Received 8 March 2016; accepted 10 June 2016; published 13 June 2016
Consider the Cauchy principle integral
where denotes a Cauchy principle value integral and s is the singular point.
There are several different definitions which can be proved equally, such as the definition of subtraction of the singularity, regularity definition, direct definition and so on. In this paper we adopt the following one
Cauchy principal value integrals have recently attracted a lot of attention  -  . The main reason for this interest is probably due to the fact that integral equations with Cauchy principal value integrals have shown to be an adequate tool for the modeling of many physical situations, such as acoustics, fluid mechanics, elasticity, fracture mechanics and electromagnetic scattering problems and so on. It is the aim of this paper to investigate the superconvergence phenomenon of rectangle rule for it and, in particular, to derive error estimates.
The superconvergence of composite Newton-Cotes rules for Hadamard finite-part integrals was studied in  -  , where the superconvergence rate and the superconvergence point were presented, respectively. Lyness  derived the Euler-Maclaurin formula for Cauchy principal value integrals. Elliott and Venturino  employed sigmoidal transformations to obtain better approximation to Cauchy principal value integrals. In the reference Avram Sidi   and  presented high-accuracy numerical quadrature methods for integrals of singular periodic functions. The classical Euler-Maclaurin summation formula  expressed the difference between a definite integral over and its approximation using the trapezoidal rule with step length as an asymptotic expansion in powers of h together with a remainder term.
The extrapolation method for the computation of Hadamard finite-part integrals on the interval and in a circle is studied in  and  which focus on the asymptotic expansion of error function. Based on the asymptotic expansion of the error functional, algorithm with theoretical analysis of the generalized extrapolation is given.
In this paper, the density function f(x) is replaced by the approximation function fC(x) while the singular kernel
is computed analysis in each subinterval, where fC(x) is the midpoint rectangle rule. This methods
may be considered as the semi-discrete methods and the order of singularity kernel can be reduced somehow. This idea was firstly presented by Linz  in the paper to calculated 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 when the singular point was always located at the middle of certain subinterval. This paper focuses on the superconvergence of mid-rectangle rule for Cauchy principle integrals. We prove both theoreti- cally and numerically that the composite mid-rectangle rule reaches the superconvergence rate when the local coordinate of the singular point s is. Then a collation methods is presented to solve certain kind of Hilbert singular integral equation.
The rest of this paper is organized as follows. In Sect. 2, after introducing some basic formulas of the rectangle rule, we present the main resluts. In Sect. 3, we perform the proof. Finally, several numerical examples are provided to validate our analysis.
2. Main Result
Let be a uniform partition of the interval with mesh size. Define by the piecewise constant interpolant for
and a linear transformation
from the reference element to the subinterval. Replacing in (2) with gives the composite rectangle rule:
where denotes the error functional and is the Cote coefficients given by
We also define
Theorem 1: Assume. For the rectangle rule defined as (5). Assume that, there exist a positive constant C, independent of h and s, such that
Proof: Let, then we have As
For the first part of (10), we have
For the second part of (10),
Combining (11) and (13) together, the proof is completed.
Lemma 1: Assume with. Let be defined by (14), then there holds that
Proof: For, by the definition of cauchy principal value integral, we have
For, taking integration by parts on the correspondent Riemann integral, we have
Now, by using the well-known identity
The proof is completed.
By the identity in 
then we get
For, by the definition of cauchy principal value integral, we have
Let be the function of the second kind associated with the Legendre polynomial, defined by (cf.  )
We also define
Then, by the definition of W,
it follows that
Theorem 2: Assume. For the rectangle rule defined as (5). Assume that, there exist a positive constant C, independent of h and s, such that
is defined as (9).
It is known that the global convergence rate of the composite rectangle rule is lower than Riemann integral.
3. Proof of the Theorem
In this section, we study the superconvergence of the composite rectangle rule for Cauchy principle integrals.
In the following analysis, C will denote a generic constant that is independent of h and s and it may have different values in different places.
Lemma 2: Under the same assumptions of Theorem 2, it holds that
Proof: Performing Taylor expansion of at the point x, we have
Combining (29) and (30) together we get the results.
Proof of Theorem 2: we have
For, we have
Putting (31) and (32) together yields
with the linear transformation from to the identity interval. As for the last part of
which can be considered as the error estimate of left rectangle rule for the definite integral. Obviously,by the Theorem, it can be expanded by the Euler-Maclaurin expansions and we have
It is easy to see that there are not relation with the singular point which can be written as
The proof is complete.
We actually obtain the error expansion of the rectangle rule and moreover, get the explicit expression of the first order term. So it is easy for us to get the superconvergence point with, which means that is the superconvergence point in subinterval not near the end of the interval.
Based on the theorem 1, we present the modify rectangle rule
4. Numerical Example
In this section, computational results are reported.
Example 1: We consider the Hilbert singular integral with. with is the superconvergence point.
From Table 1 and Table 2, we know that the superconvergence point is with the coordinate location of singular point equal zero, while for the local coordinate of singular point do not equal zero,it is not convergence in general which coincides with our analysis.
In this section, we consider the integral equation
with the compatibility condition
As in  , under the condition of (38), there exists a unique solution for the integral Equation (37). In order to get a unique solution, we adopt the following condition
By choosing the middle points, we get the composite rectangle rule to approximate the Hilbert singular integral in (37), then the following linear system is obtained
Table 1. An error estimate of the rectangle rule.
Table 2. An error estimate of the rectangle rule.
Table 3. An error estimate of the modify rectangle rule.
Table 4. An error estimate of the modify rectangle rule.
and written as the matrix expression as
here denotes the numerical solution of f at. By directly calculation, we get that is not only a symmetric Toeplitz matrix but also a circulant matrix. As for any,
from (43), we know that is singular matrix, then we cannot use system (40) or (41) to solve the integral Equation (37).
In order to get a well-conditioned definite system, we introduce a regularizing factor in (40), which leads to linear system
where defined by
Then the matrix form of system (44) can be presented as
Example 2: Now we consider an example of solving Hilbert integral equation by collocation scheme. Let, the exact solution is.
We examine the maximal nodal error, defined by
where denotes the approximation of at. Numerical results presented in Table 5 show that both the maximal nodal errors are as follow.
The work of Jin Li was supported by National Natural Science Foundation of China (Grant No. 11471195 and
Table 5. Errors for the solution of the Hilbert integral equation of first kind.
Grant No. 11101247), China Postdoctoral Science Foundation (Grant No. 2013M540541 and 2015T80703). The work of Wei Liu was supported by National Natural Science Foundation of China (Grant No. 11401289).
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