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THEORY AND NUMERICS FOR MULTI-TERM PERIODIC DELAY DIFFERENTIAL EQUATIONS: SMALL SOLUTIONS AND THEIR DETECTION

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Electronic Transactions on Numerical Analysis.

Volume 26, pp. 474-483, 2007.

Copyright2007, Kent State University.

ISSN 1068-9613.

ETNA

Kent State University [email protected]

THEORY AND NUMERICS FOR MULTI-TERM PERIODIC DELAY DIFFERENTIAL EQUATIONS: SMALL SOLUTIONS AND THEIR DETECTION

NEVILLE J. FORDyANDPATRICIA M. LUMBy

Abstract. In this paper we consider scalar linear periodic delay differential equations of the form

x 0

(t)= m

X

j=0 b

j

(t)x(t jw);x(t)=(t)fort2[0;mw); tmw (z)

wherebj,j = 0;:::;mare continuous periodic functions with periodw. We summarise a theoretical treatment that analyses whether the equation has small solutions. We consider discrete equations that arise when a numerical method with fixed step-size is applied to approximate the solution to (z) and we develop a corresponding theory.

Our results show that small solutions can be detected reliably by the numerical scheme. We conclude with some numerical examples.

Key words. delay differential equations, small solutions, super-exponential solutions, numerical methods AMS subject classifications. 34K28, 65P99, 37N30

Received September 29, 2006. Accepted for publication September 26, 2007. Recommended by F. Stenger.

yDepartment of Mathematics, University of Chester, Parkgate Road, Chester CH1 4BJ, UK (fnjford,p.lumbg@chester.ac.uk).

474

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