Clenshaw–Curtis algorithms for an efficient numerical approximation of singular and highly oscillatory Fourier transform integrals

Kayijuka, Idrissa; Ege, Şerife Müge; Konuralp, Ali; Topal, Fatma Serap

doi:10.1016/j.cam.2020.113201

Cited by 6 publications

(2 citation statements)

References 15 publications

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“…Zhang, Yang, Tang and Xu [29] proposed an orthogonal spline collocation (OSC) method to solve the fourth-order multi-term subdiffusion equation. In considering these techniques, some authors have made great contributions to the numerical solution methods for highly oscillatory problems, such as collocation methods [30,31], Filon-Clenshaw-Curtis quadratures [32,33], the Levin method [34], fast multipole methods [35], Clenshaw-Curtis algorithms [36], Clenshaw-Curtis-Filon-type methods [37], BBFM-collocation [38] and so on.…”

Section: Introductionmentioning

confidence: 99%

Multi-Effective Collocation Methods for Solving the Volterra Integral Equation with Highly Oscillatory Fourier Kernels

Wang,

Fang,

Zhang

2023

Mathematics

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In this paper, we focus on the numerical solution of the second kind of Volterra integral equation with a highly oscillatory Fourier kernel. Based on the calculation of the modified moments, we propose four collocation methods to solve the equations: direct linear interpolation, direct higher order interpolation, direct Hermite interpolation and piecewise Hermite interpolation. These four methods are simple to construct and can quickly compute highly oscillatory integrals involving Fourier functions. We present the corresponding error analysis and it is easy to see that, in some cases, our proposed method has a fast convergence rate in solving such equations. In some cases, our proposed methods have significant advantages over the existing methods. Some numerical experiments demonstrating the efficiency of the four methods are also presented.

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Section: Introductionmentioning

confidence: 99%

Multi-Effective Collocation Methods for Solving the Volterra Integral Equation with Highly Oscillatory Fourier Kernels

Wang,

Fang,

Zhang

2023

Mathematics

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“…Oscillatory integrals are the subject of ongoing research: Yang and Ma 24 , Zaman et al 25 used Levin-based approaches to calculate highly oscillatory Fourier integrals in one- and two-dimensional domains, whereas Wang und Xiang 26 applied a Levin method to singular integrands. Recent articles from Kayijuka et al 27 dealt with oscillatory Fourier integrals with singular integrands, while Zaman et al 28 used a Levin based approach to evaluate the integrals over Bessel functions. For integrals containing Bessel functions 29 , alternative methods are sometimes expedient: for example the transformation to a contour integral in the complex plane, which is then exponentially damped in an asymptotic manner.…”

Section: Introductionmentioning

confidence: 99%

The quantum-mechanical Coulomb propagator in an L2 function representation

Gersbacher

Broad

2021

Sci Rep

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The quantum-mechanical Coulomb propagator is represented in a square-integrable basis of Sturmian functions. Herein, the Stieltjes integral containing the Coulomb spectral function as a weight is evaluated. The Coulomb propagator generally consists of two parts. The sum of the discrete part of the spectrum is extrapolated numerically, while three integration procedures are applied to the continuum part of the oscillating integral: the Gauss–Pollaczek quadrature, the Gauss–Legendre quadrature along the real axis, and a transformation into a contour integral in the complex plane with the subsequent Gauss–Legendre quadrature. Using the contour integral, the Coulomb propagator can be calculated very accurately from an L$$^2$$ 2 basis. Using the three-term recursion relation of the Pollaczek polynomials, an effective algorithm is herein presented to reduce the number of integrations. Numerical results are presented and discussed for all procedures.

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Quadrature formulae of many highly oscillatory Fourier-type integrals with algebraic or logarithmic singularities and their error analysis

Kang

2023

Applied Mathematics and Computation

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Clenshaw–Curtis algorithms for an efficient numerical approximation of singular and highly oscillatory Fourier transform integrals

Cited by 6 publications

References 15 publications

Multi-Effective Collocation Methods for Solving the Volterra Integral Equation with Highly Oscillatory Fourier Kernels

Multi-Effective Collocation Methods for Solving the Volterra Integral Equation with Highly Oscillatory Fourier Kernels

The quantum-mechanical Coulomb propagator in an L2 function representation

Quadrature formulae of many highly oscillatory Fourier-type integrals with algebraic or logarithmic singularities and their error analysis

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