# Efficient computation of delay differential equations with highly oscillatory terms

Marissa Condon; Alfredo Deaño; Arieh Iserles; Karolina Kropielnicka

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

- Volume: 46, Issue: 6, page 1407-1420
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topCondon, Marissa, et al. "Efficient computation of delay differential equations with highly oscillatory terms." ESAIM: Mathematical Modelling and Numerical Analysis 46.6 (2012): 1407-1420. <http://eudml.org/doc/277850>.

@article{Condon2012,

abstract = {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.},

author = {Condon, Marissa, Deaño, Alfredo, Iserles, Arieh, Kropielnicka, Karolina},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Delay differential equations; asymptotic expansions; modulated Fourier expansions; numerical analysis; delay differential equations; numerical examples; highly oscillatory forcing terms},

language = {eng},

month = {4},

number = {6},

pages = {1407-1420},

publisher = {EDP Sciences},

title = {Efficient computation of delay differential equations with highly oscillatory terms},

url = {http://eudml.org/doc/277850},

volume = {46},

year = {2012},

}

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

DA - 2012/4//

PB - EDP Sciences

VL - 46

IS - 6

SP - 1407

EP - 1420

AB - 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.

LA - eng

KW - Delay differential equations; asymptotic expansions; modulated Fourier expansions; numerical analysis; delay differential equations; numerical examples; highly oscillatory forcing terms

UR - http://eudml.org/doc/277850

ER -

## References

top- A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford, UK (2003). Zbl1038.65058
- M.P. Calvo and J.M. Sanz-Serna, Heterogeneous multiscale methods for mechanical systems with vibrations. SIAM J. Sci. Comput.32 (2010) 2029–2046. Zbl1241.65071
- P. Chartier, A. Murua and J.M. Sanz-Serna, Higher-order averaging, formal series and numerical integration I : B-series. Found. Comput. Math.10 (2010) 695–727. Zbl1211.34052
- Y.K. Chembo, L. Larger and P. Colet, Nonlinear dynamics and spectral stability of optoelectronic microwave oscillators. IEEE J. Quant. Electron.44 (2008) 858–866.
- D. Cohen, E. Hairer and C. Lubich, Modulated Fourier expansions of highly oscillatory differential equations. Found. Comput. Math.3 (2005) 327–450. Zbl1056.34005
- M. Condon, A. Deaño and A. Iserles, On second order differential equations with highly oscillatory forcing terms. Proc. Roy. Soc. A466 (2010) 1809–1828. Zbl1194.34019
- M. Condon, A. Deaño and A. Iserles, On systems of differential equations with extrinsic oscillation. Discrete Contin. Dyn. Syst.28 (2010) 1345–1367. Zbl1207.65093
- B. Engquist, A. Fokas, E. Hairer and A. Iserles Eds., Highly Oscillatory Problems. Cambridge University Press, Cambridge, UK (2009).
- Y.N. Kyrychko and S.J. Hogan, On the use of delay equations in engineering applications. J. Vibr. Control16 (2010) 943–960. Zbl1269.70002
- V.S. Udaltsov, J.P. Goedgebuer, L. Larger, J.B. Cuenot, P. Levy and W.T. Rhodes, Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations. Phys. Lett. A308 (2003) 54–60. Zbl1008.94019
- G.D. van Wiggeren and R. Roy, Communication with chaotic lasers. Science279 (1998) 1198–1200.
- S. Wirkus and R. Rand, The dynamics of two coupled van der pol oscillators with delay coupling. Nonlinear Dyn.30 (2002) 205–221. Zbl1021.70010