Error expansion of piecewise constant interpolation rule for certain two-dimensional Cauchy principal value integrals

Jin, Li; Zhao, Qing‐Li; Huang, Hongying

doi:10.1016/j.camwa.2016.08.001

Cited by 4 publications

(3 citation statements)

References 26 publications

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“…Hypersinguler integral usually appears in boundary element methods and many physical problems [31], such as fracture mechanics, elasticity problems, acoustics and electromagnetic scattering. Numerous work such as the Gaussian method [9,10], the Newton-Cotes rule [3,12,16,17,19,[25][26][27][28], the transformation method [5,7] and some other methods [2,4,11,24,30] has been developed. These papers [33][34][35][36][37] deal precisely with the development and rigorous error analysis of trapezoidal type quadrature formulas for (i)weakly singular (algebric/logarithmic singularities), (ii) singular (Cauchy principal value), and (iii) hypersingular integrals defined on the interval, and more, and derive full asymptotic expansions for the errors in the definition of substraction the singularity.…”

Section: Introductionmentioning

confidence: 99%

Extrapolation methods for solving hypersingular integral equation of the first kind

2023

Preprint

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The hypersingular integral equation has been studied widely in boundary element methods, especially in natural boundary element methods. The asymptotic expansion of error function of composite rectangle rule for the computation of Hadamard finite-part integrals with the hypersingular kernel 1/ sin 2 (x − s) is obtained. Extrapolation algorithm is constructed. In order to solve the hypersin-gular integral equation, the superconvergence point is taken as the collocation point, then the extrapolation algorithm is presented and the convergence rate of extrapolation algorithm for hypersingular integral equation is presented. At last, some numerical results are also illustrated to confirm the theoretical results and show the efficiency of the algorithms. dt − 4f (s) cot ϵ 2 } (3) The equivalent of (2) and (3) can be similarly obtained in [32]. The composite middle rectangle rule is still valid when the singular point is located at the middle rectangle of each subintervals, in the words, the superconvergence phenomenon of the middle rectangle rule appears at this moment, we obtain the error expansion of the error functional. Moreover, we apply the asymptotic expansion results to construct a algorithm scheme for solving hypersingular integral equations. MSC Classification: 65D05 65M70.

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Section: Introductionmentioning

confidence: 99%

Extrapolation methods for solving hypersingular integral equation of the first kind

2023

Preprint

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“…Different from the idea provided by Linz in [30] to calculate the hypersingular integral on an interval, we will present a direct method to compute the Cauchy principal integral. Based on the error estimate [31][32][33], the error function is determined by a certain special function S 0 (τ). We will also give the necessary and sufficient conditions to be satisfied by the superconvergence points.…”

Section: Introductionmentioning

confidence: 99%

Extended Error Expansion of Classical Midpoint Rectangle Rule for Cauchy Principal Value Integrals on an Interval

Wei

2021

Journal of Mathematics

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The classical composite midpoint rectangle rule for computing Cauchy principal value integrals on an interval is studied. By using a piecewise constant interpolant to approximate the density function, an extended error expansion and its corresponding superconvergence results are obtained. The superconvergence phenomenon shows that the convergence rate of the midpoint rectangle rule is higher than that of the general Riemann integral when the singular point coincides with some priori known points. Finally, several numerical examples are presented to demonstrate the accuracy and effectiveness of the theoretical analysis. This research is meaningful to improve the accuracy of the collocation method for singular integrals.

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“…In [25], the classical composite rectangle (midpoint) rule for the computation of two-dimensional singular integrals is discussed, with the error functional of the rectangle rule for computing two-dimensional singular integrals, and the local coordinate of certain point and the convergence results O(h 2 ) are obtained. In [26], the classical composite trapezoidal rule for the computation of two-dimensional singular integrals is presented and the convergence results O(h 2 ) is the same as the Riemann integral convergence rate at a certain point of the classical composite trapezoidal rule. In [27], the Cauchy principal value integral is approximated by the zeros of the Chebyshev polynomials of the first kind.…”

Section: Introductionmentioning

confidence: 99%

Extrapolation Algorithm for Computing Multiple Cauchy Principal Value Integrals

Cheng

2020

Mathematical Problems in Engineering

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In this paper, the computation of multiple (including two dimensional and three dimensional) Cauchy principal integral with generalized composite rectangle rule was discussed, with the density function approximated by the middle rectangle rule, while the singular kernel was analytically calculated. Based on the expansion of the density function, the asymptotic expansion formulae of error functional are obtained. A series is constructed to approach the singular point, then the extrapolation algorithm is presented, and the convergence rate is proved. At last, some numerical examples are presented to validate the theoretical analysis.

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Error expansion of piecewise constant interpolation rule for certain two-dimensional Cauchy principal value integrals

Cited by 4 publications

References 26 publications

Extrapolation methods for solving hypersingular integral equation of the first kind

Extrapolation methods for solving hypersingular integral equation of the first kind

Extended Error Expansion of Classical Midpoint Rectangle Rule for Cauchy Principal Value Integrals on an Interval

Extrapolation Algorithm for Computing Multiple Cauchy Principal Value Integrals

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