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

Abstract

We consider a Urysohn integral operator $\mathcal{K}$ with the kernel of the type of Green's function. For $r\ge 1$, a space of piecewise polynomials of degree $\le r-1$ with respect to a uniform partition is chosen to be the approximating space, and the projection is chosen to be the orthogonal projection. The 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 is $2r$ at the partition points. 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.

KEYWORDS:

• Urysohn integral operator
• Green's kernel
• Galerkin method
• asymptotic expansion
• Richardson extrapolation

2020 AMS SUBJECT CLASSIFICATIONS:

• 45G10
• 65B05
• 65J15
• 65R20

Acknowledgments

The author Akshay S. Rane would like to thank UGC faculty recharge programme, India, for their support.

Disclosure statement

No potential conflict of interest was reported by the author(s).