• 제목/요약/키워드: Singular Integrals

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EVALUATION OF SINGULAR INTEGRALS BY HYPERBOLIC TANGENT BASED TRANSFORMATIONS

  • Yun, Beong-In
    • 대한수학회지
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    • 제48권1호
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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.

WEIGHTED Lp-BOUNDEDNESS OF SINGULAR INTEGRALS WITH ROUGH KERNEL ASSOCIATED TO SURFACES

  • Liu, Ronghui;Wu, Huoxiong
    • 대한수학회지
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    • 제58권1호
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    • pp.69-90
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    • 2021
  • In this paper, we prove weighted norm inequalities for rough singular integrals along surfaces with radial kernels h and sphere kernels Ω by assuming h ∈ △γ(ℝ+) and Ω ∈ ����β(Sn-1) for some γ > 1 and β > 1. Here Ω ∈ ����β(Sn-1) denotes the variant of Grafakos-Stefanov type size conditions on the unit sphere. Our results essentially improve and extend the previous weighted results for the rough singular integrals and the corresponding maximal truncated operators.

NORM ESTIMATES ON HARDY SPACES AND MULTIPLE SINGULAR INTEGRALS

  • Cho, Yong-Kum
    • 대한수학회지
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    • 제35권2호
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    • pp.295-314
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    • 1998
  • In this article we examine certain distinctive features regarding Hardy spaces of both classical and product notions on $R^{N}$ with our focus on their interrelations through embeddings and restrictions. Applications of our results to multiple singular integrals are included.d.

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FAMILIES OF NONLINEAR TRANSFORMATIONS FOR ACCURATE EVALUATION OF WEAKLY SINGULAR INTEGRALS

  • BEONG IN YUN
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제27권3호
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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.

GEOMETRIC AND APPROXIMATION PROPERTIES OF GENERALIZED SINGULAR INTEGRALS IN THE UNIT DISK

  • Anastassiou George A.;Gal Sorin G.
    • 대한수학회지
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    • 제43권2호
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    • pp.425-443
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    • 2006
  • The aim of this paper is to obtain several results in approximation by Jackson-type generalizations of complex Picard, Poisson-Cauchy and Gauss-Weierstrass singular integrals in terms of higher order moduli of smoothness. In addition, these generalized integrals preserve some sufficient conditions for starlikeness and univalence of analytic functions. Also approximation results for vector-valued functions defined on the unit disk are given.

Derivation of Analytic Formulas and Numerical Verification of Weakly Singular Integrals for Near-Field Correction in Surface Integral Equations

  • Rim, Jae-Won;Koh, Il-Suek
    • Journal of electromagnetic engineering and science
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    • 제17권2호
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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.