Towards a Unified Theory of Fractional and Nonlocal Vector Calculus

TL;DR

This paper develops a unified framework for nonlocal and fractional vector calculus, integrating various existing theories, establishing foundational identities, and enabling advanced modeling of complex multiscale phenomena.

Contribution

It consolidates fractional and nonlocal vector calculus into a single coherent theory, introduces an equivalence kernel for Laplacian operators, and proves Green's identity for nonlocal problems.

Findings

Fractional vector calculus is shown as a special case of nonlocal calculus.

An equivalence kernel relates unweighted and weighted Laplacian operators.

Green's identity is established for nonlocal volume-constrained problems.

Abstract

Nonlocal and fractional-order models capture effects that classical 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 that includes nonlocal and fractional gradient, divergence and Laplacian type operators, as well as tools such as Green's identities, to model 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. The goal of this work is to provide foundations for a unified vector calculus by (1) consolidating fractional vector calculus as a special case of nonlocal vector calculus, (2) relating unweighted and weighted Laplacian operators by introducing an equivalence kernel, and (3) proving a form of Green's identity to unify the corresponding variational frameworks for the resulting nonlocal volume-constrained problems. The proposed framework goes 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 the classical vector calculus.