Incorporating local boundary conditions into nonlocal theories

TL;DR

This paper develops a framework to incorporate local boundary conditions into nonlocal equations, specifically in peridynamics, by defining an abstract convolution operator that respects boundary conditions and analyzing wave equation solutions.

Contribution

It introduces an abstract convolution operator as a function of the classical operator, enabling the integration of local boundary conditions into nonlocal theories, with explicit solutions and numerical analysis.

Findings

Continuity preserved by time evolution for boundary conditions

Explicit solutions for various boundary conditions

Optimal convergence observed in numerical solutions

Abstract

We study nonlocal equations from the area of peridynamics on bounded domains. In our companion paper, we discover that, on $\mathbb{R}^n$, the governing operator in peridynamics, which involves a convolution, is a bounded function of the classical (local) governing operator. Building on this, we define an abstract convolution operator on bounded domains. The abstract convolution operator is a function of the classical operator, defined by a Hilbert basis available due to the purely discrete spectrum of the latter. As governing operator of the nonlocal equation we use a function of the classical operator, this allows us to incorporate local boundary conditions into nonlocal theories. For the homogeneous wave equation with the considered boundary conditions, we prove that continuity is preserved by time evolution. We give explicit solution expressions for the initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. In order to connect to the standard convolution, we give an integral representation of the abstract convolution operator. We present additional "simple" convolutionsbased on periodic and antiperiodic boundary conditions that lead Neumann and Dirichlet boundary conditions. We present a numerical study of the solutions of the wave equation. For discretization, we employ a weak formulation based on a Galerkin projection and use piecewise polynomials on each element which allows discontinuities of the approximate solution at the element borders. We study convergence order of solutions with respect to polynomial order and observe optimal convergence. We depict the solutions for each boundary condition.