Solving DDEs with Matlab

Ordinary differential equations (ODEs) and delay differential equations (DDEs)
are used to describe many phenomena ofph ysical interest. While ODEs contain
derivatives which depend on the solution at the present value ofthe
independent variable (“time”), DDEs contain in addition derivatives which
depend on the solution at previous times. DDEs arise in models throughout
the sciences [1]. Despite the obvious similarities between ODEs and DDEs,
solutions of DDE problems can differ from solutions for ODE problems in
several striking, and significant, ways [2] [20]. This accounts in part for the
lack of much general-purpose software for solving DDEs.

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.