Efficient methods for highly oscillatory integrals with weak and Cauchy singularities. Issue 9 (1st September 2016)

• Record Type: Journal Article
• Authors: Fang, Chunhua
• Is Part Of: International journal of computer mathematics. Volume 93:Issue 9(2016)
• Page Start: 1597
• Page End: 1610
• Publication Date: 2016-09-01
• Subjects: oscillatory Hilbert transform -- Filon–Clenshaw–Curtis method -- Gauss–Laguerre quadrature -- weak singularity -- Cauchy singularity -- steepest descent method; 65D32 -- 65D30; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2015.1067312
• Languages: English
• ISSNs: 0020-7160

Abstract

Abstract : In this paper, we present two fast and accurate numerical schemes for the approximation of highly oscillatory integrals with weak and Cauchy singularities. For analytical kernel functions, by using the Cauchy theorem in complex analysis, we transform the integral into two line integrals in complex plane, which can be calculated by some proper Gauss quadrature rules. For general kernel functions, the non-oscillatory and nonsingular part of the integrand is replaced by a polynomial interpolation in Chebyshev points, and the integral is then evaluated by using recurrence relations. Furthermore, several numerical experiments are shown to verify the validity of such methods.