On the Numerical Evaluation of Singular Integrals of Cauchy Type

1960 ◽  
Vol 8 (2) ◽  
pp. 342-353 ◽  
Author(s):  
Charles E. Stewart
Keyword(s):  
Numerical Evaluation ◽  
Singular Integrals ◽  
Cauchy Type

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 Numerical evaluation for Cauchy type singular integrals on the interval

2010 ◽  
Vol 233 (8) ◽  
pp. 1995-2001 ◽  
Author(s):  
Z.K. Eshkuvatov ◽  
N.M.A. Nik Long ◽  
M. Abdulkawi
Keyword(s):  
Numerical Evaluation ◽  
Singular Integrals ◽  
Cauchy Type

The numerical evaluation of singular integrals with coth-kernel

1987 ◽  
Vol 27 (3) ◽  
pp. 389-402 ◽  
Author(s):  
Walter Gautschi ◽  
M. A. Kovačević ◽  
Gradimir V. Milovanović
Keyword(s):  
Numerical Evaluation ◽  
Singular Integrals

scholarly journals Rational Transformations for Evaluating Singular Integrals by the Gauss Quadrature Rule

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 677
Author(s):  
Beong In Yun
Keyword(s):  
Numerical Evaluation ◽  
Singular Integrals ◽  
Quadrature Rule ◽  
Transformation Method ◽  
Gauss Quadrature ◽  
Cauchy Principal ◽  
Weakly Singular ◽  
Principal Value ◽  
Transformation Methods ◽  
Gauss Quadrature Rule

In this work we introduce new rational transformations which are available for numerical evaluation of weakly singular integrals and Cauchy principal value integrals. The proposed rational transformations include parameters playing an important role in accelerating the accuracy of the Gauss quadrature rule used for the singular integrals. Results of some selected numerical examples show the efficiency of the proposed transformation method compared with some existing transformation methods.

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.

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