Legendre spectral projection methods for weakly singular Hammerstein integral equations of mixed type

Abstract

This work provides the Legendre spectral projection (Galerkin and collocation), iterated Legendre spectral projection, Legendre spectral multi-projection and iterated Legendre spectral multi-projection methods to approximate the solution of weakly singular Hammerstein integral equations of mixed type. The convergence rates of approximate solutions to the exact solutions are obtained for all the above four methods in both $$L^{2}$$ and infinity norm. The comparison of convergence rates for all these methods have been discussed. We also have shown that iterated Galerkin improves over Galerkin, multi-Galerkin improves over iterated Galerkin and iterated multi-Galerkin improves over multi-Galerkin in $$L^2$$ norm using Legendre polynomial bases.