dde23 (MATLAB Functions)

Solve delay differential equations (DDEs) with constant delays

Syntax

sol = dde23(ddefun,lags,history,tspan)
sol = dde23(ddefun,lags,history,tspan,options)
sol = dde23(ddefun,lags,history,tspan,options,p1,p2,...)

Arguments

ddefun

Function that evaluates the right side of the differential equations

. The function must have the form

dydt = ddefun(t,y,Z)

where t corresponds to the current , y is a column vector that approximates , and Z(:,j) approximates for delay = lags(j). The output is a column vector corresponding to

.

lags
Vector of constant, positive delays .

history
Specify history in one of three ways:
A function of such that y = history(t) returns the solution for as a column vector

A constant column vector, if is constant

The solution sol from a previous integration, if this call continues that integration

tspan
Interval of integration as a vector [t0,tf] with t0 < tf.

options
Optional integration argument. A structure you create using the ddeset function. See ddeset for details.

p1,p2,...
Optional parameters that dde23 passes to ddefun, history if it is a function, and any functions you specify in options.

`ddefun`	Function that evaluates the right side of the differential equations . The function must have the form `dydt = ddefun(t,y,Z)` where `t` corresponds to the current , `y` is a column vector that approximates , and `Z(:,j)` approximates for delay = `lags(j)`. The output is a column vector corresponding to .
`lags`	Vector of constant, positive delays .
`history`	Specify `history` in one of three ways: A function of such that `y = history(t)` returns the solution for as a column vector A constant column vector, if is constant The solution `sol` from a previous integration, if this call continues that integration
`tspan`	Interval of integration as a vector `[t0,tf]` with `t0 < tf`.
`options`	Optional integration argument. A structure you create using the `ddeset` function. See `ddeset` for details.
`p1,p2,...`	Optional parameters that `dde23` passes to `ddefun`, `history` if it is a function, and any functions you specify in `options`.

Description

sol = dde23(ddefun,lags,history,tspan) integrates the system of DDEs

on the interval , where are constant, positive delays and .

dde23 returns the solution as a structure sol. Use the auxiliary function deval and the output sol to evaluate the solution at specific points tint in the interval tspan = [t0,tf].

```
yint = deval(sol,tint)
```

The structure sol returned by dde23 has the following fields.

sol.x
Mesh selected by dde23

sol.y
Approximation to at the mesh points in sol.x.

sol.yp
Approximation to at the mesh points in sol.x

sol.solver
Solver name, 'dde23'

`sol.x`	Mesh selected by `dde23`
`sol.y`	Approximation to at the mesh points in `sol.x`.
`sol.yp`	Approximation to at the mesh points in `sol.x`
`sol.solver`	Solver name, `'dde23'`

sol = dde23(ddefun,lags,history,tspan,options) solves as above with default integration properties replaced by values in options, an argument created with ddeset. See ddeset and Initial Value Problems for DDEs in the MATLAB documentation for details.

Commonly used options are scalar relative error tolerance 'RelTol' (1e-3 by default) and vector of absolute error tolerances 'AbsTol' (all components are 1e-6 by default).

Use the 'Jumps' option to solve problems with discontinuities in the history or solution. Set this option to a vector that contains the locations of discontinuities in the solution prior to t0 (the history) or in coefficients of the equations at known values of after t0.

Use the 'Events' option to specify a function that dde23 calls to find where functions

vanish. This function must be of the form

[value,isterminal,direction] = events(t,y,Z)

and contain an event function for each event to be tested. For the kth event function in events:

value(k) is the value of the kth event function.
isterminal(k) = 1 if you want the integration to terminate at a zero of this event function and 0 otherwise.
direction(k) = 0 if you want dde23 to compute all zeros of this event function, +1 if only zeros where the event function increases, and -1 if only zeros where the event function decreases.

If you specify the 'Events' option and events are detected, the output structure sol also includes fields:

sol.xe
Row vector of locations of all events, i.e., times when an event function vanished

sol.ye
Matrix whose columns are the solution values corresponding to times in sol.xe

sol.ie
Vector containing indices that specify which event occurred at the corresponding time in sol.xe

`sol.xe`	Row vector of locations of all events, i.e., times when an event function vanished
`sol.ye`	Matrix whose columns are the solution values corresponding to times in `sol.xe`
`sol.ie`	Vector containing indices that specify which event occurred at the corresponding time in `sol.xe`

sol = dde23(ddefun,lags,history,tspan,options,p1,p2,...) passes the parameters p1,p2,... to the DDE function as ddefun(t,y,z,p1,p2,...), to the history function, if there is one, as history (t,p1,p2,...), and similarly to all functions specified in options. Use options = [] as a place holder if no options are set.

Examples

This example solves a DDE on the interval [0, 5] with lags 1 and 0.2. The function ddex1de computes the delay differential equations, and ddex1hist computes the history for t <= 0.

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.

sol = dde23(@ddex1de,[1, 0.2],@ddex1hist,[0, 5]);

This code evaluates the solution at 100 equally spaced points in the interval [0,5], then plots the result.

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

ddex1 shows how you can code this problem using subfunctions. For more examples see ddex2.

Algorithm

dde23 tracks discontinuities and integrates with the explicit Runge-Kutta (2,3) pair and interpolant of ode23. It uses iteration to take steps longer than the lags.

See Also

ddeget, ddeset, deval, @ (function_handle)

References

L.F. Shampine and S. Thompson, "Solving DDEs in MATLAB," Applied Numerical Mathematics, Vol. 37, 2001, pp. 441-458.

dbup ddeadv