期刊
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
卷 23, 期 3, 页码 493-540出版社
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0218202512500546
关键词
Nonlocal operators; vector calculus; volume-constrained problems; balance laws; peridynamics; nonlocal diffusion
资金
- US Department of Energy [DE-SC0005346, DE-SC0004970, DE-AC04-94AL85000]
- US National Science Foundation [DMS-1016073, DMS-1013845]
- WCU (World Class University) program of the National Research Foundation of Korea [R31-2008-000-10049-0]
- US Department of Energy through the Office of Advanced Scientific Computing Research, DOE Office of Science [FWP-09-014290]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1016073] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1013845] Funding Source: National Science Foundation
A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoint operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. The operators of the nonlocal calculus are used to define volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application discussed is the posing of abstract nonlocal balance laws and deriving the corresponding nonlocal field equations; this is demonstrated for heat conduction and the peridynamics model for continuum mechanics.
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