Green's functions of the fractional Laplacian on a square -- boundary considerations and applications to the L\'{e}vy flight narrow capture problem

On the unit square, we introduce a method for accurately computing source-neutral Green's functions of the fractional Laplacian operator with either periodic or homogeneous Neumann boundary conditions. This method involves analytically constructing the singular behavior of the Green's function in a neighborhood around the location of the singularity, and then formulating a ``smooth'' problem for the remainder term. This smooth problem can be solved for numerically using a basic finite difference scheme. This approach allows accurate extraction of the regular part of the Green's function (and its gradient, if so desired). This new tool enables quantification of properties and characteristics of the narrow capture problem, where a particle undergoing a L\'{e}vy flight of index $\alpha \in (0,1)$ searches for small target(s) of radius $\mathcal{O}(\varepsilon)$ for $0 < \varepsilon \ll 1$ on a bounded two-dimensional domain. In particular, it allows us to show how boundary interactions and configuration of multiple targets impact expected search time. Furthermore, we are able to illustrate how a target can be ``shielded'' by obstacles, and how a L\'{e}vy flight search can be significantly superior in navigating these obstacles versus Brownian motion. All asymptotic predictions are confirmed by full numerical solutions.