  Fueled by many applications in random processes, imaging science, geophysics,
etc., fractional Laplacians have recently received significant attention. The
key driving force behind the success of this operator is its ability to capture
non-local effects while enforcing less smoothness on functions. In this paper,
we introduce a spectral method to approximate this operator employing a sinc
basis. Using our scheme, the evaluation of the operator and its application
onto a vector has complexity of $\mathcal O(N\log(N))$ where $N$ is the number
of unknowns. Thus, using iterative methods such as CG, we provide an efficient
strategy to solve fractional partial differential equations with exterior
Dirichlet conditions on arbitrary Lipschitz domains. Our implementation works
in both $2d$ and $3d$. We also recover the FEM rates of convergence on
benchmark problems. We further illustrate the efficiency of our approach by
applying it to fractional Allen-Cahn and image denoising problems.
