An Algorithm for the Numerical Evaluation of Certain Finite Part Integrals
Samir A. Ashour1,2and Hany M. Ahmed3,4
Academic Editor: Z. Huang
Received14 Mar 2011
Accepted19 Apr 2011
Published22 Jun 2011
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 Cases The 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
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.
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.
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].
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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