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August, 2018 On the Numerical Quadrature of Weakly Singular Oscillatory Integral and its Fast Implementation
Zhenhua Xu
Taiwanese J. Math. 22(4): 979-1000 (August, 2018). DOI: 10.11650/tjm/170904

Abstract

In this paper, we present a Clenshaw-Curtis-Filon-type method for the weakly singular oscillatory integral with Fourier and Hankel kernels. By interpolating the non-oscillatory and nonsingular part of the integrand at $(N+1)$ Clenshaw-Curtis points, the method can be implemented in $O(N \log N)$ operations, which requires the accurate computation of modified moments. We first give a method for the derivation of recurrence relation for the modified moments, which can be applied to the derivation of recurrence relation for the modified moments corresponding to other type oscillatory integrals. By using the recurrence relation, special functions and classic quadrature methods, the modified moments can be computed accurately and efficiently. Then, we present the corresponding error bound in inverse powers of frequencies $k$ and $\omega$ for the proposed method. Numerical examples are provided to support the theoretical results and show the efficiency and accuracy of the method.

Citation

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Zhenhua Xu. "On the Numerical Quadrature of Weakly Singular Oscillatory Integral and its Fast Implementation." Taiwanese J. Math. 22 (4) 979 - 1000, August, 2018. https://doi.org/10.11650/tjm/170904

Information

Received: 10 April 2017; Revised: 30 June 2017; Accepted: 14 September 2017; Published: August, 2018
First available in Project Euclid: 14 October 2017

zbMATH: 06965406
MathSciNet: MR3830830
Digital Object Identifier: 10.11650/tjm/170904

Subjects:
Primary: 65D30 , 65D32

Keywords: Clenshaw-Curtis-Filon-type method , error bound , modified moments , recurrence relation , weakly singular oscillatory integral

Rights: Copyright © 2018 The Mathematical Society of the Republic of China

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Vol.22 • No. 4 • August, 2018
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