On numerical solution of Fredholm and Hammerstein integral equations via Nystr\"{o}m method and Gaussian quadrature rules for splines

Nystr\"{o}m 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 Nystr\"{o}m 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 $m \rightarrow \infty$, $m$ 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.RYC-2017-2264

On numerical solution of Fredholm and Hammerstein integral equations via Nyström method and Gaussian quadrature rules for splines

The first author acknowledges partial financial support by the IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033. The second author has been partially supported by Spanish State Research Agency (Spanish Min-istry of Science, Innovation and Universities) : BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and by Ramon y Cajal with reference RYC-2017-22649. The fourth author is member of GNCS-INdAM.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.IMAG-Maria de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033Spanish State Research Agency (Spanish Min-istry of Science, Innovation and Universities) SEV-2017-0718Spanish Government RYC-2017-2264