The Mathematical Institute, University of Oxford, Eprints Archive

A non-local vector calculus,non-local volume-constrained problems,and non-local balance laws

Du, Q. and Gunzburger, M. and Lehoucq, R. B. and Zhou, K. (2011) A non-local vector calculus,non-local volume-constrained problems,and non-local balance laws. Mathematical Models and Methods in Applied Sciences . (Submitted)

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Abstract

A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoints 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 nonlocal calculus gives rise to volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application is posing abstract nonlocal balance laws and deriving the corresponding nonlocal field equations.

Item Type:Article
Subjects:D - G > General
Research Groups:Oxford Centre for Collaborative Applied Mathematics
ID Code:1437
Deposited By:Peter Hudston
Deposited On:18 Nov 2011 07:42
Last Modified:09 Feb 2012 13:53

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