Open Access

ARTICLE

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.

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Keywords

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