I'm looking to solve the Helmholtz equation $$\Delta u + k^2u = 0$$ Or in polar form $$u_{rr} + \frac{u_r}{r} + \frac{u_{\theta\theta}}{r^2}+k^2u=0$$ In the circle of radius $a$, with boundary conditions $$u(a,\theta)=f(\theta)$$ Where of course $f$ is a periodic function with $f(\theta+2\pi)=f(\theta)$ for all $\theta$.
I could easily find several sites where a solution was given for $f=0$ by using separation of variables, but is there any way of solving this for any function $f$? Is it still possible to use separation of variables with non-zero boundaries? If not, I would like to find a numerical solution to $u(r,\theta)$ by using finite differences in the polar grid and the DFT (Discrete Fourier Transform).
Thank you very much.
The separation of variables leads to an infinity of solutions which sum express the general solution. The periodic solutions are selected and the respective coefficients are computed in order to agree with the boundary condition (expressed on the form of a Fourier series) :