The Trapezoidal Rule for Computing Cauchy Principal Value Integral on Circle

Abstract

The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernelcot((x-s)/2)is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis.

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

“…Singular integrals, especially Cauchy principal value integrals, are usually encountered in the fields of Boundary Element Method (BEM) [1][2][3], for example, the fluid mechanics, the elasticity and fracture mechanics, the acoustics, and the electromagnetics. In these fields (including their related physical problems), much attention has been paid to the Cauchy principal value integrals [4][5][6][7][8]. We now consider the following integral:…”

Extrapolation Algorithm for Computing Multiple Cauchy Principal Value Integrals

“…where As an adequate tool in boundary element methods, Cauchy principal value integral equations [1][2][3][4][5][6][7] have been appeared in many engineering problems [8], such as fluid mechanics, acoustics, fracture mechanics, elasticity, and electromagnetic scattering problems for the modeling of many physical situations. Numerous works such as the Gaussian method [9], the transformation method [10], and the Newton-Cote methods [11][12][13][14][15] have been devoted to developing efficient quadrature formulae. In recent years, the superconvergence phenomenon of the hypersingular integral defined on interval and circle is studied in [16][17][18][19] and the superconvergence phenomenon of the Cauchy principal value integral is also investigated in [20][21][22].…”

Application of numerical integration in solving a reverse osmosis model

“…Three types of Riemann sums are the left Riemann sum, the right Riemann sum, and the middle Riemann sum. Some authors (see [7,8,9]) have applied some of the three numerical integration rules to find solutions to a mathematical model. General formula for the Riemann sum rule is given in [10]:…”