A Doubly Nonlocal Laplace Operator and Its Connection to the Classical Laplacian

Authors

Petronela Radu, University of Nebraska - LincolnFollow
Kelseys Wells, University of Nebraska - Lincoln

Date of this Version

2019

Citation

J. Integral Equations Applications Forthcoming (2019).

Comments

2010 Mathematics Subject Classification. Primary: 35R09, 45A05, 45P05, 35J05, Secondary: 74B99, 74A45.

Abstract

In this paper, motivated by the state-based peridynamic frame- work, we introduce a new nonlocal Laplacian that exhibits double nonlocality through the use of iterated integral operators. The operator introduces addi- tional degrees of exibility 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 connec- tions 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.