  We make the split of the 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 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 $\mathcal{O}(h^{1+\alpha-2s})$ convergence rate is obtained when
the solution $u\in C^{1,\alpha}(\bar{\Omega}^{\delta}_{n})$, where $n$ is the
dimension of the space, $\alpha\in(\max(0,2s-1),1]$, $\delta$ is a fixed
positive constant, and $h$ denotes mesh size. Finally, the performed numerical
experiments confirm the theoretical results.
