Title: Eigenvalue Problem of a Weakly Singular Compact Integral Operator by discrete Legendre Projection Methods.

Authors:  Moumita Mandal, Kapil Kant and Gnaneshwar Nelakanti

Journal: Journal of Applied Analysis and Computation

Volume: 11 Year: 2021

Publisher: Wilmington Scientific Publishers

Abstract: In this article, the discrete version of Legendre projection and iterated Legendre projection methods are considered to find the approximate eigenfunctions (eigenvalues and eigenvectors) of a weakly singular compact integral operator. Making use of a sufficiently accurate numerical quadrature rule, we establish the error bounds of the approximated eigenvalues and eigenvectors by discrete Legendre projection and iterated discrete Legendre projection methods in both 𝐿 2 and uniform norm. In particular, we obtain the optimal convergence rates O (𝑛 −𝑚) for the eigenfunctions in iterated discrete Legendre projection method in 𝐿 2 and uniform norms, where 𝑛 is the highest degree of the Legendre polynomial employed in the approximation and m is the smoothness of the eigenvectors. Numerical examples are presented to illustrate the theoretical results.