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Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels

School of Mathematics and Statistics, Central South University, Changsha 410083, China
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Symmetry 2019, 11(6), 728; https://doi.org/10.3390/sym11060728
Received: 28 April 2019 / Revised: 20 May 2019 / Accepted: 22 May 2019 / Published: 28 May 2019
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, 1 1 f ( x ) log ( x α ) e i k x x t d x , t ( 1 , 1 ) , α [ 1 , 1 ] for a smooth function f ( x ) . This method consists of evaluation of the modified moments by stable recurrence relation and Cauchy kernel is solved by steepest descent method that transforms the oscillatory integral into the sum of line integrals. Later theoretical analysis and high accuracy of the method is illustrated by some examples. View Full-Text
Keywords: Clenshaw-Curtis quadrature; steepest descent method; logarithmic singularities; Cauchy singularity; highly oscillatory integrals Clenshaw-Curtis quadrature; steepest descent method; logarithmic singularities; Cauchy singularity; highly oscillatory integrals
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SAIRA; Xiang, S. Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels. Symmetry 2019, 11, 728.

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