Asymptotic Error Analysis for the Discrete Iterated Galerkin Solution of Urysohn Integral Equations with Green’s Kernels (Advances in Applied Mathematics and Mechanics, 2024)
Consider a Urysohn integral equation $x−\mathcal{K}(x)= f,$ where $f$ and the integraloperator $\mathcal{K}$ with kernel of the type of Green’s function are given. In the computationof approximate solutions of the given integral equation by Galerkin method, all theintegrals are needed to be evaluated by some numerical integration formula. Thisgives rise to the discrete version of the Galerkin method. For $r≥1,$ a space of piecewisepolynomials of degree $≤ r−1$ with respect to a uniform partition is chosen to be theapproximating space. For the appropriate choice of a numerical integration formula,an asymptotic series expansion of the discrete iterated Galerkin solution is obtainedat the above partition points. Richardson extrapolation is used to improve the orderof convergence. Using this method we can restore the rate of convergence when theerror is measured in the continuous case. Numerical examples are given to illustratethis theory.