Efficient computation of delay differential equations with highly oscillatory terms

¹ School of Electronic Engineering, Dublin City University, Dublin 9, Ireland
marissa.condon@dcu.ie
² Dpto. de Matemáticas, Universidad Carlos III de Madrid, Avda. Universidad, 30, Leganés 28911, Madrid, Spain
³ Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, CB3 0 WA Cambridge, UK
⁴ Institute of Mathematics, University of Gdańsk, Wit Stwosz Str. 57, 80-952 Gdańsk, Poland

Received: 18 March 2011
Revised: 15 December 2011

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.