In this paper, composite trapezoidal quadrature for numerical evaluation of hypersingular integrals was first introduced. By Taylor expansion at the singular point $ y $, error functional was obtained. We know that the divergence rate of $ O(h^{-p}), p = 1, 2 $, and there were no roots of the special function for the first part in the error functional. Meanwhile, for the second part of the error functional, the divergence rate was $ O(h^{-p+1}), p = 1, 2 $, but there were roots of the special function. We proved that the convergence rate could reach $ O(h^{2}) $ at superconvergence points far from the end of the interval. Two modified trapezoidal quadratures are presented and their convergence rate can reach $ O(h^{2}) $ at certain superconvergence points or any local coordinate point. At last, several examples were presented to test our theorem.
Citation: Xiaoping Zhang, Jin Li. Composite trapezoidal quadrature for computing hypersingular integrals on interval[J]. AIMS Mathematics, 2024, 9(12): 34537-34566. doi: 10.3934/math.20241645
In this paper, composite trapezoidal quadrature for numerical evaluation of hypersingular integrals was first introduced. By Taylor expansion at the singular point $ y $, error functional was obtained. We know that the divergence rate of $ O(h^{-p}), p = 1, 2 $, and there were no roots of the special function for the first part in the error functional. Meanwhile, for the second part of the error functional, the divergence rate was $ O(h^{-p+1}), p = 1, 2 $, but there were roots of the special function. We proved that the convergence rate could reach $ O(h^{2}) $ at superconvergence points far from the end of the interval. Two modified trapezoidal quadratures are presented and their convergence rate can reach $ O(h^{2}) $ at certain superconvergence points or any local coordinate point. At last, several examples were presented to test our theorem.
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Xiaoping Zhang, Jin Li. Composite trapezoidal quadrature for computing hypersingular integrals on interval[J]. AIMS Mathematics, 2024, 9(12): 34537-34566. doi: 10.3934/math.20241645
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