Abstract
The cubic spline quadrature rule for the calculation of supersingular integral (also called “third order hypersingular integral”) is discussed. The superconvergence phenomenon exists at the midpoint of subinterval and the superconvergence point is the zero point of the special function. When \(\tau =0\), the order of convergence at the superconvergence point is higher than that at the non-superconvergence point. The superconvergence theory of the cubic spline quadrature function for the supersingular integral can be proved by hermite quadrature formula. Finally, examples are given to illustrate the effectiveness of the proposed method.
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The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.
Funding
This work is supported by the National Natural Science Foundation of China (Grant No. 11771398) and Natural Science Foundation of Hebei Province (Grant No. A2019209533).
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Communicated by Jose Alberto Cuminato.
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Li, J., Sang, Y. & Zhang, X. Cubic spline quadrature rule to calculate supersingular integral on interval. Comp. Appl. Math. 41, 302 (2022). https://doi.org/10.1007/s40314-022-02008-9
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DOI: https://doi.org/10.1007/s40314-022-02008-9