Unstable ODE

muzialis

Hello there,

I am solving numerically the ODE

$$ \dot{y} = min \, (y, A) + B\, sin(t)$$ , A,B being constant.

I obtain a very "wiggled" solution which is very fine to me actually, as it echoes the problem I am studying.
However, as the numerical solution scheme is quite "rudimentary" I am wondering if I am getting an accurate answer.

In this respect I am wondering if somebody could point me towards a suitable theory for ODE to study their well-posedness, continuity with respect to inital data, stability.
I am no expert, but I understand the problems one would encounter if trying to solve the heat equation with negative conductvity!

The ODE, in the regime $$ y(t) < A$$ is of they type $$ \dot{y} = y + f(t)$$ which is prone to diverging exponentially.
I am trying to understand if the solution I find is meanigful or just "computer noise".

Thanks

Mute

The ODE looks simple enough that you could solve it analytically for an example case or two. For example, suppose ##y(0) = y_0 < A##. Then initially your ODE is just ##\dot{y} = y(t) + B\sin(t)##, as you said, which has solution ##y(t) = (y_0+B/2)e^t - (B/2)(\sin t + \cos t)## (double-check that). This solution is valid until it grows to ##y(t_1) = A##. At this point it must satisfy the ODE ##\dot{y} = A + B\sin t##, which has solution ##y(t) = y(t_1) + A(t-t_1) - B(\cos t - \cos t_1)##. You can find ##t_1## by setting ##y(t_1) = A## in your first solution for ##t<t_1##, and ##y(t_1)## is just A. (You will probably have to solve numerically for ##t_1##). If your parameter values are such that y(t) dips below A again, you would need to find the time ##t_2## at which that happens and solve the ODE with ##\mbox{min}(y,A) = y## again, and so on.

In this way you can construct a piece-wise analytic solution for some simple parameter choices which you can test against your numerical solution.