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

2025/04/16 by Justin C. Tzou, Tzou, Justin C.
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #35C20 #35J08 #Analysis of PDEs (math.AP) #Diffusion and Search Dynamics #FOS: Mathematics #Fractional Differential Equations Solutions #stochastic dynamics and bifurcation

paper · pdf · doi:10.48550/arxiv.2504.12084

openalex publication_date 2025/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 Lévy flight of index α∈ (0,1) searches for small target(s) of radius O(ε) for 0 < ε ≪ 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 Lévy flight search can be significantly superior in navigating these obstacles versus Brownian motion. All asymptotic predictions are confirmed by full numerical solutions.

Citations

• First Hitting Time of a One-Dimensional Levy Flight to Small Targets
• Imperfect Narrow Escape problem
• Spike Solutions to the Supercritical Fractional Gierer-Meinhardt System
• Geodesic Lévy flights and expected stopping time for random searches
• The narrow capture problem on general Riemannian surfaces
• Universal kinetics of imperfect reactions in confinement
• Asymptotic analysis of extended two-dimensional narrow capture problems
• Optimization of the Mean First Passage Time in Near-Disk and Elliptical Domains in 2-D with Small Absorbing Traps
• Full distribution of first exit times in the narrow escape problem
• Simulation and Optimization of Mean First Passage Time Problems in 2-D using Numerical Embedded Methods and Perturbation Theory
• Hybrid asymptotic-numerical approach for estimating first passage time densities of the two-dimensional narrow capture problem
• Nonlocal problems with Neumann boundary conditions
• Nonlocal problems with Neumann boundary conditions
• Fractional semilinear Neumann problems arising from a fractional Keller--Segel model
• Mean first passage time for a small rotating trap inside a reflective disk
• Fractional diffusion with Neumann boundary conditions: the logistic equation
• The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray-Scott Model
• Eagle Strategy Using Lévy Walk and Firefly Algorithms For Stochastic Optimization
• From the long jump random walk to the fractional Laplacian
• Narrow Escape, Part II: The circular disk
• Narrow Escape, Part I

Related

• First Hitting Time of a One-Dimensional Levy Flight to Small Targets
• Fast integral methods for the Neumann Green's function: applications to capture and signaling problems in two dimensions
• Symmetric Lévy flights in semi-infinite domain
• Asymptotic analysis of target fluxes in the three-dimensional narrow capture problem
• A walk outside spheres for the fractional Laplacian: fields and first\n eigenvalue