Consider the following problems:
-Uxx - Uyy+k2U=0;
∂U/∂n - ikU=0, on boundary of [-1,1]x[-1,1];
where ∂U/∂n means outside normal derivative
respect to the boundary, and i=sqrt(-1).
The question is to find out k such that the Helmholtz equation have a nonzero solution.
Is there an analytical solutions?
Hi,
https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory
Go down to application to PDE, attempt to do separation of variables, but for your problem, which is not time dependent, since its elliptic. This is basically the starting point. You would just have to match the Fourier series on the 4 boundaries, this can be imposed with your normal derivative conditions. I believe your solution will be a complex valued function, similar to solution of schrodinger equation, since you have introduced the imaginary number in your boundary condition.
I sort of believe also, you could introduce a change of variable to this original problem and solve a system of first order elliptic partial differential equations. Which could or could not be easy based on what change of variables you choose.
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