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

Abstract

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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.

How to cite

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Condon, 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

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  8. B. Engquist, A. Fokas, E. Hairer and A. Iserles Eds., Highly Oscillatory Problems. Cambridge University Press, Cambridge, UK (2009).  
  9. Y.N. Kyrychko and S.J. Hogan, On the use of delay equations in engineering applications. J. Vibr. Control16 (2010) 943–960.  Zbl1269.70002
  10. 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
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