Delay Differential Equation for arbitrary functions experiencing delay

Hello everyone and Hope yoo had a nice weekend and have a great one

I have this system of equations

dy1/dt = 2t + y1(t - tau2) + y2 + g(t - tau1)

dy2/dt = 3t + f(t - tau1)

I want to solve this system using dde23. I already know this function but my problem is, I have already f(t) and g(t) as vectors obtained by another mfile and I do not know how to apply the delay to them here. I have uploaded the f, g and t data.

history can be [1;0] or anything else and tau1 = 1 and tau2 = 0.2;

thanks in advance for your precious time devoting to this.

Answer 1

[Narvik]

Hi Proman,

As per my understanding, you are trying to solve a system of delay differential equations (DDEs) "dde23" in MATLAB, with f(t) and g(t) as vectors.

To solve the system of DDEs, refer to the following steps:

• Interpolate f(t) and g(t) using the "interp1" MATLAB function to create continuous functions from your discrete data

f_interp = @(t) interp1(t, f, t, 'linear', 'extrap');
g_interp = @(t) interp1(t, g, t, 'linear', 'extrap');

• Define function that describes the system using these interpolated functions

function dydt = dde_system(t, y, Z)
    tau1 = 1;
    tau2 = 0.2;
    
    y1_tau2 = Z(:,1); % y1(t - tau2)
    
    dydt = zeros(2,1);
    dydt(1) = 2*t + y1_tau2 + y(2) + g_interp(t - tau1);
    dydt(2) = 3*t + f_interp(t - tau1);
end

• Specify the delays (tau1 = 1, tau2 = 0.2) and initial conditions

% Delays
lags = [1, 0.2];

% Initial conditions
history = @(t) [1; 0]; % History function

• Use dde23 to compute the solution over a specific time span

sol = dde23(@dde_system, lags, history, [0, 10]);

Refer to the following documentation for more information on "interp1" function:

https://www.mathworks.com/help/matlab/ref/double.interp1.html

Refer to the following documentation for more information on "dde23" function:

https://www.mathworks.com/help/matlab/ref/dde23.html

Hope this helps!