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arXiv · 2012.08879

Richardson extrapolation for the iterated Galerkin solution of Urysohn integral equations with Green's kernels

Gobinda Rakshit·Akshay S. Rane·Kshitij Patil
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Abstract

We consider a Urysohn integral operator $\mathcal{K}$ with kernel of the type of Green's function. For $r \geq 1$, a space of piecewise polynomials of degree $\leq r-1 $ with respect to a uniform partition is chosen to be the approximating space and the projection is chosen to be the orthogonal projection. Iterated Galerkin method is applied to the integral equation $x - \mathcal{K}(x) = f$. It is known that the order of convergence of the iterated Galerkin solution is $r+2$ and, at the above partition points it is $2r$. We obtain an asymptotic expansion of the iterated Galerkin solution at the partition points of the above Urysohn integral equation. Richardson extrapolation is used to improve the order of convergence. A numerical example is considered to illustrate our theoretical results.

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Gobinda Rakshit, Akshay S. Rane, Kshitij Patil. 2020-12-16. Richardson extrapolation for the iterated Galerkin solution of Urysohn integral equations with Green's kernels. https://doi.org/10.1080/00207160.2021.1986215

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