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# dde23

Solve delay differential equations (DDEs) with constant delays

## Syntax

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

## Arguments

 ddefun Function handle that evaluates the right side of the differential equations ${y}^{\prime }\left(t\right)=f\left(t,y\left(t\right),y\left(t-{\tau }_{1}\right),...,y\left(t-{\tau }_{k}\right)\right)$. The function must have the form`dydt = ddefun(t,y,Z)`where t corresponds to the current t, y is a column vector that approximates y(t), and Z(:,j) approximates y(t – τj) for delay τj = lags(j). The output is a column vector corresponding to $f\left(t,y\left(t\right),y\left(t-{\tau }_{1}\right),...,y\left(t-{\tau }_{k}\right)\right)$. lags Vector of constant, positive delays τ1, ..., τk. history Specify history in one of three ways:A function of t such that y = history(t) returns the solution y(t) for t ≤ t0 as a column vectorA constant column vector, if y(t) is constantThe solution sol from a previous integration, if this call continues that integration tspan Interval of integration from t0=tspan(1) to tf=tspan(end) with t0 < tf. options Optional integration argument. A structure you create using the ddeset function. See ddeset for details.

## Description

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

${y}^{\prime }\left(t\right)=f\left(t,y\left(t\right),y\left(t-{\tau }_{1}\right),...,y\left(t-{\tau }_{k}\right)\right)$

on the interval [t0,tf], where τ1, ..., τk are constant, positive delays and t0,tf. The input argument, ddefun, is a function handle.

Parameterizing Functions explains how to provide additional parameters to the function ddefun, if necessary.

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 y(x) at the mesh points in sol.x. sol.yp Approximation to y′(x) 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 Types of 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 t after t0.

Use the 'Events' option to specify a function that dde23 calls to find where functions $g\left(t,y\left(t\right),y\left(t-{\tau }_{1}\right),...,y\left(t-{\tau }_{k}\right)\right)$ 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

## 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 file, ddex1.mddex1.m, contains the complete code for this example. To see the code in an editor, type edit ddex1 at the command line. To run it, 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 local functions. For more examples see ddex2.

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### Algorithms

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.

## References

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

[2] Kierzenka, J., L.F. Shampine, and S. Thompson, "Solving Delay Differential Equations with DDE23," available at www.mathworks.com/dde_tutorial.