Journal
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
Volume 25, Issue 6, Pages 2488-2531Publisher
SPRINGERNATURE
DOI: 10.1007/s13540-022-00038-y
Keywords
Fractional calculus (primary); Nonlocal vector calculus; Helmholtz decomposition; Nonlocal calculus identities
Funding
- U.S. Department of Energy, Office of Advanced Scientific Computing Research under the Collaboratory on Mathematics and Physics-Informed Learning Machines for Multiscale and Multiphysics Problems (PhILMs) project [DE-SC0019453]
- Sandia National Laboratories (SNL)
- U.S. Department of Energy's National Nuclear Security Administration [DE-NA0003525]
- NSF [DMS 1910180., DMS 1937254]
Ask authors/readers for more resources
This study investigates the analytical foundations of nonlocal vector calculus and demonstrates its potential applications in various fields. The research rigorously proves the identities of nonlocal vector calculus and develops a weighted fractional Helmholtz decomposition for smooth vector fields.
Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields such as hydrology, mechanics, and image processing. In this work, we study the analytical underpinnings of these operators. We rigorously treat compositions of nonlocal operators, prove nonlocal vector calculus identities, and connect weighted and unweighted variational frameworks. We combine these results to obtain a weighted fractional Helmholtz decomposition which is valid for sufficiently smooth vector fields. Our approach identifies the function spaces in which the stated identities and decompositions hold, providing a rigorous foundation to the nonlocal vector calculus identities that can serve as tools for nonlocal modeling in higher dimensions.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available