MATLAB®

DDEs

Function Summary

• DDE Solvers

• DDE Helper Functions

• DDE Solver Options

DDE Solvers

DDE Helper Functions

DDE Solver Options

Use these functions to create, alter, or access an options structure. An options structure contains named properties, the values of which are passed to dde23 or ddesd, thus affecting the solution of the problem.

Back to Top

Initial Value Problems

The DDE dde23 solver can solve systems of ordinary differential equations, such as

where is the independent variable, is the dependent variable, and represents (derivative of y with respect to t) . The delays (lags) are positive constants. The solver ddesd allows delays that depend on and .

History and Initial Values

In an initial value problem, you seek the solution on an interval with . The DDE shows that depends on values of the solution at times prior to . In particular, depends on . Because of this, a solution on depends on its values for , i.e., its history .

Propagation of Discontinuities

Generally, the solution of an IVP for a system of DDEs has a jump in its first derivative at the initial point because the first derivative of the history function does not satisfy the DDE there. A discontinuity in any derivative propagates into the future at spacings of when the delays are constant, and in a more complicated way when they are not. For the DDEs solved by dde23 and ddesd, the solution becomes smoother as the integration proceeds.

Back to Top

Types of Solvers

This section describes:

The basic syntax for the two solvers is shown in the function reference pages for dde23 and ddesd.

DDE Solver dde23

The function dde23 solves initial value problems for DDEs with constant delays. It integrates a system of first-order differential equations

on the interval , with and given history for .

dde23 produces a solution that is continuous on . You can use the function deval and the output of dde23 to evaluate the solution at specific points on the interval of integration.

dde23 tracks low-order discontinuities and integrates the differential equations with the explicit Runge-Kutta (2,3) pair and interpolant used by ode23. The Runge-Kutta formulas are implicit for step sizes longer than the delays. When the solution is smooth enough that steps this big are justified, the implicit formulas are evaluated by a predictor-corrector iteration.

DDE Solver ddesd

The function ddesd solves initial value problems for DDEs with general delays. It integrates a system of first-order differential equations

on the interval , with , where delays can depend on both and . Use the function deval and the output of ddesd to evaluate the solution at specific points on the interval of integration.

ddesd integrates with the classic four-stage, fourth-order explicit Runge-Kutta method, and controls the size of the residual of a natural interpolant. It uses iteration to take steps that are longer than the delays. For further details, see "Solving ODEs and DDEs with Residual Control," L.F. Shampine, Applied Numerical Mathematics, 52 (2005), pp 113-127.

Back to Top

Discontinuities

dde23 performs better if it is informed of discontinuities in the history and at known locations. Discontinuities may be specified by event functions. There is a property with which you can specify a solution that is different from the value given by the history function.

Back to Top

Integrator Options

The default integration properties in the DDE solver dde23 are selected to handle common problems. In some cases, you can improve solver performance by overriding these defaults. You do this by supplying dde23 with an options structure that specifies one or more property values.

For example, to change the relative error tolerance of dde23 from the default value of 1e-3 to 1e-4,

1. Create an options structure using the function ddeset by entering

   options = ddeset('RelTol', 1e-4);
   

2. Pass the options structure to dde23 as follows:

   sol = dde23(ddefun,lags,history,tspan,options)
   

For a complete description of the available options, see the reference page for ddeset.

Back to Top

Examples

• Constant Delay

• State-Dependent Delay

• Cardiovascular Model

• Additional Examples

Constant Delay

• Solving the System

• Evaluating the Solution

Solving the System.   This example illustrates the straightforward formulation, computation, and display of the solution of a system of DDEs with constant delays. The history is constant, which is often the case. The differential equations are

The example solves the equations on [0,5] with history

for .

Note   The demo ddex1 contains the complete code for this example. To see the code in an editor, click the example name, or type edit ddex1 at the command line. To run the example type ddex1 at the command line.

1. Rewrite the problem as a first-order system. To use dde23, you must rewrite the equations as an equivalent system of first-order differential equations. Do this just as you would when solving IVPs and BVPs for ODEs. However, this example needs no such preparation because it already has the form of a first-order system of equations.

2. Identify the lags. The delays (lags) are supplied to dde23 as a vector. For the example we could use

   lags = [1,0.2];
   

   In coding the differential equations, = lags(j).

3. Code the system of first-order DDEs. Once you represent the equations as a first-order system, and specify lags, you can code the equations as a function that dde23 can use.

   This code represents the system in the function, ddex1de.

   function dydt = ddex1de(t,y,Z)
   ylag1 = Z(:,1);
   ylag2 = Z(:,2);
   dydt = [ylag1(1)
           ylag1(1) + ylag2(2)
           y(2)               ];
   

4. Code the history function. The history function for this example is

   function S = ddex1hist(t)
   S = ones(3,1);
   

5. Apply the DDE solver. The example now calls dde23 with the functions ddex1de and ddex1hist.

   sol = dde23(@ddex1de,lags,@ddex1hist,[0,5]);
   

   Here the example supplies the interval of integration [0,5] directly. Because the history is constant, we could also call dde23 by

   sol = dde23(@ddex1de,lags,ones(3,1),[0,5]);
   

6. View the results. Complete the example by displaying the results. dde23 returns the mesh it selects and the solution there as fields in the solution structure sol. Often, these provide a smooth graph.

   plot(sol.x,sol.y);
   title('An example of Wille'' and Baker');
   xlabel('time t');
   ylabel('solution y');
   legend('y_1','y_2','y_3',2)
   

   

Evaluating the Solution.   The method implemented in dde23 produces a continuous solution over the whole interval of integration . You can evaluate the approximate solution, , at any point in using the helper function deval and the structure sol returned by dde23.

Sint = deval(sol,tint)

The deval function is vectorized. For a vector tint, the ith column of Sint approximates the solution .

Using the output sol from the previous example, this code evaluates the numerical solution at 100 equally spaced points in the interval [0,5] and plots the result.

tint = linspace(0,5);
Sint = deval(sol,tint);
plot(tint,Sint);

State-Dependent Delay

This example solves a system of two DDEs with state-dependent delay that was used as a test problem by W.H. Enright and H. Hayashi [10] because it has an analytical solution. The differential equations are

The analytical solution

is used as the history for and the equations are solved on [0.1, 5]. The only thing different about solving this example is that it must be solved with ddesd rather than dde23. This is because the first factor in the second equation has the form with a delay that depends on the second component of the solution. The delay is provided to ddesd with a function like

function d = ddex3delay(t,y)
% State dependent delay function for DDEX3
d = exp(1 - y(2));

Note   The demo ddex3 contains the complete code for this example. To see the code in an editor, click the example name, or type edit ddex3 at the command line. To run the example type ddex3 at the command line.

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.

In ddex2, the fourteen physical parameters are set as fields in a structure p that is shared with nested functions. The function ddex2de for evaluating the equations begins with

function dydt = ddex2de(t,y,Z)
if t <= 600
   p.R = 1.05;
else
   p.R = 0.21 * exp(600-t) + 0.84;
end
.
.
.

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.

opts = ddeset('Jumps',600);

After defining the delay tau and the constant history, the call is

sol = dde23(@ddex2de,tau,history,[0, 1000],opts);

The demo ddex2 plots only the third component, the heart rate, which shows a sharp change at t = 600.

Restarting.   The example could have solved this problem by breaking it into two pieces

sol = dde23(@ddex2de,tau,history,[0, 600]);
sol = dde23(@ddex2de,tau,sol,[600, 1000]);

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,

Sint = deval(sol,[300,900]);

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. This is not an option with ddesd, which handles discontinuities in a different way. When you use event functions to locate such discontinuities, you must restart the integration at discontinuities.

Additional Examples

The following additional examples are available. Type

edit examplename

to view the code and

examplename

to run the example.

Additional examples are provided by "Tutorial on Solving DDEs with DDE23," available at http://www.mathworks.com/dde_tutorial.

Back to Top

ODEs

BVPs