在 Matlab 中求解矩阵值微分方程

Question

I am trying to solve a particular system of ODE's dF/dt = A*F, F_initial = eye(9) . Being a Matlab novice, I am trying to somehow use the implemented ode45 function, and I found useful advises online. However, all of them assume A to be constant, while in my case the matrix A is a function of t , in other words, A changes with each time step.

I have solved A separately and stored it in a 9x9xN array (because my grid is t = 0:dt:2 , N=2/dt is the number of time-steps, and A(:,:,i) corresponds to it's value at the i-th time step). But I can't implement this array in ode45 to eventually solve my ODE.

Any help is welcomed, and please tell me if I have missed anything important while explaining my problem. Thank you

Answer 1

First of all, F must be a column vector when using ode45. You won't ever get a result by setting F_initial = eye(9), you'd need F = ones(9,1).

Also, ode45 (documentation here, check tspan section) doesn't necessarily evaluate your function at the timesteps that you give it, so you can't pre-compute the A matrix. Here I'm going to assume that F is a column vector and A is a matrix which acts on it, which can be computed each timestep. If this is the case, then we can just include A in the function passed to ode45, like so:

F_initial = ones(9,1);
dt = 0.01;
tspan = 0:2/dt:2;
[t, F] = ode45(@(t,F) foo(t, F, Ainput), tspan, F_initial);

function f = foo(t, F, Ainput)
    A = calculate_A(t, Ainput);
    f = A*F;
end

function A = calculate_A(t, Ainput)
    %some logic, calculate A based on inputs and timestep
    A = ones(9,9)*sqrt(t)*Ainput;
end

The @(x) f(x,y) basically creates a new anonymous function which allows you to treat y as a constant in the calculation.

Hope this is helpful, let me know if I've misunderstood something or if you have other questions.