A Unified Theory of Fractional Nonlocal and Weighted Nonlocal Vector Calculus.

Abstract

Nonlocal and fractional models capture effects that classical (or standard) partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or anomalous behavior. This has driven a desire for a vector calculus based on nonlocal and fractional derivatives to derive models of, e.g., subsurface transport, turbulence, and conservation laws. In the literature, several independent definitions and theories of nonlocal and fractional vector calculus have been put forward. Some have been studied rigorously and in depth, while others have been introduced ad-hoc for specific applications. At the moment, this fragmented literature suffers from a lack of rigorous comparison and unified notation, hindering the development of nonlocal modeling. The ultimate goal of this work is to provide a new theory and to "connect all the dots" by defining a universal form of nonlocal vector calculus operators under a theory that includes, as a special case, several well-known proposals for fractional vector calculus in the limit of infinite interactions. We show that this formulation enjoys a form of Green's identity, enabling a unified variational theory for the resulting nonlocal exterior-value problems, and is consistent with several independent results in the fractional calculus literature. The proposed unified vector calculus has the potential to go beyond the analysis of nonlocal equations by supporting new model discovery, establishing theory and interpretation for a broad class of operators and providing useful analogues of standard tools from classical vector calculus.