Abstract
We present a fast and accurate numerical scheme for approximating hypersingular integrals with highly oscillatory Hankel kernels. The main idea is to first change the integration path by Cauchy’s theorem, transform the original integral into an integral on a , + ∞ , and then use the generalized Gauss Laguerre integral formula to calculate the corresponding integral. This method has the advantages of high-efficiency, fast convergence speed. Numerical examples show the effect of this method.
Highlights
In time-harmonic electromagnetic scattering, the following integral equations arise frequently [1, 2]: us ðρÞ = − ik 4 ðC q ρ′ H ð01Þkρ ρ ′dl ð1Þ where Hð01ÞðxÞ is the Hankel function of order 0 and qðxÞ is the unknown function.For scatterers with sharp edges or corners, the unknown qðxÞ should be sought in the following form [3]: qðxÞ = wðxÞφðxÞ, wðxÞ = ð1 − xÞαð1 + xÞβ, ð2Þ where φðxÞ is smooth on ð−1, 1Þ, which leads to the following integral: ð1 wðxÞHð01ÞðkφðxÞÞφðxÞdx: ð3Þ −1In addition, the following integral appears frequently in the fields of physics and engineering [4, 5]:I (f, w; c, m, k) =
In [16], using Hermite interpolation and a recurrence formula, Liu and Xiang present a method for calculating integrals (4) in combination with the numerical steepest descent method and gave the error analysis
We study the direct steepest descent method for a class of hypersingular integral
Summary
Dcm b w(x)eikx f (x)dx, a (x − c) where wðxÞ = ðx − aÞαðb − xÞβ and α > −1, β > −1 and f ðxÞ is a given function. By using Chebyshev interpolants of f ðxÞ, Hasegawa and Torii presented an efficiently uniform approximation algorithms for following integrals [12, 13]. There is a traditional method of calculating these integrals. The traditional calculation methods of these integrals have the disadvantages of low efficiency and poor accuracy and will encounter difficulties (4).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
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.
