## This content will become publicly available on October 30, 2020

# What Is the Fractional Laplacian? A Comparative Review with New Results

## Abstract

The fractional Laplacian in $$\mathbb{R}^{d}$$, which we write as (–Δ) ^{α/2} with, α ϵ (0, 2) has multiple equivalent characterizations. Furthermore, 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 must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. Yet, 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?” Beginning from first principles, we compare several commonly used definitions of the fractional Laplacian theoretically, through their stochastic interpretations as well as their analytical properties. Next, we present quantitative comparisons using a sample of state-of-the-art methods. We discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems: radial basis function collocation (for the Riesz fractional Laplacian) and nonharmonic lifting (for the spectral fractional Laplacian). In our numerical studies, we aim to compare different definitions on bounded domains using a collection of benchmark problems. We consider the fractional Poisson equation with both zero and nonzero boundary conditions, where the fractional Laplacian is defined according to the Riesz definition, the spectral definition, the directional definition, and the horizon-based nonlocal definition. We verify the accuracy of the numerical methods used in the approximations for each operator, and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions. 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.

- Authors:

- Brown Univ., Providence, RI (United States)
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Southern Methodist Univ., Dallas, TX (United States)
- Michigan State Univ., East Lansing, MI (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF); National Natural Science Foundation of China (NNSFC)

- OSTI Identifier:
- 1574478

- Report Number(s):
- SAND-2019-13611J

Journal ID: ISSN 0021-9991; 681226

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Name: Journal of Computational Physics; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Fractional Laplacian; anomalous diffusion; regularity; stable Lévy motion; nonlocal model

### Citation Formats

```
Lischke, Anna, Pang, Guofei, Gulian, Mamikon, Song, Fangying, Glusa, Christian, Zheng, Xiaoning, Mao, Zhiping, Cai, Wei, Meerschaert, Mark M., Ainsworth, Mark, and Karniadakis, George Em. What Is the Fractional Laplacian? A Comparative Review with New Results. United States: N. p., 2019.
Web. doi:10.1016/j.jcp.2019.109009.
```

```
Lischke, Anna, Pang, Guofei, Gulian, Mamikon, Song, Fangying, Glusa, Christian, Zheng, Xiaoning, Mao, Zhiping, Cai, Wei, Meerschaert, Mark M., Ainsworth, Mark, & Karniadakis, George Em. What Is the Fractional Laplacian? A Comparative Review with New Results. United States. doi:10.1016/j.jcp.2019.109009.
```

```
Lischke, Anna, Pang, Guofei, Gulian, Mamikon, Song, Fangying, Glusa, Christian, Zheng, Xiaoning, Mao, Zhiping, Cai, Wei, Meerschaert, Mark M., Ainsworth, Mark, and Karniadakis, George Em. Wed .
"What Is the Fractional Laplacian? A Comparative Review with New Results". United States. doi:10.1016/j.jcp.2019.109009.
```

```
@article{osti_1574478,
```

title = {What Is the Fractional Laplacian? A Comparative Review with New Results},

author = {Lischke, Anna and Pang, Guofei and Gulian, Mamikon and Song, Fangying and Glusa, Christian and Zheng, Xiaoning and Mao, Zhiping and Cai, Wei and Meerschaert, Mark M. and Ainsworth, Mark and Karniadakis, George Em},

abstractNote = {The fractional Laplacian in $\mathbb{R}^{d}$, which we write as (–Δ)α/2 with, α ϵ (0, 2) has multiple equivalent characterizations. Furthermore, 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 must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. Yet, 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?” Beginning from first principles, we compare several commonly used definitions of the fractional Laplacian theoretically, through their stochastic interpretations as well as their analytical properties. Next, we present quantitative comparisons using a sample of state-of-the-art methods. We discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems: radial basis function collocation (for the Riesz fractional Laplacian) and nonharmonic lifting (for the spectral fractional Laplacian). In our numerical studies, we aim to compare different definitions on bounded domains using a collection of benchmark problems. We consider the fractional Poisson equation with both zero and nonzero boundary conditions, where the fractional Laplacian is defined according to the Riesz definition, the spectral definition, the directional definition, and the horizon-based nonlocal definition. We verify the accuracy of the numerical methods used in the approximations for each operator, and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions. 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.},

doi = {10.1016/j.jcp.2019.109009},

journal = {Journal of Computational Physics},

number = ,

volume = ,

place = {United States},

year = {2019},

month = {10}

}