The Trapezoidal Rule for Computing Cauchy Principal Value Integral on Circle

Jin Li

doi:10.1155/2015/918083

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.

Highlights

Consider the Cauchy principal integral I (f; s) = c+2π ∫− c cot x − 2 s f (x) dx g (s) (1)s ∈ (0, 2π), where ∫−cc+2π denotes a Cauchy principal value integral and s is the singular point
We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate
Numerous work has been devoted in developing efficient quadrature formulas, such as the Gaussian method [2,3,4,5,6,7,8], the Newton-Cotes methods [9,10,11,12,13], spline methods [14, 15], and some other methods [16,17,18,19,20,21,22]

Summary

Introduction

S ∈ (0, 2π) , where ∫−cc+2π denotes a Cauchy principal value integral and s is the singular point. Numerous work has been devoted in developing efficient quadrature formulas, such as the Gaussian method [2,3,4,5,6,7,8], the Newton-Cotes methods [9,10,11,12,13], spline methods [14, 15], and some other methods [16,17,18,19,20,21,22] It is the aim of this paper to investigate the superconvergence phenomenon of trapezoidal rule and, in particular, to derive error estimates. This method is different from the semidiscrete methods and the order of singularity kernel can be reduced somehow which was firstly presented by Linz in the paper to calculate the hypersingular integral on interval He used the trapezoidal rule and Simpson rule to approximate the density function f(x) and the convergence rate was O(hk), k = 1, 2, when the singular point is always located at the middle of certain subinterval. Several numerical examples are provided to validate our analysis

Main Result

Proof of the Theorem

Numerical Example