Efficient methods and analysis for computing integrals of composite oscillatory functions

Hongchao Kang and Peiying Zhao

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 241, issue PB, 393-410

Abstract: In this paper, we propose and analyze three different methods to efficiently calculate the composite highly oscillatory integral ∫abf(x)Jv(Csin(ωx))dx, where v∈Z,C∈R+. Firstly, we utilize the series expansion of Bessel functions to transform the integral. On this basis, we derive asymptotic expansions of the integral, and then truncate the resulting asymptotic series to get the first method. These asymptotic expansions play a crucial role in the construction of numerical methods and error analysis. Some useful explicit formulae of the important integral ∫0yxαeiωxdx are derived. The second method entails interpolating f(x) using a polynomial at equally spaced nodes. The third method involves performing Hermite interpolation polynomials at Clenshaw–Curtis points leading to the interpolatory quadrature method II. Particularly, we analyze error estimates of the presented quadrature methods. Finally, numerical examples are used to validate the error analysis and compare the three methods under the same conditions.

Keywords: Asymptotic expansions; Asymptotic methods; Interpolatory quadrature methods; Error analysis (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:241:y:2026:i:pb:p:393-410

DOI: 10.1016/j.matcom.2025.10.022

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