I have a PDE of the form $\partial_t y(x,t)+\partial_x(y(x) y'''(x)-f(x))=0$, where $f(x)=\cos(x)$.
Suppose a stable equilibrium exists, and I want to find the steady-state solution $y(x) y'''(x)=f(x)$ and I expect the physical solution to be periodic (as the source term is also periodic). How do I solve this equation (or 3-order nonlinear ODE in general) numerically (or analytically) in Scipy/Matlab/Mathematica?
For the boundary conditions, for the original problem I can just specify $y(x,0)$ and evolve the system, and the steady-state solution can be obtained for large $t$. But if I want to solve the steady-state ODE directly, do I need to figure out three equations for the boundary conditions for it to be solvable?