Finite Difference Method for Inhomogeneous Fractional Dirichlet Problem

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