Approximation of Cauchy-type singular integrals with high frequency Fourier kernel

2021 ◽  
Vol 130 ◽  
pp. 209-219
Author(s):  
Suliman Khan ◽  
Sakhi Zaman ◽  
Siraj-ul -Islam
Keyword(s):  
High Frequency ◽  
Singular Integrals ◽  
Cauchy Type ◽  
Fourier Kernel

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

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 728 ◽  
Author(s):  
SAIRA ◽  
Shuhuang Xiang
Keyword(s):  
Singular Integrals ◽  
Oscillatory Integral ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Cauchy Kernel ◽  
Cauchy Type ◽  
Highly Oscillatory ◽  
Logarithmic Singularities ◽  
Highly Oscillatory Integrals ◽  
Modified Moments

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.

scholarly journals Modified Gauss-Legendre, Lobatto and Radau cubature formulas for the numerical evaluation of 2-D singular integrals

1983 ◽  
Vol 6 (3) ◽  
pp. 567-587 ◽  
Author(s):  
P. S. Theocaris
Keyword(s):  
Numerical Evaluation ◽  
Singular Integrals ◽  
Numerical Technique ◽  
Logarithmic Singularity ◽  
Complete Analysis ◽  
Cauchy Type ◽  
Two Dimensional ◽  
One Dimensional ◽  
Cubature Formulas ◽  
Principal Value

A numerical technique, first reported in 1979 in refs.[1] and [2], for the numerical evaluation of two-dimensional Cauchy-type principal-value integrals, is extended in this paper to include several cubature formlas of the Radau and Lobatto types. For the construction of such a cubature formula the 2-D singular integral is considered as an iterated one, and the second-order pole involved in this integral analyzed into a pair of complex poles. Based on this procedure, the methods of numerical integration, valid for one-dimensional singular integrals, are extanded to the case of two-dimensional singular integrals. The cubature formulas of the Lobatto- and Radau-type are now formulated to include the cases where some of the desired abscissas may be chosen accordins to any appropriate criterion.Moreover, the theory developed is enlarged to include the case of a 2-D principal-value integral, containing a logarithmic singularity. The validity of the results is illustrated by considering certain numerical examples. Furthermore, a complete analysis of the convergence and the construction of error estimates is also presented.

scholarly journals Symbolic Computation Applied to Cauchy Type Singular Integrals

2021 ◽  
Vol 27 (1) ◽  
pp. 3
Author(s):  
Ana C. Conceição ◽  
Jéssica C. Pires
Keyword(s):  
Operator Theory ◽  
Electromagnetic Waves ◽  
Singular Integrals ◽  
Transport Theory ◽  
Source Code ◽  
Cauchy Type ◽  
Linear Differential ◽  
First Time ◽  
Linear Transport Theory ◽  
Mathematics And Physics

The development of operator theory is stimulated by the need to solve problems emerging from several fields in mathematics and physics. At the present time, this theory has wide applications in the study of non-linear differential equations, in linear transport theory, in the theory of diffraction of acoustic and electromagnetic waves, in the theory of scattering and of inverse scattering, among others. In our work, we use the computer algebra system Mathematica to implement, for the first time on a computer, analytical algorithms developed by us and others within operator theory. The main goal of this paper is to present new operator theory algorithms related to Cauchy type singular integrals, defined in the unit circle. The design of these algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Several nontrivial examples computed with the algorithms are presented. The corresponding source code of the algorithms has been made available as a supplement to the online edition of this article.

Computing some classes of Cauchy type singular integrals with Mathematica software

2012 ◽  
Vol 39 (2) ◽  
pp. 273-288 ◽  
Author(s):  
Ana C. Conceição ◽  
Viktor G. Kravchenko ◽  
José C. Pereira
Keyword(s):  
Singular Integrals ◽  
Cauchy Type ◽  
Mathematica Software
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