MATLAB Mathematics - SERC - Index of

Initial Value Problems for DDEs
Using a History to Specify the Solution of Interest
In an initial value problem, we seek the solution on an interval [ t 0 , t f ]. with
t 0 < t f . The DDE shows that y′ () t depends on values of the solution at times
prior to t. In particular, y′ ( t 0 ) depends on yt ( 0 – τ 1 ),…,
yt ( 0 – τ k ). Because of
this, a solution on [ t 0 , t f ] depends on its values for t ≤ t 0 , i.e., its history St ().
Propagation of Discontinuities
Generally, the solution yt () of an IVP for a system of DDEs has a jump in its
first derivative at the initial point t 0 because the first derivative of the history
function does not satisfy the DDE there.
– +
S′ ( t 0 ) ≠ y′ ( t 0 ) = ft ( 0 , yt ( 0 ), St ( 0 – τ 1 ), …,
St ( 0 – τ k ))
A discontinuity in any derivative propagates into the future at spacings of
τ 1 , τ 2 , …,
τ k .
For reliable and efficient integration of DDEs, a solver must track
discontinuities in low order derivatives and deal with them. For DDEs with
constant lags, the solution gets smoother as the integration progresses, so after
a while the solver can stop tracking a discontinuity. See “Discontinuities” on
page 5-57 for more information.
DDE Solver
This section describes:
• The DDE solver, dde23
• DDE solver basic syntax
The DDE Solver
The function dde23 solves initial value problems for delay differential
equations (DDEs) with constant delays. It integrates a system of first-order
differential equations
y′ () t = ftyt ( , ()yt , ( – τ 1 ),…,
yt ( – τ k ))
on the interval [ t 0 , t f ], with t 0 < t f and given history yt () = St () for t ≤ t 0 .
5-51