Using Runge-Kutta in Matlab

2023 8 23
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I am trying to solve an equation dx/dt = f(x,t) where f(x,t) is a velocity. I want to solve for the positions x to obtain the trajectory x(t). I've seen that the time evolution of the positions x can be calculated using Runge-Kutta, but Matlab's implementation via ode45 (or similar) looks like it requires a function in the first argument. Here, I instead have a starting position x0 and compute the velocity at that initial point, f(x0,t0). How do I use Runge-Kutta to determine the time evolution of the position based on the velocities? If it matters, I can calculate an arbitrary instantaneous velocity f(x,t) from the forces acting on a position x. To give more context, I am calculating the velocity fxt = S*Fx where the matrix S and the force Fx both depend on the particle position, so the velocity that I calculate is a scalar.

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Bruno Luong
Bruno Luong 2023-8-23
编辑：Bruno Luong 2023-8-23

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" I can calculate an arbitrary velocity f(x,t) from the forces acting on a position x."
Good this is a very good starting point. If you have f(x,t) in MATLAB function form,
then simply call
sol = ode45(@(t,x) f(x,t), [t0 tend], x0)
tend is the last time where you want the solution to be computed.
Read the doc of ode45

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L'O.G.
L'O.G. 2023-8-23
编辑：L'O.G. 2023-8-23
@Bruno Luong The problem is the velocity f(x,t) is just something I calculate at a specific point. It's not in "function form", if I'm understanding you correctly: it's a scalar. It depends on the forces which themselves depend on the position. Doesn't that complicate things? To give context, I am calculating the velocity fxt = S*Fx where the matrix S and the force Fx both depend on the particle position.
Bruno Luong
Bruno Luong 2023-8-23
编辑：Bruno Luong 2023-8-23
then simply write down
function v = f(x,t)
v = S(x).*Fx(x); % you said you know how to compute S(x) and Fx(x)
end
L'O.G.
L'O.G. 2023-8-23
编辑：L'O.G. 2023-8-23
@Bruno Luong Thank you, and then pass that into ode45? How do I choose the time step for the method?
Torsten
Torsten 2023-8-23
dx/dt = v, and v is a function of the position that is given to you in the input variables (t,x). What is the problem ?
L'O.G.
L'O.G. 2023-8-23
@Torsten I only know the initial position and velocity. As mentioned, I want to determine the particle trajectory. I hope that makes sense.
Bruno Luong
Bruno Luong 2023-8-23
ode45 adjusts the time step for you automatically. If you need solution at specific time, set tspan argument. Please read the doc in the link above.
Torsten
Torsten 2023-8-23
编辑：Torsten 2023-8-23
I only know the initial position and velocity. As mentioned, I want to determine the particle trajectory. I hope that makes sense.
ode45 has x - the actual position at time t - as input to your function where you define dx/dt.
L'O.G.
L'O.G. 2023-8-23
Thank you, I have a follow up question that I will ask

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