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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?

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  • $\begingroup$ If you assume that your solution is periodic over the space domain, could you write $y(x)$ and $f(x)$ as a Fourier expansion, work out the derivative and multiplication in the LHS and apply the Fourier integral left and right? If you assume $f(x)$ to have a Fourier expansion, that would leave only the Fourier expansion coefficients at the RHS. This transforms your non-linear ode in a large set of equations for the Fourier coefficients for $y(x)$. But as I write this down, I realise that this might not necessarily make things easier.... $\endgroup$ – GertVdE Jul 2 at 6:27
  • $\begingroup$ Does y depend on t or not? $\endgroup$ – David Ketcheson Jul 3 at 8:04
  • $\begingroup$ To solve it numerically, you can use finite different scheme or finite element scheme based on B-Splines (since you need higher continuities). $\endgroup$ – Chenna K Jul 3 at 13:39

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