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Error Reduction in Gauss-Jacobi-Nyström Quadraturefor Fredholm Integral Equations of the Second Kind

M. A. Kelmanson1 and M. C. Tenwick1
1 Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK.

Computer Modeling in Engineering & Sciences 2010, 55(2), 191-210. https://doi.org/10.3970/cmes.2010.055.191

Abstract

A method is presented for improving the accuracy of the widely used Gauss-Legendre Nyström method for determining approximate solutions of Fredholm integral equations of the second kind on finite intervals. The authors' recent continuous-kernel approach is generalised in order to accommodate kernels that are either singular or of limited continuous differentiability at a finite number of points within the interval of integration. This is achieved by developing a Gauss-Jacobi Nyström method that moreover includes a mean-value estimate of the truncation error of the Hermite interpolation on which the quadrature rule is based, making it particularly accurate at low orders. A theoretical framework of the new technique is developed, implemented and validated on test problems with known exact solutions, and degenerate cases of the new Gauss-Jacobi scheme are corroborated against standard Gauss-Legendre and first- and second-kind Gauss-Chebyshev methods (i.e. using tabulated weights and abscissae). Significant error reductions over standard methods are observed, and all results are explained in the context of the new theory.

Keywords

Fredholm integral equations, Nyström method, numerical quadrature, Gauss-Jacobi polynomials, error analysis.

Cite This Article

Kelmanson, M. A. (2010). Error Reduction in Gauss-Jacobi-Nyström Quadraturefor Fredholm Integral Equations of the Second Kind. CMES-Computer Modeling in Engineering & Sciences, 55(2), 191–210.



This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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