Analytical solution of diffusion PDE with Robin boundary condition

Question

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!

Answer 1

Yes, there is an associated Green's function for your problem, which is explicitly known (at least for complex $\alpha$, $\beta$ with positive real parts). More precisely, the "Robin Green's function", in which you are interested into, can be represented by the sum of an integral expression (= Green's function on $\mathbb{R}^2$) and a Fourier series that contains a product of several modified Bessel functions (= corrector function). The expression is very complicated, but it is known, which even allows you to consider inhomogeneous Robin boundary conditions.

Unfortunately, I do not know any reference where you can find the Green's function for your problem. However, the case $\alpha=1$ and $\beta=0$ has deeply been studied in the literature. The Greens function you are looking for should be called the Robin Green's function for the perturbed Poisson equation on the unit circle (disk).

Good luck.