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The modified composite Gauss type rules for singular integrals using Puiseux expansions


Authors: Tongke Wang, Zhifang Liu and Zhiyue Zhang
Journal: Math. Comp. 86 (2017), 345-373
MSC (2010): Primary 65D30
DOI: https://doi.org/10.1090/mcom/3105
Published electronically: March 28, 2016
MathSciNet review: 3557802
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Abstract: This paper is devoted to designing some modified composite Gauss type rules for the integrals involving algebraic and logarithmic endpoint singularities, at which the integrands possess the Puiseux expansions in series. Firstly, the error asymptotic expansion of a general composite quadrature rule is obtained directly by using the asymptotic expansions of the partial sum of the Hurwitz zeta function and its higher derivatives. Secondly, the deduced error asymptotic expansion is applied to the composite Gauss-Legendre and Gauss-Kronrod rules. By simplifying the evaluations of the Hurwitz zeta function and its derivatives, two modified composite Gaussian rules and their error estimates are obtained. The methods can also effectively deal with infinite range singular integrals, the singular and oscillatory Fourier transforms and the Cauchy principal value integrals by simple variable transformations. The advantage of the practical algorithms is that three ways can be used to increase the accuracy of the algorithms, which are decreasing the step length of the composite rule, increasing the orders of the Puiseux expansions and increasing the number of nodes of the Gaussian quadrature rules. Finally, the excellent performance of the proposed methods is demonstrated through several typical numerical examples. It is shown that the algorithms can be used to automatically evaluate a wide range of singular integrals over finite or infinite intervals.


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Additional Information

Tongke Wang
Affiliation: School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, People’s Republic of China
Email: wangtke@sina.com

Zhifang Liu
Affiliation: School of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, People’s Republic of China
Email: liuzhifang0628@gmail.com

Zhiyue Zhang
Affiliation: Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, People’s Republic of China
Email: zhangzhiyue@njnu.edu.cn

DOI: https://doi.org/10.1090/mcom/3105
Keywords: Integral with algebraic and logarithmic singularity, oscillatory Fourier transform, Cauchy principal value integral, Puiseux series, modified composite Gauss-Legendre rule, Gauss-Kronrod rule, error asymptotic expansion.
Received by editor(s): November 21, 2014
Received by editor(s) in revised form: May 22, 2015, and July 31, 2015
Published electronically: March 28, 2016
Additional Notes: This project was partially supported by the National Natural Science Foundation of China (grant No. 11471166), Natural Science Foundation of Jiangsu Province (grant No. BK20141443) and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD)
Article copyright: © Copyright 2016 American Mathematical Society