# 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 `k`th event function in `events`:

• `value(k)` is the value of the `k`th 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.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`.