Affiliation:
Department of Mathematics, University of Oviedo, E.P.I. Campus of Viesques, Gijón, 33203, Spain

**M. M. Ruiz**

Affiliation:
Department of Mathematics, University of Oviedo, E.P.I. Campus of Viesques, Gijón, 33203, Spain

**P. M. Suárez**

Affiliation:
Department of Mathematics, University of Oviedo, E.P.I. Campus of Viesques, Gijón, 33203, Spain

DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11340-1

Keywords:
Ordinary differential equation,
optimal control,
cyclic coordinate descent

Received by editor(s):
December 14, 2010

Received by editor(s) in revised form:
June 3, 2011

Published electronically:
April 24, 2012

Additional Notes:
This work was supported by the Government of Principality of Asturias through PCTI: FICYT IB09-085 and by the Spanish Government (MICINN, project: MTM2010-15737)

Communicated by:
Yingfei Yi

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.