Computation of Eigenvalues for Nonlocal Models by Spectral Methods

TL;DR

This paper develops spectral methods using Fourier and Chebyshev polynomials to approximate eigenvalues of nonlocal operators, providing convergence analysis and numerical comparisons.

Contribution

It introduces a spectral approach for eigenvalue computation of nonlocal models, with convergence proofs and numerical validation.

Findings

Spectral methods achieve high accuracy in eigenvalue approximation.

Fourier and Chebyshev basis methods show comparable convergence rates.

Numerical simulations confirm theoretical convergence orders.

Abstract

The purpose of this work is to study spectral methods to approximate the eigenvalues of nonlocal integral operators. Indeed, even if the spatial domain is an interval, it is very challenging to obtain closed analytical expressions for the eigenpairs of peridynamic operators. Our approach is based on the weak formulation of eigenvalue problem and we consider as orthogonal basis to compute the eigenvalues a set of Fourier trigonometric or Chebyshev polynomials. We show the order of convergence for eigenvalues and eigenfunctions in $L^2$-norm, and finally, we perform some numerical simulations to compare the two proposed methods.