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Zhang, A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem, J. Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput. Lynch, Fractional diffusion in plasma turbulence, Phys. Pang, A new definition of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction, J. Pasciak, Numerical approximation of the integral fractional Laplacian, Numer. Rowe, A Galerkin Radial Basis Function Method for Nonlocal Diffusion, Lect. Meerschaert, Tempered stable Lévy motion and transient super-diffusion, J. Glusa, Hybrid finite element-spectral method for the fractional Laplacian: Approximation theory and efficient solver, SIAM J. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM J. Borthagaray, A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian, Comput. Compared to the recently proposed Wendland RBF method, our method exactly incorporates the Dirichlet boundary conditions into the scheme and is free of the Gibbs phenomenon as observed in the wendland method.ġ. We show that our method has spectral accuracy and can achieve good approximation even with a small number of points. Extensive numerical experiments are provided to study the performance of our method in both approximating the Dirichlet Laplace operators and solving classical and fractional PDE problems.

Moreover, our method is simple and easy when handling complex geometries and local refinements, and its computer program implementation remains the same for any dimension $d \ge 1$. These important merits distinguish our method from other existing methods for fractional PDEs. The proposed method takes advantage of the analytical Laplacian of Gaussian functions so as to accommodate the discretization of the classical and fractional Laplacians in a single framework and avoid the large computational cost for numerical evaluation of the fractional derivatives. In this paper, we propose a meshfree method based on the Gaussian radial basis function (RBF) to solve both classical and fractional PDEs.