Computation of integrals with oscillatory and singular integrands using Chebyshev expansions

Kang, Hongchao; Xiang, Shuhuang; He, Guo

doi:10.1016/j.cam.2012.10.016

Cited by 20 publications

(13 citation statements)

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“…It can combine a fixed computational cost and very high asymptotic order with numerical convergence. Of course, the presented method is also much more efficient than the Chebyshev expansions method proposed in [32]. An outline of this paper is as illustrated below.…”

Section: Introductionmentioning

confidence: 98%

“…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.…”

mentioning

confidence: 93%

“…In 2011, the authors of [31] presented a Filontype method and a Clenshaw-Curtis-Filon-type method for computing the integral (1), the error of which satisfies ( − − −2 ), where is the highest multiplicity of the Hermite interpolation at the endpoints and = min{ , }. In 2013, the recent literature [32] gave a widely used Chebyshev expansions method depending on the frequency for computing many types of singular oscillatory integrals, one of which is such integral (1). Based on these relevant background literatures above, in Section 2 of this paper, thanks to analytic continuation, special contours, and generalized Gauss-Laguerre quadrature rule, we devise efficient method to compute the class of integrals (1).…”

Section: Introductionmentioning

confidence: 99%

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Fast Computation of Singular Oscillatory Fourier Transforms

Kang

Shao

2014

Abstract and Applied Analysis

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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.

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Section: Introductionmentioning

confidence: 98%

mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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Fast Computation of Singular Oscillatory Fourier Transforms

Kang

Shao

2014

Abstract and Applied Analysis

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“…Also, Kang and Xiang in [12] introduced a Clenshaw-CurtisFilon-type method, which was based on a special Hermite interpolation polynomial at ClenshawCurtis-points. Afterwards, a new method based on Chebyshev expansions was presented in [13].…”

Section: Introductionmentioning

confidence: 99%

Efficient methods for highly oscillatory integrals with weak and Cauchy singularities

Fang

2015

International Journal of Computer Mathematics

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In this paper, we present two fast and accurate numerical schemes for the approximation of highly oscillatory integrals with weak and Cauchy singularities. For analytical kernel functions, by using the Cauchy theorem in complex analysis, we transform the integral into two line integrals in complex plane, which can be calculated by some proper Gauss quadrature rules. For general kernel functions, the non-oscillatory and nonsingular part of the integrand is replaced by a polynomial interpolation in Chebyshev points, and the integral is then evaluated by using recurrence relations. Furthermore, several numerical experiments are shown to verify the validity of such methods .

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“…We note in passing that the computation of singular highly oscillatory integrals has been already considered in literature [1,9,15,19]. For the weak singularities mentioned in this paper, although there are some estimates provided, such as [24], asymptotic analysis is incomplete, as is the design of effective quadrature methods.…”

Section: Introductionmentioning

confidence: 99%

Quadrature Methods for Highly Oscillatory Singular Integrals

Jing¹

2021

JCM

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Computation of integrals with oscillatory and singular integrands using Chebyshev expansions

Cited by 20 publications

References 20 publications

Fast Computation of Singular Oscillatory Fourier Transforms

Fast Computation of Singular Oscillatory Fourier Transforms

Efficient methods for highly oscillatory integrals with weak and Cauchy singularities

Quadrature Methods for Highly Oscillatory Singular Integrals

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