Fractional Laplacian in Bounded Domains
Mohsen Zayernouri, Li-Lian Wang, Jie Shen, George Em Karniadakis · Cambridge University Press eBooks · 2024
The fractional Laplacian has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. Although the study of the fractional Laplacian is far from complete, this chapter can serve as a proper educational/research starting point for students/researchers in order to employ these operators to model complex anomalous systems. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. By contrast, the spectral definition requires only the standard local boundary condition. We compare several commonly used definitions of the fractional Laplacian theoretically, through their stochastic interpretations as well as their analytical properties. Then, we present quantitative comparisons using a sample of state-of-the-art methods. We finally discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems.