This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory problem. This leads to methods which, counter-intuitively to those developed according to standard numerical reasoning, exhibit improved performance with growing frequency of oscillation.
Mots clés : Delay differential equations, asymptotic expansions, modulated Fourier expansions, numerical analysis
@article{M2AN_2012__46_6_1407_0,
author = {Condon, Marissa and Dea\~no, Alfredo and Iserles, Arieh and Kropielnicka, Karolina},
title = {Efficient computation of delay differential equations with highly oscillatory terms},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1407--1420},
publisher = {EDP-Sciences},
volume = {46},
number = {6},
year = {2012},
doi = {10.1051/m2an/2012004},
mrnumber = {2996333},
zbl = {1270.65032},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2012004/}
}
TY - JOUR AU - Condon, Marissa AU - Deaño, Alfredo AU - Iserles, Arieh AU - Kropielnicka, Karolina TI - Efficient computation of delay differential equations with highly oscillatory terms JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 1407 EP - 1420 VL - 46 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012004/ DO - 10.1051/m2an/2012004 LA - en ID - M2AN_2012__46_6_1407_0 ER -
%0 Journal Article %A Condon, Marissa %A Deaño, Alfredo %A Iserles, Arieh %A Kropielnicka, Karolina %T Efficient computation of delay differential equations with highly oscillatory terms %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 1407-1420 %V 46 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012004/ %R 10.1051/m2an/2012004 %G en %F M2AN_2012__46_6_1407_0
Condon, Marissa; Deaño, Alfredo; Iserles, Arieh; Kropielnicka, Karolina. Efficient computation of delay differential equations with highly oscillatory terms. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 1407-1420. doi : 10.1051/m2an/2012004. http://www.numdam.org/articles/10.1051/m2an/2012004/
[1] and , Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford, UK (2003). | MR | Zbl
[2] and , Heterogeneous multiscale methods for mechanical systems with vibrations. SIAM J. Sci. Comput. 32 (2010) 2029-2046. | MR | Zbl
[3] , and , Higher-order averaging, formal series and numerical integration I : B-series. Found. Comput. Math. 10 (2010) 695-727. | MR | Zbl
[4] , and , Nonlinear dynamics and spectral stability of optoelectronic microwave oscillators. IEEE J. Quant. Electron. 44 (2008) 858-866.
[5] , and , Modulated Fourier expansions of highly oscillatory differential equations. Found. Comput. Math. 3 (2005) 327-450. | MR | Zbl
[6] , and , On second order differential equations with highly oscillatory forcing terms. Proc. Roy. Soc. A 466 (2010) 1809-1828. | MR | Zbl
[7] , and , On systems of differential equations with extrinsic oscillation. Discrete Contin. Dyn. Syst. 28 (2010) 1345-1367. | MR | Zbl
[8] , , and Eds., Highly Oscillatory Problems. Cambridge University Press, Cambridge, UK (2009). | MR | Zbl
[9] and , On the use of delay equations in engineering applications. J. Vibr. Control 16 (2010) 943-960. | MR | Zbl
[10] , , , , and , Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations. Phys. Lett. A 308 (2003) 54-60. | MR | Zbl
[11] and , Communication with chaotic lasers. Science 279 (1998) 1198-1200.
[12] and , The dynamics of two coupled van der pol oscillators with delay coupling. Nonlinear Dyn. 30 (2002) 205-221. | MR | Zbl
Cité par Sources :
