DISCRETE MULTI-GALERKIN METHODS FOR DERIVATIVE-DEPENDENT HAMMERSTEIN TYPE WEAKLY SINGULAR NONLINEAR FREDHOLM INTEGRAL EQUATIONS WITH GREEN'S KERNEL

In this article, we study discrete multi-Galerkin and iterated discrete multi-Galerkin methods for solving the derivative-dependent nonlinear Hammerstein type Fredholm integral equations, where the nonlinear function within the integration is dependent on the derivative, and the kernel function is of Green's type. We achieve the error bounds by substituting all integrals in the multi-Galerkin method with numerical quadrature and obtain the superconvergence results for derivative-dependent Fredholm-Hammerstein integral equations using piecewise polynomials as basis functions. By applying the numerical quadrature rule, we prove that the iterated discrete multi-Galerkin method provides superior convergence rates over the discrete multi-Galerkin method with $\mathbb{O}(h^{\min(d+1,~m+2{m_{1}},~m+2{m_{2}})})$, where $h$ represents the norm of the partitions. Numerical results are presented to validate the theoretical findings, with figures illustrating a comparison of the error analysis between the proposed methods and those discussed in [<span class="xref"><a href="#b22" ref-type="bibr">22</a></span>].