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Efficient computation of delay differential equations with highly oscillatory terms

Published online by Cambridge University Press:  19 April 2012

Marissa Condon ,
Alfredo Deaño ,
Arieh Iserles  and
Karolina Kropielnicka
Marissa Condon
Affiliation:
School of Electronic Engineering, Dublin City University, Dublin 9, Ireland. marissa.condon@dcu.ie
Alfredo Deaño
Affiliation:
Dpto. de Matemáticas, Universidad Carlos III de Madrid, Avda. Universidad, 30, Leganés 28911, Madrid, Spain
Arieh Iserles
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, CB3 0WA Cambridge, UK
Karolina Kropielnicka
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, CB3 0WA Cambridge, UK Institute of Mathematics, University of Gdańsk, Wit Stwosz Str. 57, 80-952 Gdańsk, Poland
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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.

Keywords

Delay differential equationsasymptotic expansionsmodulated Fourier expansionsnumerical analysis
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012 , pp. 1407 - 1420
Copyright
© EDP Sciences, SMAI, 2012

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