Open Access

ARTICLE

Collocation Methods to Solve Certain Hilbert Integral Equation with Middle Rectangle Rule

by

1 School of Sciences, Shandong Jianzhu University, Jinan 250101, China.
2 School of Mathematics, Shandong University, Jinan 250100, P. R.China.
3 School of Mathematical Sciences, Xiamen University, Xiamen 361005, China.
4 LSEC, ICMSEC, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing 100190, China.

Computer Modeling in Engineering & Sciences 2014, 102(2), 103-126. https://doi.org/10.3970/cmes.2014.102.103

Abstract

The generalized composite middle rectangle rule for the computation of Hilbert integral is discussed. The pointwise superconvergence phenomenon is presented, i.e., when the singular point coincides with some a priori known point, the convergence rate of the rectangle rule is higher than what is global possible. We proved that the superconvergence rate of the composite middle rectangle rule occurs at certain local coordinate of each subinterval and the corresponding superconvergence error estimate is obtained. By choosing the superconvergence point as the collocation points, a collocation scheme for solving the relevant Hilbert integral equation is presented and an error estimate is established. At last, some numerical examples are provided to validate the theoretical analysis.

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Keywords

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