DDE Solver :: Differential Equations (Mathematics)

Mathematics

DDE Solver

This section describes:

The DDE solver, dde23
DDE solver basic syntax

The DDE Solver

The function dde23 solves initial value problems for delay differential equations (DDEs) with constant delays. It integrates a system of first-order differential equations

on the interval [t sub 0, t sub f] , with t sub 0 < t sub f and given history y(t) = S(t) for t less than or equal t sub 0 .

dde23 produces a solution that is continuous on [t sub 0, t sub f] . You can use the function deval and the output of dde23 to evaluate the solution at specific points on the interval of integration.

dde23 tracks discontinuities and integrates the differential equations with the explicit Runge-Kutta (2,3) pair and interpolant used by ode23. The Runge-Kutta formulas are implicit for step sizes longer than the delays. When the solution is smooth enough that steps this big are justified, the implicit formulas are evaluated by a predictor-corrector iteration.

DDE Solver Basic Syntax

The basic syntax of the DDE solver is

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

The input arguments are

ddefun

Handle to a function that evaluates the right side of the differential equations. The function must have the form

dydt = ddefun(t,y,Z)

where the scalar t is the independent variable, the column vector y is the dependent variable, and Z(:,j) is y(t minus tau sub j) for = lags(j). See Function Handles in the MATLAB Programming documentation for more information.

lags
A vector of constant positive delays tau sub 1 through tau sub k .

history
Handle to a function of that evaluates the solution y(t) for t less than or equal t sub 0 . The function must be of the form
S = history(t)
where S is a column vector. Alternatively, if is constant, you can specify history as this constant vector.
If the current call to dde23 continues a previous integration to t0, use the solution sol from that call as the history.

tspan
The interval of integration as a two-element vector [t0,tf] with t0 < tf.

`ddefun`	Handle to a function that evaluates the right side of the differential equations. The function must have the form `dydt = ddefun(t,y,Z)` where the scalar `t` is the independent variable, the column vector `y` is the dependent variable, and `Z(:,j)` is for = `lags(j)`. See Function Handles in the MATLAB Programming documentation for more information.
`lags`	A vector of constant positive delays .
`history`	Handle to a function of that evaluates the solution for . The function must be of the form S = history(t) where `S` is a column vector. Alternatively, if is constant, you can specify `history` as this constant vector. If the current call to `dde23` continues a previous integration to `t0`, use the solution `sol` from that call as the history.
`tspan`	The interval of integration as a two-element vector `[t0,tf]` with `t0 < tf`.

For more advanced applications, you can specify solver options by passing an input argument options.

options
Structure of optional parameters that change the default integration properties. You can create the structure options using odeset. The odeset reference page describes the properties you can specify.

`options`	Structure of optional parameters that change the default integration properties. You can create the structure `options` using `odeset`. The `odeset` reference page describes the properties you can specify.

The output argument sol is a structure created by the solver. It has fields:

sol.x
Nodes of the mesh selected by dde23

sol.y
Approximation to y(t) at the mesh points of sol.x

sol.yp
Approximation to y prime (t) at the mesh points of sol.x

sol.solver
'dde23'

`sol.x`	Nodes of the mesh selected by `dde23`
`sol.y`	Approximation to at the mesh points of `sol.x`
`sol.yp`	Approximation to at the mesh points of `sol.x`
`sol.solver`	`'dde23'`

To evaluate the numerical solution at any point from [t0,tf], use deval with the output structure sol as its input.

Introduction to Initial Value DDE Problems Solving DDE Problems