Nonlocal vector calculus

Abstract

Nonlocal vector calculus, introduced in generalizes differential operators' calculus to nonlocal calculus of integral operators. Nonlocal vector calculus has been applied to many fields including peridynamics, nonlocal diffusion, and image analysis. In this report, we present a vector calculus for nonlocal operators such as a nonlocal divergence, a nonlocal gradient, and a nonlocal Laplacian. In Chapter 1, we review the local (differential) divergence, gradient, and Laplacian operators. In addition, we discuss their adjoints, the divergence theorem, Green's identities, and integration by parts. In Chapter 2, we define nonlocal analogues of the divergence and gradient operators, and derive the corresponding adjoint operators. In Chapter 3, we present a nonlocal divergence theorem, nonlocal Green's identities, and integration by parts for nonlocal operators. In Chapter 4, we establish a connection between the local and nonlocal operators. In particular, we show that, for specific integral kernels, the nonlocal operators converge to their local counterparts in the limit of vanishing nonlocality.