Green's functions of the fractional Laplacian on a square: Boundary considerations and applications to the Lévy flight narrow capture problem

For a particle undergoing a Lévy flight of index <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>s</a:mi><a:mo>∈</a:mo><a:mo>(</a:mo><a:mn>0</a:mn><a:mo>,</a:mo><a:mn>1</a:mn><a:mo>)</a:mo></a:mrow></a:math> in the unit square, we analyze the first hitting time to a set of small targets of radius <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi mathvariant="script">O</b:mi><b:mo>(</b:mo><b:mi>ɛ</b:mi><b:mo>)</b:mo></b:mrow></b:math> for <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mn>0</d:mn><d:mo>&lt;</d:mo><d:mi>ɛ</d:mi><d:mo>≪</d:mo><d:mn>1</d:mn></d:mrow></d:math>. In particular, we show how boundary interactions and the configuration of targets within the unit square impact the expected first hitting time. Furthermore, we illustrate how a target can be “shielded” by absorbing obstacles, and how a Lévy flight search can be significantly superior in navigating these obstacles versus Brownian motion. As part of this analysis, we introduce a method for accurately computing source-neutral Green's functions of the fractional Laplacian operator on the unit square with either periodic or homogeneous Neumann boundary conditions, the latter of which we formulate and interpret using a method of images-type argument. Our approach 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 and leads directly to accurate extraction of the regular part of the Green's function (and its gradient, if so desired). Incorporating this new method for computing Green's functions into a matched asymptotic analysis framework enables us to provide new insights into the 2-D Lévy flight narrow capture problem beyond those of leading order theory. All asymptotic predictions are confirmed by full numerical solutions.