An Efficient Quadrature Rule for Highly Oscillatory Integrals with Airy Function

Guidong Liu, Zhenhua Xu and Bin Li ()
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Guidong Liu: School of Mathematics, Nanjing Audit University, Nanjing 211815, China
Zhenhua Xu: College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China
Bin Li: School of Mathematics and Big Data, Guizhou Education University, Guiyang 550018, China

Mathematics, 2024, vol. 12, issue 3, 1-14

Abstract: 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 ∫ 0 b x α f ( x ) Ai ( − ω x ) d x over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when ω ≫ 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 , + ∞ ) , 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.

Keywords: Airy function; highly oscillatory integrals; complex integration method; Filon-type method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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