Nonlocal diffusion models with consistent local and fractional limits

. For some spatially nonlocal diffusion models with a finite range of nonlocal interac- tions measured by a positive parameter δ , we review their formulation defined on a bounded domain subject to various conditions that correspond to some inhomogeneous data. We consider their con- sistency to similar inhomogeneous boundary value problems of classical partial differential equation (PDE) models as the nonlocal interaction kernel gets localized in the local δ → 0 limit, and at the same time, for rescaled fractional type kernels, to corresponding inhomogeneous nonlocal boundary value problems of fractional equations in the global δ → ∞ limit. Such discussions help to delineate issues related to nonlocal problems defined on a bounded domain with inhomogeneous data.