What is the fractional Laplacian? A comparative review with new results

By Anna Lischke, Guofei Pang, Mamikon A. Gulian, Fangying Song, Christian Glusa, Xiaoning Zheng, Zhiping Mao, Wei Cai, Mark M. Meerschaert, Mark Ainsworth and 1 more. Journal of Computational Physics, 2018

476 citations recorded by the source corpus.

The fractional Laplacian in R^d 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. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function u(x) must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. In contrast, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: "What is the fractional Laplacian?" We compare several commonly used definitions of the fractional Laplacian (the Riesz, spectral, directional, and horizon-based nonlocal definitions), and we use a quantitative approach to identify their practical differences. In this work, we provide a quantitative assessment of new numerical methods as well as available state-of-the-art methods for discretizing the fractional Laplacian, and we present new results on the differences in features, regularity, and boundary behaviors of solutions to equations posed with these different definitions. We present stochastic interpretations and demonstrate the equivalence between some recent formulations. Through our efforts, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/8530ee42e49fe89db5e770866f4d1f9a1a4b4243

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Lexing Ying
2. Maizar Raissi
3. G. Em Karniadakis
4. Paris Perdikaris
5. Victor Minden
6. Mamikon A. Gulian

In Computational Theory and Mathematics

1. G. Em Karniadakis
2. Paris Perdikaris
3. Maizar Raissi
4. Lexing Ying
5. Victor Minden
6. Mamikon A. Gulian
7. Yibo Yang
8. Xuhui Meng

In Numerical Analysis

1. G. Em Karniadakis
2. Lexing Ying
3. Maizar Raissi
4. Paris Perdikaris
5. Victor Minden

Citations in the corpus

• Machine Learning of Space-Fractional Differential Equations (2018, background): “Moreover, unlike standard derivatives, fractional derivatives, whether in R, R or on bounded subsets, are nonlocal operators typically defined by singular integrals or eigenfunction expansions that are difficult and expensive to discretize [14], [18].”
• A Simple Solver for the Fractional Laplacian in Multiple Dimensions (2018, background): “General references for fractional Laplacians on bounded domains include, e.g., Ros-Oton [43], D’Elia and Gunzburger [12], Felsinger [14], and Lischke et al. [31].”