On numerical solution of Fredholm and Hammerstein integral equations via Nyström method and Gaussian quadrature rules for splines
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AuthorBarrera Rosillo, Domingo
Fredholm integral equationHammerstein integral equationNyström methodNumerical integrationGaussian quadrature rules for splines
D. Barrera... [et al.]. On numerical solution of Fredholm and Hammerstein integral equations via Nyström method and Gaussian quadrature rules for splines, Applied Numerical Mathematics, Volume 174, 2022, Pages 71-88, ISSN 0168-9274, [https://doi.org/10.1016/j.apnum.2022.01.009]
SponsorshipIMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033; Spanish State Research Agency (Spanish Min-istry of Science, Innovation and Universities) SEV-2017-0718; Spanish Government RYC-2017-22649
Nyström method is a standard numerical technique to solve Fredholm integral equations of the second kind where the integration of the kernel is approximated using a quadrature formula. Traditionally, the quadrature rule used is the classical polynomial Gauss quadrature. Motivated by the observation that a given function can be better approximated by a spline function of a lower degree than a single polynomial piece of a higher degree, in this work, we investigate the use of Gaussian rules for splines in the Nyström method. We show that, for continuous kernels, the approximate solution of linear Fredholm integral equations computed using spline Gaussian quadrature rules converges to the exact solution for m →∞, m being the number of quadrature points. Our numerical results also show that, when fixing the same number of quadrature points, the approximation is more accurate using spline Gaussian rules than using the classical polynomial Gauss rules. We also investigate the non-linear case, considering Hammerstein integral equations, and present some numerical tests.