Error expansion of trapezoidal rule for certain two-dimensional Cauchy principal value integrals

Jin, Li; Rui, Hongxing

doi:10.1016/j.camwa.2017.09.025

Cited by 5 publications

(2 citation statements)

References 20 publications

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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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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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“…In [24], a principal value definition of the basic hypersingular integral in the fundamental integral equation for two-dimensional cracks in three-dimensional isotropic elasticity is proposed. 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.…”

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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Superconvergence of Newton–Cotes rule for computing hypersingular integral on a circle

Jin

Cheng

2022

Comp. Appl. Math.

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Error expansion of trapezoidal rule for certain two-dimensional Cauchy principal value integrals

Cited by 5 publications

References 20 publications

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

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

Superconvergence of Newton–Cotes rule for computing hypersingular integral on a circle

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