Numerical solution of eigenvalue problems for a compact integral operator with Green’s kernels

This paper presents spectral projection and modified projection methods for approximating the eigenelements of a compact integral operator with Green’s function-type kernels. The projection can either be the orthogonal projection or the interpolatory projection using Legendre polynomials. To the best of our knowledge, this paper is the first to consider the eigenvalue problem with Green’s kernels by global polynomials. We analyze the convergence of these methods and their iterated versions, and we establish superconvergence results. The effectiveness of the proposed approach is illustrated through various numerical tests.