How to solve two differential equations simultaneously with one having a matrix form of initial condition?

I have a set of two differential equations with initial conditions and was wondering how am I able to solve them simulateously using ODE 45? The equations are as shown below,

with initial conditions,

where is the Jacobian matrix. I want to integerate these equations simulationeously (42 ODEs). How is it possible since one initial condition is in a vector form and the other is a matrix.

Answer 1

[Peter O]

Hi Adham,

Just to be clear, it's an ordinary differential equation, is /? If so, and each matrix term is separable, then you can just reorder them into a vector for the integration.

So if the below is an accurate representation of your problem:

phi = [x1, x2;
       x3, x4]
phi_prime = [dx1/dt, dx2/dt
             dx3/dt, dx4/dt]

Then

         function dx = myDeriv(t,x)
         actual_x = x(1);
         actual_x_prime = f(actual_x, t);
         phi_vector = x(2:end);
         phi_matrix = reshape(phi_vector,6,7) % Assuming 6R x 7C here...
         A = getA(t);
         dphidt_matrix = A*phi_matrix; % 
         dphidt_vector = dphidt_matrix(:); % Reshape to column. MATLAB is column major ordering.
         dx = [actual_x_prime;
               dphidt_vector];
         
         end
         
         function A = getA(t)
         % do whatever
         end

And calling it:

         phi0 = eye(6);
         x0 = [actual_x0; phi0(:)];
         tspan = [0 10];
         [t,x] = ode45(@myDeriv,tspan, x0)
         

Finally reshape out your variables for each time point, because x will by n_t by 43 instead of the scalar vs time and matrix vs time variables you want:

actual_x_integrated = x(:,1);
phi_integrated = nan(6,7,length(t));
for ix =1:length(t)
phi_integrated(:,:,ix) = reshape(x(ix,2:end),6,7);
end

I believe that should do what you're after.

[Adham Alkhaja]

Perfect! Exactly what I was looking for. Thank you!