An Efficient Quadrature Rule for Highly Oscillatory Integrals with Airy Function

被引:0
|
作者
Liu, Guidong [1 ]
Xu, Zhenhua [2 ]
Li, Bin [3 ]
机构
[1] Nanjing Audit Univ, Sch Math, Nanjing 211815, Peoples R China
[2] Zhengzhou Univ Light Ind, Coll Math & Informat Sci, Zhengzhou 450002, Peoples R China
[3] Guizhou Educ Univ, Sch Math & Big Data, Guiyang 550018, Peoples R China
关键词
Airy function; highly oscillatory integrals; complex integration method; Filon-type method; NUMERICAL QUADRATURE;
D O I
10.3390/math12030377
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form integral (b)(0) x(alpha)f(x)Ai(-omega x)dx over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when omega >> 1. The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: [0,1] and [1,b]. For integrals over the interval [0,1], we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over [1,b], we transform the Airy function into the first kind of Bessel function. By applying Cauchy's integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over [0,+infinity), which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method.
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页数:14
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