Computation of Eigenvalues for Nonlocal Models by Spectral Methods

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 in order to compute the eigenvalues we consider an orthogonal basis consisting of a set of Fourier trigonometric or Chebyshev polynomials. We show the order of convergence for eigenvalues and eigenfunctions in L2-norm, and finally, we perform some numerical simulations to compare the two proposed methods.