How to solve a differential equation with a time-varying term?

Hi all,

I'm trying to solve the following differential equation: x' = (A-B*K)*x, where A, B, and K are matrices. A and B are constant, but K is composed of different values (each corresponding to a different value in time).

This is the ode45 call I have:

[T,X] = ode45(@(t,x) sysfun(T,x,A,B,K,T),T,[1;0]);

Where T is my time vector I previously defined.

This is the sysfun function I defined:

function dx = sysfun(t,x,A,B,K,T)
dx = (A-B*K)*x;
end

My question is, how can I vary K? I'd like to be able to index it, but I don't understand how to do that. I saw some similar questions/answers but they didn't really help. I did find the following example from here

k = find(t >= timeserie,1);
k = max(1,k-1);
a = A(k);

However when I tried applying this to my code by doing the following, I just kept getting that k was always = 1 (instead of getting k =1, then k = 2, then k =3, etc).

function dx = sysfun(t,x,A,B,K,T)

k = find(t>=T)
k = max(1,k-1);
Kk = K(k,:);
dx = (A-B*Kk)*x;
end

Any help would be greatly appreciated. Thanks!

[Sam Chak]

Hi @Julian

Do you have the time-varying functions of the elements in the matrix K?

It looks like the matrix K depends on some kind of a schedule.

[Julian]

Yes, I calculated the values of Ki(t) using some values I obtained from another differential equation that is solved earlier in my code.

Answer 1

[Torsten]

Use "interp1" to interpolate the elements of K to the time t requested by the ODE integrator.

[Julian]

You mean to use interp1(K,t) ? Could you explain where this would go? I don't really understand why we're interpolating when we already have a value of K for each value of t

[Torsten]

You don't have values for each value of t since the MATLAB solver ode45 does not only use the values for t you specify as output values in tspan, but also values in between.

Thus use

Kt = interp1(T,K,t)

to interpolate K(T) to a time t requested by the solver.

[Julian]

Oh that makes sense, thank you. I tried adding that to my code like this:

function dx = sysfun(t,x,A,B,K,T)
Kt = interp1(T,K,t);
dx = (A-B*Kt)*x;
end

But I get a bunch of errors. The errors come from the B*Kt part because my code is still treating Kt like it's still a matrix composed of all rows of K instead of each individual row.

[Torsten]

K can be an array with its first dimension equal to the length of the T vector.

So the elements of K don't need to be interpolated separately.

[Julian]

Ohh, so

[T,X] = ode45(@(t,x) sysfun(T,x,A,B,K,T),T,[1;0]);

becomes

[T,X] = ode45(@(t,x) sysfun(T,x,A,B,interp1(T,K,t),T),T,[1;0]);

and everything works. Thank you so much for your help!