Numerical approximations of highly oscillatory Hilbert transforms
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Numerical approximations of highly oscillatory Hilbert transforms Ruyun Chen1 · Di Yu1 · Juan Chen1 Received: 4 December 2019 / Revised: 3 May 2020 / Accepted: 14 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we are concerned with the numerical approximations of one-sided Hilbert transforms with oscillatory kernel by means of the multiple integrals. This type of Hilbert transform has two computing difficulties: singularity and oscillation. To avoid the singularity, we transfer the Hilbert transform to an individual oscillatory integral which can be analytically calculated and a non-singular integral which can be well evaluated by the multiple integrals. Numerical examples are provided to illustrate the advantages of the proposed methods. Keywords Hilbert transform · Multiple integral · Oscillatory kernel · Singularity · Numerical approximation · Error order Mathematics Subject Classification 65D30 · 65D32
1 Introduction The numerical evaluation of one-sided highly oscillatory Hilbert transforms of form: ∞
H + (t α f (t)eiωt )(x) = 0
t α eiωt
f (t) dt, x ≥ 0, −1 < α ≤ 0 and ω 1 t−x
Communicated by Hui Liang. The work is supported by Natural Science Foundation of Guangdong Province, China (No. 2015A030313615).
B
Ruyun Chen [email protected]; [email protected] Di Yu [email protected] Juan Chen [email protected]
1
School of Mathematics and Computer, Guangdong Ocean University, Zhanjiang 524088, Guangdong, China 0123456789().: V,-vol
123
(1.1)
180
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plays an important role in many areas of science and engineering, such as astronomy, optics, electromagnetics, and seismology image processing, where f (t) is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin (Davis and Rabinowitz 1984; Arfken 1985; Bao and Sun 2005; Wang et al. 2013). In fact, we can obtain different integral for different values x in (1.1): (i) when x = 0, the integral is a Hadamard finite part integral; (ii) when x > 0, the integral is a Cauchy principal value integral. Computing this type of transform will encounter two difficulties: singularity and oscillation, that is, the integrand is singular at the points t = x and t = 0, and highly oscillatory for ω 1, so that many standard methods will not be suitable for computing it (see Wang and Xiang 2009; Xu 2018). Fortunately, for large frequency ω, many efficient methods have been developed for numerically evaluating the highly oscillatory integral: ˆ b f (x)eiωg(x) dx (1.2) a
over the past few years, for instance, Filon method (Filon 1928), asymptotic method, and Filon-type method (Iserles and Nørsett 2005). As we all know, the moments first need be evaluated when Filon method and Filon-type method are used to evaluate (1.2). However, in general, the moments is difficult to obtain the explicit expression. Although the moments is free for Levin-type method and generalized quadrature rule, nevertheless, the
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