The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator

We analyze a nonlocal diffusion operator having as special cases the fractional Laplacian and fractional differential operators that arise in several applications. In our analysis, a nonlocal vector calculus is exploited to define a weak formulation of the nonlocal problem. We demonstrate that, when sufficient conditions on certain kernel functions hold, the solution of the nonlocal equation converges to the solution of the fractional Laplacian equation on bounded domains as the nonlocal interactions become infinite. We also introduce a continuous Galerkin finite element discretization of the nonlocal weak formulation and we derive a priori error estimates. Through several numerical examples we illustrate the theoretical results and we show that by solving the nonlocal problem it is possible to obtain accurate approximations of the solutions of fractional differential equations circumventing the problem of treating infinite-volume constraints.