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The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions

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Abstract

A general framework is constructed for efficiently and stably evaluating the Hadamard finite-part integrals by composite quadrature rules. Firstly, the integrands are assumed to have the Puiseux expansions at the endpoints with arbitrary algebraic and logarithmic singularities. Secondly, the Euler-Maclaurin expansion of a general composite quadrature rule is obtained directly by using the asymptotic expansions of the partial sums of the Hurwitz zeta function and the generalized Stieltjes constant, which shows that the standard numerical integration formula is not convergent for computing the Hadamard finite-part integrals. Thirdly, the standard quadrature formula is recast in two steps. In step one, the singular part of the integrand is integrated analytically and in step two, the regular integral of the remaining part is evaluated using the standard composite quadrature rule. In this stage, a threshold is introduced such that the function evaluations in the vicinity of the singularity are intentionally excluded, where the threshold is determined by analyzing the roundoff errors caused by the singular nature of the integrand. Fourthly, two practical algorithms are designed for evaluating the Hadamard finite-part integrals by applying the Gauss-Legendre and Gauss-Kronrod rules to the proposed framework. Practical error indicator and implementation involved in the Gauss-Legendre rule are addressed. Finally, some typical examples are provided to show that the algorithms can be used to effectively evaluate the Hadamard finite-part integrals over finite or infinite intervals.

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Authors and Affiliations

  1. School of Mathematical Sciences, Tianjin Normal University, Tianjin, 300387, China

    Tongke Wang

  2. Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Zhiyue Zhang

  3. School of Mathematical Sciences, Tianjin Normal University, Tianjin, 300387, China

    Zhifang Liu

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  1. Tongke Wang
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  2. Zhiyue Zhang
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  3. Zhifang Liu
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Correspondence to Tongke Wang.

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Communicated by: Martin Stynes

This project is 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).

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Wang, T., Zhang, Z. & Liu, Z. The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions. Adv Comput Math 43, 319–350 (2017). https://doi.org/10.1007/s10444-016-9487-7

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