ISRN Applied MathematicsVolume 2011 (2011), Article ID 571583, 21 pagesdoi:10.5402/2011/571583

Research Article

An Algorithm for the Numerical Evaluation of Certain Finite Part Integrals

Samir A. Ashour1,2 and Hany M. Ahmed3,4

¹Department of Mathematics, Faculty of Sciences, Cairo University, Giza, Egypt²Department of Mathematics, Faculty of Education, King Abdulaziz University, Jeddah 2356, Saudi Arabia³Department of Mathematics, Faculty of Industrial Education, Helwan University, Cairo, Egypt⁴Department of Mathematics, Faculty of Sciences, Shaqra University, Shaqra, Saudi Arabia

Received 14 March 2011; Accepted 19 April 2011

Academic Editor: Z. Huang

Copyright © 2011 Samir A. Ashour and Hany M. Ahmed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable. In this work we present some formulae to evaluate the known Gaussian quadrature rules for finite part integrals , and extend Clenshow's algorithm to evaluate these integrals in a stable way.

1. Introduction

In the theory of the numerical approximation of Cauchy principal value integrals, basically two kinds of Gaussian quadrature formulae have been investigated. For the modified Gaussian formula, we refer to [1–7] and the literature cited therein. Results on the Gaussian quadrature formula in the strict sense can be found, for example, in [1, 3, 7–13]. In [14] Diethelm proposed a quadrature formula which is called Gaussian quadrature formula of the third kind. Many algorithms that have been proposed for the numerical evaluation of Cauchy principal value integrals are numerically unstable. In particular, this is true for the standard algorithm proposed in [3, Section 3] for the evaluation of the modified Gaussian formula [15, Section 4]. In this paper, we present some formulae to evaluate the known Gaussian quadrature rules for finite part integrals and extend Clenshow's algorithm [16] to evaluate these integrals in a stable way.

This paper is organized as follows: in Section 1, we introduce some known quadrature rules for finite part integrals and we analyze the relations among these rules. In Section 2, we extend Clenshow's algorithm [16] to evaluate any sum of derivatives of functions. In Section 3, we apply the extended algorithm to compute Lagrange interpolation polynomial and its derivatives, to evaluate some known singular integrals exactly and quadrature rules for finite part integrals in a stable way. Finally, we present a numerical example.

Some hypersingular integrals can be found by successive differentiation; in some cases evaluation of these integrals becomes tedious and the formulas are lengthily. The evaluation of the such singular integrals is a necessary step for the numerical approach to the integral equations in crack problems. One of the main tasks of the proposed algorithm is evaluate these integrals. The appendix, with closed form solutions for class of these integrals, supplements the paper.

1.1. Guassian Quadrature Rule

Let be an integrable weight function on the interval , and let be orthogonal system of polynomials associated with which satisfiesa three-term recurrence relation: and hence satisfy a recurrence relation: The known Gaussian quadrature rule for is defined as

1.2. Modified Gaussian Quadrature Rules for Singular Integrals

Definition 1.1.

The finite part integral is defined as follows.

Definition 1.2.

The modified Gaussian quadrature rule for can be defined as follows.

Definition 1.3.

Since , then by differentiating both sides -times, using Leibniz formula and Definition 1.1, we have the following the lemma.

Lemma 1.4.

Lemma 1.5. Let ; then

Proof. Since the quadrature rule (1.7) is Gaussian: then This implies that Differentiate both sides -times to have (1.9).

Thus the modified Gaussian quadrature rule for can be written as

Special CasesThe first case: The second case: which are the known formulae

1.3. The Gaussian Quadrature Formula in the Strict Sense for Singular Integrals

In the integral , if we interpolate by a polynomial of degree using the zeros of , that is, the nodes of the classical Gauss formula, as interpolation nodes and use as an approximation for , we obtain the Gaussian quadrature formula in the strict sense for singular integrals: so due to (1.16), one obtains the following lemma.

Lemma 1.6.

1.4. Gaussian Quadrature Rule of the Third Kind

Let . According to [14], the integral can be approximated by which is called Gaussian quadrature rule of the third kind:

for ,

Lemma 1.7.

1.5. The Connection between the Gaussian Quadrature Rules

Lemma 1.8.

Proof. From (1.16), Since , one obtains (1.22).

Corollary 1.9. If , then

Proof. From Definition 1.3, and since then Lemma has been proved.

Lemma 1.10.

2. Sum Series Algorithm

Theorem 2.1. The sum of any series of functions which satisfy a linear recurrence relation is given by where

Moreover, if satisfies the linear recurrence relation then where

Proof. Using (2.1), we get (2.2). Moreover, if satisfies (2.4), we can similarly deduce (2.5).

In a similar way, we can prove the following theorem.

Theorem 2.2. where Moreover, if satisfies the linear recurrence relation (2.4), then where

Now we extend the previous theorems to evaluate .

Suppose that satisfies (2.1) with and that satisfies the following recurrence relation:

Theorem 2.3. Let ; then where

Proof. Let Due to Theorem 2.1, we obtain , By induction, we have , and by using Theorem 2.1, we obtain where Finally so

The alternative algorithm that incorporate 's in an upward direction can be deduced as follows.

Theorem 2.4. Let ; then where

Proof. Let Due to Theorem 2.2, we obtain , By induction, we have , and by using Theorem 2.2, we obtain where Finally so

3. Computational Aspects and Numerical Examples

In this section, we use the previous algorithm to evaluate the known Gaussian quadrature rules for the integrals in (1.6).

Lemma 3.1. Let ; then where is the weight of the classical Gaussian quadrature formula corresponding to the node .

Proof. Since then multiply both sides by , and integrate over , we derive by using the orthogonality of , we get and by knowing that , we obtain (3.1).

Now, since satisfy (2.1) which and , the following statement follows from Theorem 2.3.

Lemma 3.2. where

Since satisfies recurrence relation (2.12), so we can apply Theorem 2.3 to have the following lemma.

Lemma 3.3. where , , are given in Theorem 2.3.

Since obeys recurrence relation (2.12), so we can evaluate the pervious Gaussian quadrature rules introduced in Sections 1.2–1.4 by the following theorems.

Theorem 3.4. where where ’s are given by Theorem 2.3.

Theorem 3.5. If , then where (using Theorem 2.3) where , which fulfill a three-term recurrence (1.2), and are given by Theorem 2.3.

Proof. Using Lemma 1.8, we have Then differentiating both sides -times, we obtain relation (3.10).

Theorem 3.6. If , then

Proof. Using Corollary 1.9, we have and by differentiating both sides -times, we obtain relation (3.13).

A direct result of Lemma 1.10 and Theorem 2.3 is the following theorem.

Theorem 3.7. where and are given by Theorem 2.3.

4. A Numerical Example

Hui* and Shia [17] consider the modified Gaussian quadrature where .

In Table 1, numerical results are obtained for by using the quadrature rule ; in this case, ; and are the zeros of Chebyshev polynomial of the second kind ; then we have Then applying the algorithm in Theorem 3.6 for , , , we obtain the following results comparing with these of Hui* and Shia [17].

Table 1

Equation (4.1)
Equation (3.13)

4
−2.33956
−2.33956

5
NA
−2.33942

6
−2.33956
−2.33956

7
NA
−2.33956

8
−2.33956
−2.33956

The results indicate that the algorithm is indeed stable even when coincides with one of the nodes of the quadrature formula (in this case, this node is located at the origin). However, in deriving the modified method [17] there are no numerical values presented for and 7 that is why it is assumed that the roots of the orthogonal polynomials are distinct from .

Appendix

A. Evaluation for Some Finite Part Integrals Using the Proposed Algorithm

The goal of this appendix is to provide the evaluations for some integrals using the proposed algorithm with suitable points

A.1. ,

A.2. ,

A.3. ,

A.4. ,

A.5. ,

A.6. ,

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