Using Runge-Kutta in Matlab

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.

Answer 1

[Bruno Luong]

" 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

[L'O.G.]

@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]

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.]

@Bruno Luong Thank you, and then pass that into ode45? How do I choose the time step for the method?

[Torsten]

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.]

@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]

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]

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.]

Thank you, I have a follow up question that I will ask