A collocation scheme for a certain Cauchy singular integral equation based on the superconvergence analysis
A collocation scheme for a certain Cauchy singular integral equation based on the superconvergence analysis
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DOI：
10.1016/j.amc.2012.11.034
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Dongjie Liu;Xiaoping Zhang;Jiming Wu
中科院分区：
文献类型：
--
作者：
Dongjie Liu;Xiaoping Zhang;Jiming Wu
本文研究了区间上柯西主值积分求值的复合中点规则，并将重点放在区间的点向超收敛现象上。得到了该规则的误差展开式，表明在局部坐标为某函数零点的每一子区间上都存在超收敛现象。然后，应用中点规则近似Cauchy主值积分，选择超收敛点作为配点，得到求解某Cauchy奇异积分方程的配点方案。更有趣的是，所得到的线性系统的系数矩阵具有一些良好的性质，由此我们可以得到最优的误差估计。最后，通过数值算例验证了理论分析的正确性。
In this paper, we investigate the composite midpoint rule for the evaluation of Cauchy principal value integral in an interval and place the key point on its pointwise superconvergence phenomenon. The error expansion of the rule is obtained, which shows that the superconvergence phenomenon occurs at the points of each subinterval whose local coordinate is the zeros of some function. Then, by applying the midpoint rule to approximate the Cauchy principal value integral and choosing the superconvergence points as the collocation points, we obtain a collocation scheme for solving a certain Cauchy singular integral equation. The more interesting thing is that the coefficient matrix of the resulting linear system possesses some good properties, from which we obtain an optimal error estimate. Finally, some numerical examples are provided to validate the theoretical analysis.