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In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ ($0 < \alpha < 2$) in hypersingular integral form. The proposed finite difference methods provide a fractional analogue of the central difference schemes to the fractional Laplacian, and as $\alpha \to 2^-$, they collapse to the central difference schemes of the classical Laplace operator $-\Delta$. We prove that our methods are consistent if $u \in C^{\lfloor\alpha\rfloor, \alpha-\lfloor\alpha\rfloor+\epsilon}({\mathbb R}^d)$, and the local truncation error is ${\mathcal O}(h^\epsilon)$, with $\epsilon > 0$ a small constant and $\lfloor \cdot \rfloor$ denoting the floor function. If $u \in C^{2+\lfloor\alpha\rfloor, \alpha-\lfloor\alpha\rfloor+\epsilon}({\mathbb R}^d)$, they can achieve the second order of accuracy for any $\alpha \in (0, 2)$. These results hold for any dimension $d \ge 1$ and thus improve the existing error estimates for the finite difference method of the one-dimensional fractional Laplacian. Extensive numerical experiments are provided and confirm our analytical results. We then apply our method to solve the fractional Poisson problems and the fractional Allen-Cahn equations. Numerical simulations suggest that to achieve the second order of accuracy, the solution of the fractional Poisson problem should {\it at most} satisfy $u \in C^{1,1}({\mathbb R}^d)$. One merit of our methods is that they yield a multilevel Toeplitz stiffness matrix, an appealing property for the development of fast algorithms via the fast Fourier transform (FFT). Our studies of the two- and three-dimensional fractional Allen-Cahn equations demonstrate the efficiency of our methods in solving the high-dimensional fractional problems., Comment: 24 pages, 6 figures, and 6 tables

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 \alpha \to 2. The eigenvalues and eigenfunctions of these four operators are different, and the k-th (for k \in N) eigenvalue of the spectral fractional Laplacian is always larger than those of the fractional Laplacian and regional fractional Laplacian. For any \alpha \in (0, 2), the peridynamic operator can provide a good approximation to the fractional Laplacian, if the horizon size \delta is sufficiently large. We find that the solution of the peridynamic model converges to that of the fractional Laplacian model at a rate of O(\delta^{-\alpha}). In contrast, although the regional fractional Laplacian can be used to approximate the fractional Laplacian as \alpha \to 2, it generally provides inconsistent result from that of the fractional Laplacian if \alpha \ll 2. Moreover, some conjectures are made from our numerical results, which could contribute to the mathematics analysis on these operators., Comment: 25 pages, 15 figures, 1 table

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 non-locality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrodinger equation either analytically or numerically. We first introduce a fractional gradient flow with discrete normalization and then discretize it by using the trapezoidal type quadrature rule in space and the semi-implicit Euler method in time. Our method can be used to compute the ground and first excited states not only in the linear cases but also in the nonlinear cases. Moreover, it can be generalized to solve the fractional partial differential equations (PDEs) with Riesz fractional derivatives in space. Our numerical results suggest that the eigenfunctions of the fractional Schrodinger equation in an infinite potential well are significantly different from those of the standard (non-fractional) Schrodinger equation. In addition, 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.