A doubly nonlocal Laplace operator and its connection to the classical Laplacian. (English) Zbl 1429.35189

Radu, Petronela; Wells, Kelsey

J. Integral Equations Appl. 31, No. 3, 379-409 (2019).

Summary: Motivated by the state-based peridynamic framework, we introduce a new nonlocal Laplacian that exhibits double nonlocality through the use of iterated integral operators. The operator introduces additional degrees of flexibility that can allow for better representation of physical phenomena at different scales and in materials with different properties. We study mathematical properties of this state-based Laplacian, including connections with other nonlocal and local counterparts. Finally, we obtain explicit rates of convergence for this doubly nonlocal operator to the classical Laplacian as the radii for the horizons of interaction kernels shrink to zero.

MSC:

35R09	Integro-partial differential equations
45A05	Linear integral equations
45P05	Integral operators
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q74	PDEs in connection with mechanics of deformable solids
74A45	Theories of fracture and damage

Keywords:

state-based peridynamics; nonlocal models; nonlocal Laplacian; horizon of interaction; convolution; convergence

References:

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