# Is it possible to explicitly solve the inhomogeneous Helmholtz equation in a rectangle?

Consider the following Helmholtz equation in a rectangle $\Omega$ and Neumann boundary conditions: \begin{align} \Delta u + k^2 u = \delta_y, \quad \quad x \in \Omega, \\ \frac{\partial u}{\partial \nu} = 0, \quad \quad x \in \partial \Omega. \end{align} Here $\delta_y$ is some point source emitted from the point $y\in \Omega$. Can an explicit solution be found for this equation?

If you know a free-space solution of $$\Delta v+k^2v=\delta_y$$ then you can solve for $w$ such that $$\Delta w+k^2w=0 \\ \frac{\partial w}{\partial n}=\frac{\partial v}{\partial n}$$ and the solution you want will be $v-w$.
• Why does $u=v-w$? How do you know that?
• @majormaki : $\Delta(v-w)+k^2(v-w)=(\Delta v+k^2v)-(\Delta w+k^2w)=\delta_y$ and $\frac{\partial}{\partial n}(v-w)=0$, which are the equations for $u$. Feb 21, 2018 at 23:17