Towards a Unified theory of Fractional and Nonlocal Vector Calculus

D’Elia, Marta; Gulian, Mamikon; Olson, Hayley; Karniadakis, George Em

doi:10.1515/fca-2021-0057

Towards a Unified theory of Fractional and Nonlocal Vector Calculus

Journal Article · Thu Oct 28 04:00:00 EDT 2021 · Fractional Calculus and Applied Analysis

DOI:https://doi.org/10.1515/fca-2021-0057· OSTI ID:1897197

D’Elia, Marta ^[1]; Gulian, Mamikon ^[2]; Olson, Hayley ^[3]; Karniadakis, George Em ^[4]

Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Univ. of Nebraska, Lincoln, NE (United States)
Brown Univ., Providence, RI (United States)
Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

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. Here, 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.

View Accepted Manuscript (DOE)

Research Organization:: Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE National Nuclear Security Administration (NNSA); US Army Research Office (ARO)

Grant/Contract Number:: AC05-76RL01830; NA0003525

OSTI ID:: 1897197

Alternate ID(s):: OSTI ID: 2282020

Report Number(s):: PNNL-SA--178937

Journal Information:: Fractional Calculus and Applied Analysis, Journal Name: Fractional Calculus and Applied Analysis Journal Issue: 5 Vol. 24; ISSN 1311-0454

Publisher:: de GruyterCopyright Statement

Country of Publication:: United States

Language:: English

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