Galerkin and multi-Galerkin methods for weakly singular Volterra–Hammerstein integral equations and their convergence analysis

Abstract

In this paper, we consider the Galerkin and iterated Galerkin methods to approximate the solution of the second kind nonlinear Volterra integral equations of Hammerstein type with the weakly singular kernel of algebraic type, using piecewise polynomial basis functions based on graded mesh. We prove that the Galerkin method converges with the order of convergence $${\mathcal {O}}(n^{-m})$$, whereas iterated Galerkin method converges with the order $${\mathcal {O}}(n^{-2m})$$ in uniform norm. We also prove that in iterated multi-Galerkin method the order of convergence is $${\mathcal {O}}(n^{-3m}) $$. Numerical results are provided to justify the theoretical results.