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Symmetry 2018, 10(7), 239; https://doi.org/10.3390/sym10070239

On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels

1,2,* and 1,2,*
1
School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China
2
Guizhou Provincial Key Laboratory of Public Big Data, Guizhou University, Guiyang 550025, China
*
Authors to whom correspondence should be addressed.
Received: 13 June 2018 / Revised: 23 June 2018 / Accepted: 23 June 2018 / Published: 25 June 2018
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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Abstract

Lubich’s convolution quadrature rule provides efficient approximations to integrals with special kernels. Particularly, when it is applied to computing highly oscillatory integrals, numerical tests show it does not suffer from fast oscillation. This paper is devoted to studying the convergence property of the convolution quadrature rule for highly oscillatory problems. With the help of operational calculus, the convergence rate of the convolution quadrature rule with respect to the frequency is derived. Furthermore, its application to highly oscillatory integral equations is also investigated. Numerical results are presented to verify the effectiveness of the convolution quadrature rule in solving highly oscillatory problems. It is found from theoretical and numerical results that the convolution quadrature rule for solving highly oscillatory problems is efficient and high-potential. View Full-Text
Keywords: highly oscillatory; convolution quadrature rule; volterra integral equation; Bessel kernel; convergence highly oscillatory; convolution quadrature rule; volterra integral equation; Bessel kernel; convergence
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Ma, J.; Liu, H. On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels. Symmetry 2018, 10, 239.

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