How to solve for PDE (Greens function) for mixed Neumann-Dirichlet boundary value problem?

Question

I want to know if there is a way to solve mixed Neumann-Dirichlet boundary value problem using Greens function. For instance, let us assume an infinite boundary with reflecting surface ($\partial \Omega_r$) but a small portion of the boundary is absorbing ($\partial \Omega_a$). I want to derive the PDF of particle motion in such an environment using a diffusion differential equation. I searched for literature and get Narrow escape problem but there they assume that the absorbing boundary is quite small as compared to the reflective one. What if the absorbing boundary is not that small, is there a reference that deals specifically with mixed boundary value problems.

Answer 1

Actually, if you look at the derivation of the narrow escape PDE, for instance in a paper written by Xinfu Chen & Carey Caginalp. The "size" of the window $\varepsilon > 0 $ can be as large as possible. The name "narrow escape" refers to the fact that $\varepsilon$ is tiny, whence various perturbation technique can be used.