# ddensd

Solve delay differential equations (DDEs) of neutral type

## Syntax

## Description

integrates
a system of delay differential equations of neutral type, that has
the form `sol`

= ddensd(`ddefun`

,`dely`

,`delyp`

,`history`

,`tspan`

)

y '(t) = f(t, y(t), y(dy_{1}),..., y(dy_{p}), y '(dyp_{1}),..., y '(dyp_{q})) | (1) |

*t*is the independent variable representing time.*dy*_{i}is any of*p*solution delays.*dyp*_{j}is any of*q*derivative delays.

## Examples

### Neutral DDE with Two Delays

Solve the following neutral DDE, presented by Paul, for .

The solution history is for .

Create a new program file in the editor. This file will contain a main function and four local functions.

Define the first-order DDE as a local function named `ddefun`

.

function yp = ddefun(t,y,ydel,ypdel) yp = 1 + y - 2*ydel^2 - ypdel; end

Define the solution delay as a local function named `dely`

.

function dy = dely(t,y) dy = t/2; end

Define the derivative delay as a local function named `delyp`

.

function dyp = delyp(t,y) dyp = t-pi; end

Define the solution history as a local function named `history`

.

function y = history(t) y = cos(t); end

Define the interval of integration and solve the DDE using `ddensd`

. Add this code to the main function.

tspan = [0 pi]; sol = ddensd(@ddefun,@dely,@delyp,@history,tspan);

Evaluate the solution at 100 equally spaced points between and . Add this code to the main function.

tn = linspace(0,pi); yn = deval(sol,tn);

Plot the results. Add this code to the main function.

plot(tn,yn); xlim([0 pi]); ylim([-1.2 1.2]); xlabel('time t'); ylabel('solution y');

Run your entire program to calculate the solution and display the plot. The file `ddex4.m`

contains the complete code for this example. To see the code in an editor, type `edit ddex4`

at the command line.

## Input Arguments

`ddefun`

— Derivative function

function handle

Derivative function, specified as a function handle whose syntax
is `yp = ddefun(t,y,ydel,ypdel)`

. The arguments for `ddefun`

are
described in the table below.

ddefun Argument | Description |
---|---|

`t` | A scalar value representing the current value of time, t. |

`y` | A vector that represents y(t)
in Equation 1. The size
of this vector is `n` -by-`1` , where `n` is
the number of equations in the system you want to solve. |

`ydel` | A matrix whose columns, `ydel(:,i)` , represent y(dy_{i}).
The size of this matrix is `n` -by-`p` ,
where `n` is the number of equations in the system
you want to solve, and `p` is the number of y(dy)
terms in Equation 1. |

`ypdel` | A matrix whose columns, `ypdel(:,j)` represent y '(dyp_{j}).
The size of this matrix is `n` -by-`q` ,
where `n` is the number of equations in the system
you want to solve, and `q` is the number of y '(dyp)
terms in Equation 1. |

`yp` | The result returned by `ddefun` . It is an `n` -by-`1` vector
whose elements represent the right side of Equation 1. |

`dely`

— Solution delays

function handle | vector

Solution delays, specified as a function handle, which returns *dy*_{1},..., *dy*_{p} in Equation 1.
Alternatively, you can pass constant delays in the form of a vector.

If you specify `dely`

as a function handle,
the syntax must be `dy = dely(t,y)`

. The arguments
for this function are described in the table below.

dely Argument | Description |
---|---|

`t` | A scalar value representing the current value of time, t. |

`y` | A vector that represents y(t)
in Equation 1. The size
of this vector is `n` -by-`1` , where `n` is
the number of equations in the system you want to solve. |

`dy` | A vector returned by the `dely` function whose
values are the solution delays, dy_{i} ,
in Equation 1. The size
of this vector is `p` -by-`1` , where `p` is
the number of solution delays in the equation. Each element must be
less than or equal to t. |

If you want to specify constant solution delays having the form *dy*_{i} = *t* – *τ*_{i}, then `dely`

must be a vector, where `dely(i)`

= *τ*_{i}.
Each value in this vector must be greater than or equal to zero.

If *dy* is not present in the problem, set `dely`

to `[]`

.

**Data Types: **`function_handle`

| `single`

| `double`

`delyp`

— Derivative delays

function handle | vector

Derivative delays, specified as a function handle, which returns *dyp*_{1},..., *dyp*_{q} in Equation 1.
Alternatively, you can pass constant delays in the form of a vector.

If `delyp`

is a function handle, its syntax
must be `dyp = delyp(t,y)`

. The arguments for this
function are described in the table below.

delyp Argument | Description |
---|---|

`t` | A scalar value representing the current value of time, t. |

`y` | A vector that represents y(t)
in Equation 1. The size
of this vector is `n` -by-`1` , where `n` is
the number of equations in the system you want to solve. |

`dyp` | A vector returned by the `delyp` function
whose values are the derivative delays, dyp_{j},
in Equation 1. The size
of this vector must be `q` -by-`1` ,
where `q` is the number of solution delays, dyp_{j},
in the equation. Each element of `dyp` must be less
than t. There is one exception to this restriction:
if you are solving an initial value DDE, the value of `dyp` can
equal t at t = t_{0}.
For more information, see Initial Value Neutral Delay Differential Equations. |

If you want specify constant derivative delays having the form *dyp*_{j} = *t* – *τ*_{j}, then `delyp`

must be a vector, where `delyp(j)`

= *τ*_{j}.
Each value in this vector must be greater than zero. An exception
to this restriction occurs when you solve initial value problems for
DDEs of neutral type. In such cases, a value in `delyp`

can
equal zero at *t* = *t*_{0}.
See Initial Value Neutral Delay Differential Equations for more information.

If *dyp* is not present in the problem, set `delyp`

to `[]`

.

**Data Types: **`function_handle`

| `single`

| `double`

`history`

— Solution history

function handle | column vector | structure (`sol`

, from previous integration) | `1`

-by-`2`

cell array

Solution history, specified as a function handle, column vector, `sol`

structure
(from a previous integration), or a cell array. This is the solution
at *t* ≤ *t*_{0}.

If the history varies with time, specify the solution history as a function handle whose syntax is

`y = history(t)`

. This function returns an`n`

-by-`1`

vector that approximates the solution,*y*(*t*), for*t*<=*t*_{0}. The length of this vector,`n`

, is the number of equations in the system you want to solve.If

*y*(*t*) is constant, you can specify`history`

as an`n`

-by-`1`

vector of the constant values.If you are calling

`ddensd`

to continue a previous integration to*t*_{0}, you can specify history as the output,`sol`

, from the previous integration.If you are solving an initial value DDE, specify history as a cell array,

`{y0, yp0}`

. The first element,`y0`

, is a column vector of initial values,*y*(*t*_{0}). The second element,*yp*_{0}, is a column vector whose elements are the initial derivatives,*y*'(*t*_{0}). These vectors must be consistent, meaning that they satisfy Equation 1 at*t*_{0}. See Initial Value Neutral Delay Differential Equations for more information.

**Data Types: **`function_handle`

| `single`

| `double`

| `struct`

| `cell`

`tspan`

— Interval of integration

`1`

-by-`2`

vector

Interval of integration, specified as the vector ```
[t0
tf]
```

. The first element, `t0`

, is the initial
value of *t*. The second element, `tf`

,
is the final value of *t*. The value of `t0`

must
be less than` tf`

.

**Data Types: **`single`

| `double`

`options`

— Optional integration parameters

structure returned by `ddeset`

## Output Arguments

`sol`

— Solution

structure

Solution, returned as a structure containing the following fields.

`sol.x` | Mesh selected by `ddensd` . |

`sol.y` | An approximation to y(t)
at the mesh points. |

`sol.yp` | An approximation to y '(t)
at the mesh points. |

`sol.solver` | A character vector identifying the solver, `'ddensd'` . |

You can pass `sol`

to the `deval`

function to evaluate the solution
at specific points. For example, ```
y = deval(sol, 0.5*(sol.x(1)
+ sol.x(end)))
```

evaluates the solution at the midpoint of
the interval of integration.

## More About

### Initial Value Neutral Delay Differential Equations

An initial value DDE has *dy*_{i}≥*t*_{0} and *dyp*_{j}≥*t*_{0},
for all i and j. At *t* = *t*_{0},
all delayed terms reduce to *y*(*dy*_{i})
= *y*(*t*_{0}) and *y* '(*dyp*_{j})
= * y* '(*t*_{0}):

y '(t_{0})
= f(t_{0}, y(t_{0}), y(t_{0}),..., y(t_{0}), y '(t_{0}),..., y '(t_{0})) | (2) |

*t*>

*t*

_{0}, all derivative delays must satisfy

*dyp*< t.

When you solve initial value neutral DDEs, you must supply *y* '(*t*_{0})
to `ddensd`

. To do this, specify `history`

as
a cell array `{Y0,YP0}`

. Here, `Y0`

is
the column vector of initial values, *y*(*t*_{0}), and `YP0`

is a column vector of
initial derivatives, *y* '(*t*_{0}). These vectors must be consistent, meaning that they
satisfy Equation 2 at *t*_{0}.

## Algorithms

For information about the algorithm used in this solver, see Shampine [2].

## References

[1] Paul, C.A.H. “A Test Set of Functional
Differential Equations.” *Numerical Analysis Reports*.
No. 243. Manchester, UK: Math Department, University of Manchester,
1994.

[2] Shampine, L.F. “Dissipative Approximations
to Neutral DDEs.” *Applied Mathematics & Computation*.
Vol. 203, Number 2, 2008, pp. 641–648.

**Introduced in R2012b**

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