understanding DDE23 function format

I look at this example https://www.mathworks.com/help/matlab/math/dde-with-constant-delays.html

DDE23 function has a format of

ddex1de(t,y,Z)

How should one understand Z?

Answer 1

[Guru Mohanty]

I understand you are trying to solve system of differential equation using ‘dde23’. To do this you need to do following steps.

1. Define constant delays.

In this system of equation there are two lags i.e. (t-1) and (t-0.2).

        lags = [1 0.2];

2. Define Solution History.

It Defines the first solution from which the solver starts iterations.

        function s = history(t)
            s = ones(3,1);
        end

3. Form the equation.

        function dydt = ddex1de(t,y,Z)
            ylag1 = Z(:,1);
            ylag2 = Z(:,2);
            dydt = [ylag1(1); 
                    ylag1(1)+ylag2(2); 
                    y(2)];
        end

Here In the call to ddex1de,

‘t’ is a scalar indicates the current ‘t’ in the equation

‘y’ is a column vector approximates y(t)

‘Z’ is a column vector approximates y(t – αj) for delay αj= lags(J).

In the following example Solution history is [1;1;1]. So Approximate values of lag ie ‘Z’ will be a matrix of size 3x2.In ‘Z’ Row defines the number of equations and column defines Number of lags.

4. Solve using ‘dde23’.

        tspan = [0 5];
        sol = dde23(@ddefun, lags, @history, tspan);

[alpedhuez]

Thank you.

1. Would you write down the content of this 3*2 matrix Z?

2. Another question:

function dydt = ddex1de(t,y,Z)
            ylag1 = Z(:,1);
            ylag2 = Z(:,2);
            dydt = [ylag1(1); 
                    ylag1(1)+ylag2(2); 
                    y(2)];
        end

I would think intuitively we define ddex1de by dydt so it should be like

function ddex1de = dydt(t,y,Z)
            ylag1 = Z(:,1);
            ylag2 = Z(:,2);
            dydt = [ylag1(1); 
                    ylag1(1)+ylag2(2); 
                    y(2)];
end

Woud you explain why this is not the case?

[Guru Mohanty]

As it is a System of 3 Equations, 'y', 'dydt', 'ylag1' 'ylag2' have dimension 3x1. There are two constant Delay present, So The dimention of 'Z' is 3x2.