Legendre spectral projection methods for Hammerstein integral equations with weakly singular kernel (Q1627736)

This paper is devoted to the Fredholm-Hammerstein integral equation of the second kind \[ u(s)-\int\limits_{-1}^{1}k(s,t)\psi(t,u(t))dt=f(s), \; -1\leq s\leq 1, \tag{1} \] with respect to the unknown function \(u(s)\) defined in a Banach space \(\mathbb{X}=C[-1,1]\subset L^2[-1,1]\). The kernel \(k(s,t)\) is of weakly singular type and has the form \[ k(s,t)=m(s,t)g_{\alpha}(|s-t|), \] where \(m(s,t)\in C([-1,1]\times C[-1,1])\) and \[ g_{\alpha}(x)=\begin{cases} x^{\alpha-1} & \text{if } 0<\alpha<1 ;\\ \log x & \text{if } \alpha=1. \end{cases} \] The Legendre-Galerkin and Legendre collocation methods for solving (1) are studied. The convergence rates for both methods in both \(L^2\) and infinity-norm are evaluated. To improve the convergence rates, iterated Legendre-Galerkin and iterated Legendre collocation methods are considered. It is proved that iterated Legendre-Galerkin methods converge faster than Legendre-Galerkin methods in both \(L^2\) and infinity-norm. Numerical examples are presented to validate the theoretical estimate.