Asymptotic Error Analysis for the Discrete Iterated Galerkin Solution of Urysohn Integral Equations with Green’s Kernels
Year: 2025
Author: Gobinda Rakshit
Advances in Applied Mathematics and Mechanics, Vol. 17 (2025), Iss. 1 : pp. 175–206
Abstract
Consider a Urysohn integral equation $x−\mathcal{K}(x)= f,$ where $f$ and the integral operator $\mathcal{K}$ with kernel of the type of Green’s function are given. In the computation of approximate solutions of the given integral equation by Galerkin method, all the integrals are needed to be evaluated by some numerical integration formula. This gives rise to the discrete version of the Galerkin method. For $r≥1,$ a space of piecewise polynomials of degree $≤ r−1$ with respect to a uniform partition is chosen to be the approximating space. For the appropriate choice of a numerical integration formula, an asymptotic series expansion of the discrete iterated Galerkin solution is obtained at the above partition points. Richardson extrapolation is used to improve the order of convergence. Using this method we can restore the rate of convergence when the error is measured in the continuous case. Numerical examples are given to illustrate this theory.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2023-0004
Advances in Applied Mathematics and Mechanics, Vol. 17 (2025), Iss. 1 : pp. 175–206
Published online: 2025-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 32
Keywords: Urysohn integral equation Green’s kernel iterated Galerkin method Nyström approximation Richardson extrapolation.