Numerical approximation to ODEs using the error functional
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- by L. Bayón, J. M. Grau, M. M. Ruiz and P. M. Suárez PDF
- Proc. Amer. Math. Soc. 140 (2012), 4295-4308 Request permission
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.References
- Sergio Amat and Pablo Pedregal, A variational approach to implicit ODEs and differential inclusions, ESAIM Control Optim. Calc. Var. 15 (2009), no. 1, 139–148. MR 2488572, DOI 10.1051/cocv:2008020
- Uri M. Ascher, Robert M. M. Mattheij, and Robert D. Russell, Numerical solution of boundary value problems for ordinary differential equations, Classics in Applied Mathematics, vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. Corrected reprint of the 1988 original. MR 1351005, DOI 10.1137/1.9781611971231
- L. Bayón, J. M. Grau, M. M. Ruiz, and P. M. Suárez, A Bolza problem in hydrothermal optimization, Appl. Math. Comput. 184 (2007), no. 1, 12–22. MR 2295455, DOI 10.1016/j.amc.2005.09.108
- L. Bayón, J. M. Grau, M. M. Ruiz, and P. M. Suárez, A constrained and non-smooth hydrothermal problem, Appl. Math. Comput. 209 (2009), no. 1, 10–18. MR 2493281, DOI 10.1016/j.amc.2008.06.013
- R. Burden and J. Faires, Numerical Analysis, Thomson Brooks/Cole, Belmont, CA, 2005.
- F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth analysis and control theory, Graduate Texts in Mathematics, vol. 178, Springer-Verlag, New York, 1998. MR 1488695
- S. Effati and H. Roohparvar, Iterative dynamic programming for solving linear and nonlinear differential equations, Appl. Math. Comput. 175 (2006), no. 1, 247–257. MR 2216338, DOI 10.1016/j.amc.2005.07.065
- S. Hassani, Mathematical Methods Using Mathematica, Springer, NY, 2003.
- Z. Q. Luo and P. Tseng, On the convergence of the coordinate descent method for convex differentiable minimization, J. Optim. Theory Appl. 72 (1992), no. 1, 7–35. MR 1141764, DOI 10.1007/BF00939948
- Rein Luus, Iterative dynamic programming, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 110, Chapman & Hall/CRC, Boca Raton, FL, 2000. MR 1750212, DOI 10.1201/9781420036022
- Higinio Ramos and Jesús Vigo-Aguiar, An almost $L$-stable BDF-type method for the numerical solution of stiff ODEs arising from the method of lines, Numer. Methods Partial Differential Equations 23 (2007), no. 5, 1110–1121. MR 2340663, DOI 10.1002/num.20212
- Lawrence F. Shampine, Numerical solution of ordinary differential equations, Chapman & Hall, New York, 1994. MR 1261869
- L. F. Shampine, I. Gladwell, and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, Cambridge, 2003. MR 1985643, DOI 10.1017/CBO9780511615542
- Richard Vinter, Optimal control, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2000. MR 1756410
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
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 4295-4308
- MSC (2010): Primary 65L05, 47J30
- DOI: https://doi.org/10.1090/S0002-9939-2012-11340-1
- MathSciNet review: 2957220