Error Reduction in Gauss-Jacobi-Nyström Quadrature
for Fredholm Integral Equations of the Second Kind
M. A. Kelmanson; and M. C. Tenwick 

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 55, No. 2, pp. 191-210, 2010
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Keywords Fredholm integral equations, Nyström method, numerical quadrature, Gauss-Jacobi polynomials, error analysis.
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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