  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.
