Abstract
We consider the problem of the numerical evaluation of singular oscillatory Fourier transforms ∫abx-aαb-xβf(x)eiωxdx, whereα>-1 and β>-1. Based on substituting the original interval of integration by the paths of steepest descent, iffis analytic in the complex regionGcontaining [a,b], the computation of integrals can be transformed into the problems of integrating two integrals on [0, ∞) with the integrand that does not oscillate and decays exponentially fast, which can be efficiently computed by using the generalized Gauss Laguerre quadrature rule. The efficiency and the validity of the method are demonstrated by both numerical experiments and theoretical results. More importantly, the presented method in this paper is also a great improvement of a Filon-type method and a Clenshaw-Curtis-Filon-type method shown in Kang and Xiang (2011) and the Chebyshev expansions method proposed in Kang et al. (2013), for computing the above integrals.
Highlights
In many areas of science and engineering one often encounters the problem of computing rapidly oscillatory integrals due to their frequent occurrences in wide fields ranging from quantum chemistry, image analysis, electrodynamics, and computerized tomography to fluid mechanics
From the error formula (23) it can be seen that more accurate approximations can be obtained for the case of the fixed number of nodes and increasing frequency ω, and for the case of the fixed frequency ω and increasing the number of nodes
For an analytic and not increasing fast f in G = {z ∈ C | a ≤ R(z) ≤ b, I(z) ≥ 0}, we present an efficient method for handling the integral (1) based on analytical continuation and special contours
Summary
In many areas of science and engineering one often encounters the problem of computing rapidly oscillatory integrals due to their frequent occurrences in wide fields ranging from quantum chemistry, image analysis, electrodynamics, and computerized tomography to fluid mechanics. In 2013, the recent literature [32] gave a widely used Chebyshev expansions method depending on the frequency ω for computing many types of singular oscillatory integrals, one of which is such integral (1) The results differ from those in previous research in the sense that the constructed rules are asymptotically optimal; that is, among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence, relative to the required number of evaluations of the integrand It can combine a fixed computational cost and very high asymptotic order with numerical convergence.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
