Efficient computation of highly oscillatory integrals with weak singularities by Gauss-type method. Issue 1 (2nd January 2016)

• Record Type: Journal Article
• Authors: He, Guo; Xiang, Shuhuang; Zhu, Enwen
• Is Part Of: International journal of computer mathematics. Volume 93:Issue 1(2016)
• Page Start: 83
• Page End: 107
• Publication Date: 2016-01-02
• Subjects: Hankel function -- oscillatory integrals -- weakly singular -- steepest descent method -- Gauss quadrature; 65D32 -- 65D30; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2014.987761
• Languages: English
• ISSNs: 0020-7160

Abstract

Abstract : In this paper, we consider the numerical computation of Hankel transform with weak singularities at the endpoints. By using the analytic continuation, we transform the integral into two line integrals in complex plane, which can be efficiently evaluated by some proper Gauss quadrature rules. The asymptotic orders of the errors on k and ω are derived. These asymptotic orders are demonstrated by several numerical experiments either for fixed k or for fixed ω, which show that the efficiency and accuracy of this method significantly improve as the frequencies increase.