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Efficient computational methods of highly oscillatory Bessel transforms with a singular point of Cauchy type and a nonlinear special oscillator

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2024年 130卷

作者： Xu, Qi Kang, Hongchao Zhou, Yujing Hangzhou Dianzi Univ Sch Sci Dept Math Hangzhou 310018 Zhejiang Peoples R China

In this article, we propose two efficient combined methods to compute the integral integral(1)(0) f(x)/x-c J(m) (omega x(r)) dx with a special and nonlinear oscillator x(r), where 0 < c < 1, m >= 0, 2r is an element of N+. One is established by combining the complex integration method with the two-point taylor interpolation method. The other is to combine two complex integration methods. The explicit formula of the integral. +8 integral(+infinity)(0) 1/x-c J(m)(omega x(r))dx is expressed in terms of the Meijer.. function. Importantly, the rigorous error analysis is performed by a large amount of theoretical analysis. The asymptotic error estimates in inverse powers of omega are achieved. Finally, numerical experiments show the validity of theoretical analysis and the efficiency of the proposed methods. In addition, we make numerical comparisons between the two proposed combined methods and another combined method.

关键词： Bessel functions Singular point of Cauchy type Oscillatory integrals two-point taylor interpolation method Complex integration method Error analysis

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

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APPLIED MATHEMATICS AND COMPUTATION 2023年 440卷

作者： Kang, Hongchao Wang, Ruoxia Zhang, Meijuan Xiang, Chunzhi Hangzhou Dianzi Univ Sch Sci Dept Math Hangzhou 310018 Zhejiang Peoples R China

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

关键词： Weak singularities Contour integration method two-point taylor interpolation method Generalized Gaussian-Laguerre quadrature rule

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