# Documentation

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# ddesd

Solve delay differential equations (DDEs) with general delays

## Syntax

```sol = ddesd(ddefun,delays,history,tspan) sol = ddesd(ddefun,delays,history,tspan,options) ```

## Arguments

 `ddefun` Function handle that evaluates the right side of the differential equations y′(t) = f(t,y(t),y(d(1),...,y(d(k))). 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(d(j)) for delay d(j) given as component j of `delays(t,y)`. The output is a column vector corresponding to f(t,y(t),y(d(1),...,y(d(k))). `delays` Function handle that returns a column vector of delays d(j). The delays can depend on both t and y(t). `ddesd` imposes the requirement that d(j) ≤ t by using `min`(d(j),t).If all the delay functions have the form d(j) = t – τj, you can set the argument `delays` to a constant vector `delays`(j) = τj. With delay functions of this form, `ddesd` is used exactly like `dde23`. `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 = ddesd(ddefun,delays,history,tspan)` integrates the system of DDEs

`${y}^{\prime }\left(t\right)=f\left(t,y\left(t\right),y\left(d\left(1\right)\right),...,y\left(d\left(k\right)\right)\right)$`

on the interval [t0,tf], where delays d(j) can depend on both t and y(t), and t0 < tf. Inputs `ddefun` and `delays` are function handles. See Create Function Handle for more information.

Parameterizing Functions explains how to provide additional parameters to the functions `ddefun`, `delays`, and `history`, if necessary.

`ddesd` 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 `ddesd` has the following fields.

 `sol.x` Mesh selected by `ddesd` `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, `'ddesd'`

`sol = ddesd(ddefun,delays,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 for more information.

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 `'Events'` option to specify a function that `ddesd` calls to find where functions g(t,y(t),y(d(1)),...,y(d(k))) 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 `ddesd` 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

The equation

`sol = ddesd(@ddex1de,@ddex1delays,@ddex1hist,[0,5]);`

solves a DDE on the interval `[0,5]` with delays specified by the function `ddex1delays` and differential equations computed by `ddex1de`. The history is evaluated for t ≤ 0 by the function `ddex1hist`. The solution is evaluated at 100 equally spaced points in `[0,5]`:

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

and plotted with

`plot(tint,yint);`

This problem involves constant delays. The `delay` function has the form

```function d = ddex1delays(t,y) %DDEX1DELAYS Delays for using with DDEX1DE. d = [ t - 1 t - 0.2];```

The problem can also be solved with the syntax corresponding to constant delays

```delays = [1, 0.2]; sol = ddesd(@ddex1de,delays,@ddex1hist,[0, 5]);```

or using `dde23`:

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

For more examples of solving delay differential equations see `ddex2` and `ddex3`.

## References

[1] Shampine, L.F., “Solving ODEs and DDEs with Residual Control,” Applied Numerical Mathematics, Vol. 52, 2005, pp. 113-127.