How to solve time dependent ODEs of the form (dx/dt = f(x,t))using ODE45

I have a complex time dependent dynamic system of order 4. I want to solve this system using ODE45 in matrix form.

My dynamic equation is of the form below

I need to solve this time dependent ODEs in Matrix form. I am getting dimension mismatch error for the below function;

Can anybody help me with this?

Thanks

function xdot = TV_dynamic(tspan, x, k)
%%% k is the time dependent variable 
con_exp = (0.1/2) * exp(x(1,1) + x(2,1)+ x(3,1));

Delta_z = [(x(1,1) - (sin(0.1*k)/2) + con_exp + (x(4,1)*cos(0.1*k)));
           (x(2,1) - (sin(0.1*k)/2) + con_exp + (x(4,1)*sin(0.1*k)));
           (x(3,1) - (sin(0.1*k)/2) + con_exp + x(4,1));
           ((cos(0.1*k)*x(1,1)) +  (sin(0.1*k)*x(2,1)) + x(3,1)- 1)];
       
Delta_zz = [1+con_exp   con_exp     con_exp     cos(0.1*k);
            con_exp     1+con_exp   con_exp     sin(0.1*k);
            con_exp     con_exp     1+con_exp   1;
            cos(0.1*k)  sin(0.1*k)  1           0];

Delta_zt = [((-0.1*cos(0.1*k)/2) - (0.1 * x(4,1) * sin(0.1*k)));
            ((-0.1*cos(0.1*k)/2) + (0.1 * x(4,1) * cos(0.1*k)));
            (-0.1*cos(0.1*k)/2) ; 
            ((-0.1* sin(0.1*k)* x(1,1)) + (0.1* cos(0.1*k))*x(2,1))];
        
xdot(:,1) =   (- inv(Delta_zz)) * ( P * Delta_z(:,1) + Delta_zt );
end

[darova]

Attach the whole code

Answer 1

[Ameer Hamza]

It depends on how the time-dependent term is defined. The following shows an example if the term 'k' depends on time.

k_fun = @(t) sin(t); % k as function of time
tspan = [0 1];
ic = [1; 0; 0; 0];
[t, y] = ode45(@(t, x) TV_dynamic(t, x, k_fun(t)), tspan, ic);
plot(t, y);

function xdot = TV_dynamic(tspan, x, k)
%%% k is the time dependent variable 
P = rand(4); % for example

con_exp = (0.1/2) * exp(x(1,1) + x(2,1)+ x(3,1));
Delta_z = [(x(1,1) - (sin(0.1*k)/2) + con_exp + (x(4,1)*cos(0.1*k)));
           (x(2,1) - (sin(0.1*k)/2) + con_exp + (x(4,1)*sin(0.1*k)));
           (x(3,1) - (sin(0.1*k)/2) + con_exp + x(4,1));
           ((cos(0.1*k)*x(1,1)) +  (sin(0.1*k)*x(2,1)) + x(3,1)- 1)];
       
Delta_zz = [1+con_exp   con_exp     con_exp     cos(0.1*k);
            con_exp     1+con_exp   con_exp     sin(0.1*k);
            con_exp     con_exp     1+con_exp   1;
            cos(0.1*k)  sin(0.1*k)  1           0];
Delta_zt = [((-0.1*cos(0.1*k)/2) - (0.1 * x(4,1) * sin(0.1*k)));
            ((-0.1*cos(0.1*k)/2) + (0.1 * x(4,1) * cos(0.1*k)));
            (-0.1*cos(0.1*k)/2) ; 
            ((-0.1* sin(0.1*k)* x(1,1)) + (0.1* cos(0.1*k))*x(2,1))];
        
xdot(:,1) =   (- inv(Delta_zz)) * ( P * Delta_z(:,1) + Delta_zt );
end

Note that I have defined P as a random 4x4 matrix.

[Bhagyashree Umathe]

Thank you for your reply. I want to confirm that in main code, "k" is not reflecting anywhere while in function file, time dependent variable is "k". Is it okay?

Also, "k_fun = @(t) sin(t)" will it take care of cos(t) in dynamic equation?

[Ameer Hamza]

Yes, it is fine. 'k' itself does not need to appear in the main script. Also, k_fun = @(t) sin(t), is just an example. It indicates that k is a function of time. In this case, I used sin() as an example. You can vary 'k' however you want.