This example shows how to use `ddensd`

to
solve the initial value DDE presented by Jackiewicz [1] for
0 ≤ *t* ≤ 0.1.

Click `ddex5.m`

or
type `edit ddex5.m`

in a command window to view the
code for this example in an editor.

The equation is

*y* '(*t*)
= 2cos(2*t*) *y*(*t*/2)^{2cos(t) }+
log(*y* '(*t*/2)) – log(2cos(*t*))
– sin(*t*).

This is an initial value DDE because the delays are zero at *t*_{0}.
The initial conditions are:

*y*(0) =
1

*y* '(0) = *s*,

where *s* is
the solution of:

2 + log(*s*) –
log(2) = 0.

This equation is satisfied by *s*_{1} =
2 and *s*_{2} = 0.4063757399599599.

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

Define the DDE as a local function.

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

Define the solution delay and derivative delay. Add this line to the main function.

delay = @(t,y) t/2;

You can use one anonymous function to handle both delays since they are the same in the equation.

Define the initial conditions,

`y0`

and`s1`

, and the interval of integration,`tspan`

. Add this code to the main function.y0 = 1; s1 = 2; tspan = [0 0.1];

Solve the DDE for 0 ≤

*t*≤ 0.1, with initial conditions*y*(0) = 1, and*y*'(0) = 2. Add this code to the main function.sol1 = ddensd(@ddefun,delay,delay,{y0,s1},tspan);

Solve the equation again, this time using

*y*'(0) = 0.4063757399599599. Add this code to the main function.s2 = 0.4063757399599599; sol2 = ddensd(@ddefun,delay,delay,{y0,s2},tspan);

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

figure plot(sol1.x,sol1.y,sol2.x,sol2.y); legend('y''(0) = 2','y''(0) = .40638','Location','NorthWest'); xlabel('time t'); ylabel('solution y'); title('Two solutions of Jackiewicz''s initial-value NDDE');

Run your program to calculate and plot the solutions for each value of

*s*.

[1] Jackiewicz, Z. "One step Methods
of any Order for Neutral Functional Differential Equations." *SIAM
J. Numer. Anal.* Vol. 21, Number 3. 1984. pp. 486–511.

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