How to code a pulse function in differential equations

Hello everyone,

could you help me with my problem described below?

assuming I have a simple model dX/dt = -k*X/V, where X is amount, t is time (between 0 and 24 hours), V is volume, and k is rate.

I would like to add a certain amount of mass (M) into my system every 1 hour, in other words, adding mass on 1, 2, 3...23 hour. It does not need to be exactly on every hour, but could be a small interval (tau) around 1, 2, 3, ... so equation becomes something like

dX/dt = -k*X/V + M/tau, M is zero when t is out of these small intervals.

I just cannot figure out a smart way to code this, and then solve the ODE. Could you give me some idea? Thanks in advance!

Rui

[태훈]

If you can't get explicit solutions, I guess you could try "a series of odes" using a for loop. For example,...

tv=[0:1:100]

y0=1

for k=1:length(t)-1

tspan=[tv(k), tv(k+1)];

[t,y]=ode45(@yourfun, tspan...)

y0=y(length(y))+1 % e.g., adding 1 at every 1 hour

end

Answer 1

[Ayush Aniket]

Hi Rui,

Assuming 'tau' to be a small interval around the hour marks, you can define a custom 'odefun' to pass as argument to the 'ode45' MATLAB function. This custom function should add the mass 'M' if the evalutaion time lies in the 'tau' interval around the hour marks as shown below:

function dXdt = modelODE(t, X, k, V, M, tau, additionTimes)
    % Check if t is within any of the mass addition intervals
    isAddingMass = any(abs(t - additionTimes) <= tau/2);
    
    % Compute dX/dt
    if isAddingMass
        dXdt = -k*X/V + M/tau;
    else
        dXdt = -k*X/V;
    end
end

The 'additionTimes' argument specifies the hour marks around which the mass should be added to the model. For your problem it can be defined as below:

additionTimes = 1:23; % Times to add mass
% Time span
tSpan = [0 24];

% Solve the ODE
[t, X] = ode45(@(t, X) modelODE(t, X, k, V, M, tau, additionTimes), tSpan, X0);