Discrete projection methods for Hammerstein integral equations on the half-line

In this paper, we study discrete projection methods for solving the Hammerstein integral equations on the half-line with a smooth kernel using piecewise polynomial basis functions. We show that discrete Galerkin/discrete collocation methods converge to the exact solution with order \({\mathcal {O}}(n^{-min\{r, d\}}),\) whereas iterated discrete Galerkin/iterated discrete collocation methods converge to the exact solution with order \({\mathcal {O}}(n^{-min\{2r, d\}}),\) where \(n^{-1}\) is the maximum norm of the graded mesh and r denotes the order of the piecewise polynomial employed and \(d-1\) is the degree of precision of quadrature formula. We also show that iterated discrete multi-Galerkin/iterated discrete multi-collocation methods converge to the exact solution with order \({\mathcal {O}}(n^{-min\{4r, d\}})\). Hence by choosing sufficiently accurate numerical quadrature rule, we show that the convergence rates in discrete projection and discrete multi-projection methods are preserved. Numerical examples are given to uphold the theoretical results.