Finite Difference Method for Inhomogeneous Fractional Dirichlet Problem

Abstract

We make the split of the integral fractional Laplacian as $$(−∆)^su = (−∆)(−∆)^{s−1}u,$$ where $s ∈ (0,\frac{1}{2}) ∪ (\frac{1}{2}, 1).$ Based on this splitting, we respectively discretize the one- and two-dimensional integral fractional Laplacian with the inhomogeneous Dirichlet boundary condition and give the corresponding truncation errors with the help of the interpolation estimate. Moreover, the suitable corrections are proposed to guarantee the convergence in solving the inhomogeneous fractional Dirichlet problem and an $\mathcal{O}(h ^{1+α−2s})$ convergence rate is obtained when the solution $u ∈ C ^{1,α}(\overline{Ω}^ δ_n),$ where $n$ is the dimension of the space, $α ∈ ({\rm max}(0, 2s − 1), 1], δ$ is a fixed positive constant, and $h$ denotes mesh size. Finally, the performed numerical experiments confirm the theoretical results.