Collocation Methods to Solve Certain Hilbert IntegralEquation with Middle Rectangle Rule
Collocation Methods to Solve Certain Hilbert IntegralEquation with Middle Rectangle Rule
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DOI：
10.3970/cmes.2014.102.103
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发表时间：
2014-11
影响因子：
2.4
通讯作者：
Jin Li;Dehao Yu
中科院分区：
文献类型：
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作者：
Jin Li;Dehao Yu
讨论了希尔伯特积分计算的广义复合中矩形法则。提出了逐点超收敛现象，即当奇异点与某个先验已知点重合时，矩形规则的收敛速度高于全局可能的收敛速度。我们证明了复合中间矩形规则的超收敛率出现在每个子区间的某个局部坐标处，并得到了相应的超收敛误差估计。通过选择超收敛点作为配置点，给出了求解相关希尔伯特积分方程的配置方案，并建立了误差估计。最后，给出了一些数值例子来验证理论分析。
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.