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I'm trying to solve the Helmholtz equation on the unit disk with a prescribed boundary condition of the third kind (Robin) on the circle.

The Helmholtz equation in polar coordinates is given by

$\Delta p(r,\varphi)+k^2\cdot p(r,\varphi)=0$

and the particular boundary condition shall be

$p_r(1,\varphi)=f(\varphi)\cdot p(1,\varphi)$.

In particular, a solution for $f(\varphi)=-\sin(\varphi)$ is desired. Parameter $k$ has to be found such that the boundary condition is satisfied.

Solution attempt: Separation of variables fails for this example because the boundary condition is not separable with $p(r,\varphi)= R(r)\cdot \Phi(\varphi)$ because then

$\Phi(\varphi)\cdot R'(1) = f(\varphi)\cdot \Phi(\varphi)\cdot R(1) \Leftrightarrow R'(1)/R(1) = f(\varphi)\cdot \Phi(\varphi)$

which indeed would require $f(\varphi)\cdot \Phi(\varphi)=\mathrm{const.}$ and thus is not possible.

Questions:

  1. Do you know of other approaches that are suitable for solving the above problem?
  2. Can you name a specific term for referring to the type of above boundary condition (for me it is a spatially dependent Robin boundary condition)?
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This can be solved without numerical methods. Look at

https://en.m.wikipedia.org/wiki/Poisson_kernel

And try to write down the solution explicitly ;) hope it works

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  • $\begingroup$ Thank you for your suggestion. The wikipedia page you cited says that the Poisson Kernel is used to solve the Laplace equation with Dirichlet boundary conditions. Here, we have some kind of Robin boundary condition. Can you be a little more explicit how to write down the solution explicitly and which steps to apply. $\endgroup$ – apfel Jul 5 '17 at 13:37
  • $\begingroup$ Sorry I roughly thought it was dirichlet. My mistake. $\endgroup$ – user459312 Jul 5 '17 at 14:26

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