I need to find the analytical solution of the time-independent diffusion equation with constant coefficients on the unit disk $\Omega$ with subject to Robin boundary conditions. The formulation is as follows:

$ - \alpha \Delta\varphi (\mathbf{x}) + \beta \varphi(\mathbf{x}) = q(\mathbf{x}), ~ \mathbf{x}\in \Omega$

$ \varphi(\mathbf{x}) + 2\alpha (\nabla \varphi(\mathbf{x}) \cdot \mathbf{n}) = 0, ~ \mathbf{x} \in \partial\Omega$

where $\alpha$ and $\beta$ are not dependent on $\mathbf{x} \in \Re^2$, $\mathbf{n}$ is the normal to $\partial\Omega$.

Is it possible to derive a closed form analytical solution (e.g. Green function) for this case at all? Is it already described anywhere in the literature?

I would be very grateful for any help. Thank you very much!