Efficient and accurate quadrature methods of Fourier integrals with a special oscillator and weak singularities

作     者：Kang, Hongchao Wang, Ruoxia Zhang, Meijuan Xiang, Chunzhi 

作者机构：Hangzhou Dianzi Univ Sch Sci Dept Math Hangzhou 310018 Zhejiang Peoples R China 

出 版 物：《APPLIED MATHEMATICS AND COMPUTATION》 (应用数学和计算)

年 卷 期：2023年第440卷

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Zhejiang Provincial Natural Science Foundation of China National Natural Science Foundation of China LY22A010002 LY18A010009 11301125 11971138 

主　　题：Weak singularities Contour integration method Two-point Taylor interpolation method Generalized Gaussian-Laguerre quadrature rule 

摘      要：The recent article (J. Math. Anal. Appl. 494 (2021), Article number: 124 4 48) presented an asymptotic Filon-type method for computing the oscillatory integral with a special oscilla-tor and weak singularities, f0b x alpha (b - x)fi f (x)eiwxrdx, -1 alpha , fi 0 , 0 b + 00 , w E R , r E N +. In this article, we propose and analyze two different efficient and accurate quadrature methods for this singularly oscillatory integral. First, we give a two-point Taylor interpo-lation method by using a two-point Taylor polynomial instead of f (x ) . In addition, we propose a more efficient contour integration method. By exploiting the Taylor polynomial of the function f at x = 0 , and then based on the additivity of the integration interval, we change the considered integral into two integrals. One integral can be efficiently com-puted by the contour integration method based on Cauchy Residue Theorem and gener-alized Gaussian-Laguerre quadrature rule. The other integral can be explicitly calculated by special functions. Specifically, we perform the rigorous error analysis of the proposed methods and obtain asymptotic error estimates in inverse powers of the frequency param-eter w. Ultimately, the proposed methods are compared with the asymptotic Filon-type method given in this work (J. Math. Anal. Appl. 494 (2021), Article number: 124 4 48) and the modified Filon-type method. At the same computational cost, the two-point Taylor in-terpolation method and the asymptotic Filon-type method have a very close accuracy level. Their accuracy is higher than that of the modified Filon-type method, and the precision of the contour integration method is much higher than that of the asymptotic Filon-type method, the modified Filon-type method, and the two-point Taylor interpolation method. We verify error analyses of the proposed methods by experimental results. Numerical ex-periments can also verify the efficiency and precision of the proposed methods. (c) 2022 Elsevier Inc. All