A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem

In this paper, we develop a novel finite difference method to discretize the fractional Laplacian (−Δ)α/2 in hypersingular integral form. By introducing a splitting parameter, we formulate the fractional Laplacian as the weighted integral of a weak singular function, which is then approximated by the weighted trapezoidal rule. Compared to other existing methods, our method is more accurate and simpler to implement, and moreover it closely resembles the central difference scheme for the classical Laplace operator. We prove that for u∈C3,α/2(R), our method has an accuracy of O(h2)uniformly for anyα∈(0,2), while for u∈C1,α/2(R), the accuracy is O(h1−α/2). The convergence behavior of our method is consistent with that of the central difference approximation of the classical Laplace operator. Additionally, we apply our method to solve the fractional Poisson equation and study the convergence of its numerical solutions. The extensive numerical examples that accompany our analysis verify our results, as well as give additional insights into the convergence behavior of our method.