Finite Difference Method for Inhomogeneous Fractional Dirichlet Problem

We make the split of the integral fractional Laplacian as (-Delta)(s)u = (-Delta)(-Delta()s-1)u, where s is an element of (0, 1/2) boolean OR (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 O(h(1+alpha-2s)) convergence rate is obtained when the solution u is an element of C-1,C-alpha((Omega) over bar (delta)(n)), where n is the dimension of the space, alpha is an element of (max(0, 2s - 1), 1], delta is a fixed positive constant, and h denotes mesh size. Finally, the performed numerical experiments confirm the theoretical results.

Automated summary

In this paper, the integral fractional Laplacian is decomposed and discretized in one and two dimensions. The convergence in solving the inhomogeneous fractional Dirichlet problem is guaranteed by suitable corrections, and a convergence rate is obtained. Numerical experiments confirm the theoretical results.