# Documentation

## Types of DDEs

### Constant Delay DDEs

A system of differential equations (DDEs) with constant delays has the following form:

 ${y}^{\prime }\left(t\right)=f\left(t,y\left(t\right),y\left(t-{\tau }_{1}\right),\dots ,y\left(t-{\tau }_{k}\right)\right).$ (11-1)

Here, t is the independent variable, y is a column vector of dependent variables, and y ′ represents the first derivative of y with respect to t. The delays, τ1,…,τk, are positive constants.

The `dde23` function solves DDEs of the form given by Equation 11-1 with history y(t) = S(t) for t <t0.

The solutions of DDEs are generally continuous, but they have discontinuities in their derivatives. The `dde23` function tracks discontinuities in low-order derivatives. It integrates the differential equations with the same explicit Runge-Kutta (2,3) pair and interpolant used by `ode23`. The Runge-Kutta formulas are implicit for step sizes bigger than the delays. When y(t) is smooth enough to justify steps this big, the implicit formulas are evaluated by a predictor-corrector iteration.

### Time-Dependent and State-Dependent DDEs

Equation 11-1 is a special case of

 ${y}^{\prime }\left(t\right)=f\left(t,y\left(t\right),y\left(d{y}_{1}\right),...,y\left(d{y}_{p}\right)\right)$ (11-2)

that involves delays, dy1,..., dyk, which can depend on both time, t, and state, y. The delays, dyj(t, y), must satisfy dyj(t, y) ≤ t on the interval [t0, tf] with t0 < tf.

The `ddesd` function finds the solution, y(t), for DDEs of the form given by Equation 11-2 with history y(t) = S(t) for t < t0. The `ddesd` function integrates with the classic four-stage, fourth-order explicit Runge-Kutta method, and it controls the size of the residual of a natural interpolant. It uses iteration to take steps that are longer than the delays.

### DDEs of Neutral Type

Delay differential equations of neutral type involve delays in y ′ as well as y:

 ${y}^{\prime }\left(t\right)=f\left(t,y\left(t\right),y\left(d{y}_{1}\right),...,y\left(d{y}_{p}\right),{y}^{\prime }\left(dy{p}_{1}\right),...,{y}^{\prime }\left(dy{p}_{q}\right)\right).$ (11-3)

The delays in the solution must satisfy dyi(t,y) ≤ t. The delays in the first derivative must satisfy dypj(t,y) < t so that y ′ does not appear on both sides of the equation.

The `ddensd` function solves DDEs of neutral type by approximating them with DDEs of the form given by Equation 11-2. For more information, see Shampine [1].

### Evaluating the Solution at Specific Points

Use the `deval` function and the output from any of the DDE solvers to evaluate the solution at specific points in the interval of integration. For example, ```y = deval(sol, 0.5*(sol.x(1) + sol.x(end)))``` evaluates the solution at the midpoint of the interval of integration.

### History and Initial Values

When you solve a DDE, you approximate the solution on an interval [t0,tf] with t0 < tf. The DDEs show how y ′(t) depends on values of the solution (and possibly its derivative) at times prior to t. For example, Equation 11-1 shows that y ′(t0) depends on y(t0τ1),…,y(t0τk) for positive constants τj. Because of this, a solution on [t0, tk] depends on values it has at tt0. You must define these values with a history function, y(t) = S(t) for t <t0.

### Propagation of Discontinuities

Generally, the first derivative of the solution has a jump at the initial point. This is because the first derivative of the history function, S(t), generally does not satisfy the DDE at this point. A discontinuity in any derivative of y(t) propagates into the future at spacings of τ1,…, τk when the delays are constant, as in Equation 11-1. If the delays are not constant, the propagation of discontinuities is more complicated. For neutral DDEs of the form given by Equation 11-1 or Equation 11-2, the discontinuity appears in the next higher order derivative each time it is propagated. In this sense, the solution gets smoother as the integration proceeds. Solutions of neutral DDEs of the form given by Equation 11-3 are qualitatively different. The discontinuity in the solution does not propagate to a derivative of higher order. In particular, the typical jump in y ′(t) at t0 propagates as jumps in y ′(t) throughout [t0, tf].

## References

[1] Shampine, L.F. "Dissipative Approximations to Neutral DDEs." Applied Mathematics & Computation, Vol. 203, 2008, pp. 641–648.