期刊
FRACTIONAL CALCULUS AND APPLIED ANALYSIS
卷 25, 期 6, 页码 2488-2531出版社
SPRINGERNATURE
DOI: 10.1007/s13540-022-00038-y
关键词
Fractional calculus (primary); Nonlocal vector calculus; Helmholtz decomposition; Nonlocal calculus identities
资金
- 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]
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.
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