Fractional Differential Equations

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

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 joint theoretical and computational approach to examining their different characteristics by studying solutions of related fractional Poisson equations formulated on bounded domains. 
In this work, we provide new numerical methods as well as a self-contained discussion of 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.

What Is the Fractional Laplacian

Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

/pdf/numerical-methods-for-nonlocal-and-fractional-models-2o0wqhc7i9.pdf

Numerical methods for nonlocal and fractional models

We propose three Fourier spectral methods, i.e.,źthe split-step Fourier spectral (SSFS), the Crank-Nicolson Fourier spectral (CNFS), and the relaxation Fourier spectral (ReFS) methods, for solving the fractional nonlinear Schrodinger (NLS) equation. All of them are mass conservative and time reversible, and they have the spectral order accuracy in space and the second-order accuracy in time. In addition, the CNFS and ReFS methods are energy conservative. The performance of these methods in simulating the plane wave and soliton dynamics is discussed. The SSFS method preserves the dispersion relation, and thus it is more accurate for studying the long-time behaviors of the plane wave solutions. Furthermore, our numerical simulations suggest that the SSFS method is better in solving the defocusing NLS, but the CNFS and ReFS methods are more effective for the focusing NLS.

Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation

A Novel and Accurate Finite Difference Method for the Fractional Laplacian and the Fractional Poisson Problem

Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications

In this paper, we numerically study the ground and first excited states of the fractional Schrodinger equation in an infinite potential well. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrodinger equation analytically. We first introduce a normalized fractional gradient flow and then discretize it by a quadrature rule method in space and the semi-implicit Euler method in time. Our numerical results suggest that the eigenfunctions of the fractional Schrodinger equation in an infinite potential well differ from those of the standard (non-fractional) Schrodinger equation. We find that the strong nonlocal interactions represented by the fractional Laplacian can lead to a large scattering of particles inside of the potential well. Compared to the ground states, the scattering of particles in the first excited states is larger. Furthermore, boundary layers emerge in the ground states and additionally inner layers exist in the first excited states of the fractional nonlinear Schrodinger equation. Our simulated eigenvalues are consistent with the lower and upper bound estimates in the literature.

Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well

In this paper, we study four nonlocal diffusion operators, including the fractional Laplacian, spectral fractional Laplacian, regional fractional Laplacian, and peridynamic operator. These operators represent the infinitesimal generators of different stochastic processes, and especially their differences on a bounded domain are significant. We provide extensive numerical experiments to understand and compare their differences. We find that these four operators collapse to the classical Laplace operator as \begin{document}$ \alpha \to2 $\end{document} . The eigenvalues and eigenfunctions of these four operators are different, and the \begin{document}$ k $\end{document} -th (for \begin{document}$ k \in {\mathbb N} $\end{document} ) eigenvalue of the spectral fractional Laplacian is always larger than those of the fractional Laplacian and regional fractional Laplacian. For any \begin{document}$ \alpha \in (0, 2) $\end{document} , the peridynamic operator can provide a good approximation to the fractional Laplacian, if the horizon size \begin{document}$ \delta $\end{document} is sufficiently large. We find that the solution of the peridynamic model converges to that of the fractional Laplacian model at a rate of \begin{document}$ {\mathcal O}(\delta ^{-\alpha }) $\end{document} . In contrast, although the regional fractional Laplacian can be used to approximate the fractional Laplacian as \begin{document}$ \alpha \to2 $\end{document} , it generally provides inconsistent result from that of the fractional Laplacian if \begin{document}$ \alpha \ll 2 $\end{document} . Moreover, some conjectures are made from our numerical results, which could contribute to the mathematics analysis on these operators.

Siwei Duo

Papers

Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation

A Novel and Accurate Finite Difference Method for the Fractional Laplacian and the Fractional Poisson Problem

Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications

Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well

A Comparative Study on Nonlocal Diffusion Operators Related to the Fractional Laplacian