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 – τ_j, you can set the argument `delays` to a constant vector `delays`(j) = τ_j. 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 vector A constant column vector, if y(t) is constant The 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

on the interval [t₀,t_f], where delays d(j) can depend on both t and y(t), and t₀ < t_f. Inputs ddefun and delays are function handles. See Function Handles in the MATLAB Programming documentation for more information.

Parameterizing Functions in the MATLAB Mathematics documentation, 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 Delay Differential Equations 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 '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 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 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.

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