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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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 nth 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
PII: S 0002-9939(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.