Solving Delay Differential Equations with dde23

Solving Delay Differential Equations with
dde23

dde23 aims to make it as easy as possible to solve effectively delay-
differential equations (DDEs) with constant delays in matlab. In this
paper we discuss some of its features, including discontinuity tracking,
iteration for short delays, and event location. We also develop some the-
oretical results that underlie the solver, including convergence, error esti-
mation, and the effects of short delays on the evaluation of formulas and
stability. Some examples illustrate the use of dde23 and show it to be a
capable DDE solver that is exceptionally easy to use for a wide range of
complex problems.
1 Introduction
Our goal is to make it as easy as possible to solve effectively a large class of delay-
differential equations (DDEs). We restrict ourselves to systems of equations of
the form
y0(x) = f(x; y(x); y(x ? ?1); y(x ? ?2); : : : ; y(x ? ?k)) (1)
for constant delays ?j such that ? = min(?1; : : : ; ?k) > 0. The equations are to
hold on a ? x ? b, which requires the history y(x) = S(x) to be given for x ? a.
Although DDEs with delays (lags) of more general form are important, this is
a large and useful class of DDEs. Indeed, Baker, Paul, and Will′e [1] write that
“The lag functions that arise most frequently in the modelling literature are
constants.” Restricting ourselves to this class of DDEs makes possible a simpler
user interface and more robust numerical solution.

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