• Journal of the Korean Mathematical Society
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• v.48 no.1
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• pp.133-146
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• 2011

We employ a hyperbolic tangent function to construct nonlinear transformations which are useful in numerical evaluation of weakly singular integrals and Cauchy principal value integrals. Results of numerical implementation based on the standard Gauss quadrature rule show that the present transformations are available for the singular integrals and, in some cases, give much better approximations compared with those of existing non-linear transformation methods.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.27 no.3
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• pp.194-206
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• 2023

We present families of nonlinear transformations useful for numerical evaluation of weakly singular integrals. First, for end-point singular integrals, we define a prototype function with some appropriate features and then suggest a family of transformations. In addition, for interior-point singular integrals, we develop a family of nonlinear transformations based on the aforementioned prototype function. We take some examples to explore the efficiency of the proposed nonlinear transformations in using the Gauss-Legendre quadrature rule. From the numerical results, we can find the superiority of the proposed transformations compared to some existing transformations, especially for the integrals with high singularity strength.

• Journal of electromagnetic engineering and science
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• v.17 no.2
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• pp.91-97
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• 2017

An accurate and efficient evaluation for hypersingular integrals (HIs), strongly singular integrals (SSIs), and weakly singular integrals (WSIs) plays an essential role in the numerical solutions of 3D electromagnetic scattering problems. We derive analytic formulas for WSIs based on Stokes' theorem, which can be expressed in elementary functions. Several numerical examples are presented to validate these analytic formulas. Then, to show the feasibility of the proposed formulations for numerical methods, these formulations are used with the existing analytical expressions of HIs and SSIs to correct the near-field interaction in an iterative physical optics (IPO) scheme. Using IPO, the scattering caused by a dihedral reflector is analyzed and compared with the results of the method of moments and measurement data.

• Journal of the Korean Mathematical Society
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• v.41 no.6
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• pp.957-976
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• 2004

It is well known that the application of the nonlinear coordinate transformations is useful for efficient numerical evaluation of weakly singular integrals. In this paper, we consider the trapezoidal rule combined with a nonlinear transformation $\Omega$$_{m}$(b;$\chi$), containing a parameter b, proposed first by Yun [14]. It is shown that the trapezoidal rule with the transformation $\Omega$$_{m}$(b;$\chi$), like the case of the Gauss-Legendre quadrature rule, can improve the asymptotic truncation error by using a moderately large b. By several examples, we compare the numerical results of the present method with those of some existing methods. This shows the superiority of the transformation $\Omega$$_{m}$(b;$\chi$).TEX>).