^{1}

^{2}

^{3}

The classical composite rectangle (constant) rule for the computation of Cauchy principle value integral with the singular kernel

Consider the Cauchy principle integral

where

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 superconvergence of composite Newton-Cotes rules for Hadamard finite-part integrals was studied in [

The extrapolation method for the computation of Hadamard finite-part integrals on the interval and in a circle is studied in [

In this paper, the density function f(x) is replaced by the approximation function f_{C}(x) while the singular kernel

_{C}(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 [

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.

Let

and a linear transformation

from the reference element

where

We also define

Theorem 1: Assume

where

Proof: Let

For the first part of (10), we have

For the second part of (10),

Combining (11) and (13) together, the proof is completed.

Setting

Lemma 1: Assume

Proof: For

For

Now, by using the well-known identity

and

The proof is completed.

By the identity in [

then we get

and

For

Let

We also define

Then, by the definition of W,

it follows that

Theorem 2: Assume

where

It is known that the global convergence rate of the composite rectangle rule is lower than Riemann integral.

In this section, we study the superconvergence of the composite rectangle rule for Cauchy principle integrals.

PreliminariesIn 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

where

Proof: Performing Taylor expansion of

and

Combining (29) and (30) together we get the results.

Proof of Theorem 2: we have

For

Putting (31) and (32) together yields

Here

with the linear transformation from

which can be considered as the error estimate of left rectangle rule for the definite integral

It is easy to see that there are not relation with the singular point

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

Based on the theorem 1, we present the modify rectangle rule

In this section, computational results are reported.

Example 1: We consider the Hilbert singular integral with

From

For the modify classical rectangle rule, from

In this section, we consider the integral equation

with the compatibility condition

As in [

By choosing the middle points

8 | −1.7764e−015 | 0 | 6.6963e−001 | 1.7335e+000 |

16 | −1.7764e−015 | 0 | 2.2690e+000 | 4.1887e+000 |

32 | 0 | 0 | 2.9882e+000 | 5.2932e+000 |

64 | −5.3291e−015 | 2.6645e−015 | 3.3190e+000 | 5.8039e+000 |

128 | −4.4409e−014 | −4.3521e−014 | 3.4762e+000 | 6.0477e+000 |

256 | 1.3323e−014 | −9.7700e−015 | 3.5526e+000 | 6.1665e+000 |

8 | −1.1102e−015 | −6.2172e−015 | 5.0863e+000 | 8.7150e+000 |

16 | −3.1086e−015 | −7.1054e−015 | 4.6011e+000 | 7.8365e+000 |

32 | −8.8818e−015 | −4.3521e−014 | 4.1701e+000 | 7.1371e+000 |

64 | −2.6645e−015 | −1.7764e−014 | 3.9119e+000 | 6.7284e+000 |

128 | −2.6645e−015 | −3.8192e−014 | 3.7729e+000 | 6.5102e+000 |

256 | 2.9310e−014 | 1.0658e−013 | 3.7010e+000 | 6.3978e+000 |

8 | −1.7764e−015 | 0 | −1.7764e−015 | 3.5527e−015 |

16 | −1.7764e−015 | 0 | −5.3291e−015 | 7.1054e−015 |

32 | 0 | 0 | −3.5527e−015 | 2.3093e−014 |

64 | −5.3291e−015 | 2.6645e−015 | 1.0658e−014 | 3.0198e−014 |

128 | −4.4409e−014 | −4.3521e−014 | −4.7962e−014 | −1.3323e−013 |

256 | 1.3323e−014 | −9.7700e−015 | −6.0396e−014 | 2.8422e−014 |

8 | −1.1102e−015 | −6.2172e−015 | −8.8818e−016 | 0 |

16 | −3.1086e−015 | −7.1054e−015 | −2.8866e−015 | −3.5527e−015 |

32 | −8.8818e−015 | −4.3521e−014 | −9.5479e−015 | −1.5099e−014 |

64 | −2.6645e−015 | −1.7764e−014 | −7.3275e−015 | −3.6637e−015 |

128 | −2.6645e−015 | −3.8192e−014 | 4.4409e−015 | −2.6645e−014 |

256 | 2.9310e−014 | 1.0658e−013 | 3.1974e−014 | 1.2212e−014 |

and written as the matrix expression as

where

here

from (43), we know that

In order to get a well-conditioned definite system, we introduce a regularizing factor

where

Then the matrix form of system (44) can be presented as

where

and

Example 2: Now we consider an example of solving Hilbert integral equation by collocation scheme. Let

We examine the maximal nodal error, defined by

where

The work of Jin Li was supported by National Natural Science Foundation of China (Grant No. 11471195 and

n | |
---|---|

32 | 2.2204e−16 |

64 | 1.9984e−15 |

128 | 3.9968e−15 |

256 | 8.8818e−15 |

512 | 1.4433e−14 |

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).

Lianju Chen,Xue Su,Yuhong Li,Jin Li,1 1,Benxue Gong,Wei Liu, (2016) The Rectangle Rule for Computing Cauchy Principal Value Integral on Circle. American Journal of Computational Mathematics,06,98-107. doi: 10.4236/ajcm.2016.62011