Mathematics 
Discontinuities
dde23
can solve problems with discontinuities in the history or discontinuities in coefficients of the equations. It provides properties that enable you to supply locations of known discontinuities and a different initial value.
Example: Cardiovascular Model
This example solves a cardiovascular model due to J. T. Ottesen [6]. The equations are integrated over the interval [0,1000]. The situation of interest is when the peripheral pressure is reduced exponentially from its value of 1.05 to 0.84 beginning at = 600.
This is a problem with one delay, a constant history, and three differential equations with fourteen physical parameters. It has a discontinuity in a low order derivative at t = 600
.
Note
The demo ddex2 contains the complete code for this example. To see the code in an editor, click the example name, or type edit ddex2 at the command line. To run the example type ddex2 at the command line. See DDE Solver Basic Syntax for more information.

In ddex2
, the fourteen physical parameters are set as fields in a structure p
that dde23
passes to ddex2de
as an additional argument. The function ddex2de
for evaluating the equations begins with
function dydt = ddex2de(t,y,Z,p) if t <= 600 p.R = 1.05; else p.R = 0.21 * exp(600t) + 0.84; end . . .
Solve Using the Jumps Property. The peripheral pressure is a continuous function of , but it does not have a continuous derivative at t = 600
. The example uses the Jumps
property to inform dde23
about the location of this discontinuity.
After defining the delay tau
and the constant history
, the call is
The demo ddex2
plots only the third component, the heart rate, which shows a sharp change at t = 600
.
Solve by Restarting. The example could have solved this problem by breaking it into two pieces
The solution structure sol
on the interval [0,600]
serves as history for restarting the integration at t = 600
. In the second call, dde23
extends sol
so that on return the solution is available on the whole interval [0,1000]
. That is, after this second return,
evaluates the solution obtained in the first integration at t = 300
, and the solution obtained in the second integration at t = 900
.
When discontinuities occur in low order derivatives at points known in advance, it is better to use the Jumps
property. When you use event functions to locate such discontinuities, you must restart the integration at discontinuities.
Solving DDE Problems  Changing DDE Integration Properties 