Abstract
The approximate solutions for the semibounded Hadamard type hypersingular integrals (HSIs) for smooth density function are investigated. The automatic quadrature schemes (AQSs) are constructed by approximating the density function using the third and fourth kinds of Chebyshev polynomials. Error estimates for the semibounded solutions are obtained in the class of . Numerical results for the obtained quadrature schemes revealed that the proposed methods are highly accurate when the density function is any polynomial or rational functions. The results are in line with the theoretical findings.
1. Introduction
In an attempt to solve the Cauchy’s problem for hyperbolic partial differential equations, Hadamard [1] introduced the concept of hypersingular integrals. He defined the hypersingular integrals to be the finite part of a divergent integral in which singularities are arranged at the endpoints of the interval for one-dimensional integrals and on the domain boundary for multidimensional integrals. These forms of integrals are later identified as Hadamards finite part integrals or simply Hadamards integrals. Numerous applications of the approximate solutions of Hadamard integrals have been found in mechanics, electrodynamics, aerodynamics, and acoustics with various numerical approaches [2–5]. Some authors applied various regularization methods [2, 4, 5] for the transformation of the hypersingular integrals into singular or weakly singular integrals, while others applied the direct numerical computation of the finite part integrals by using various quadrature or cubature formulae [4, 6]. Hypersingular integral represents a natural extension of singular integrals in the Cauchy principal value: where
Hui and Shia [7] developed Gaussian quadrature formula for hypersingular integrals with second-order singularities which have been extensively used for the solution of elasticity problems. Chen [8] used the hypersingular integral equation approach to study the plane elastic problem for crack problem. Nik Long and Eshkuvatov [9] formulated the multiple curved crack problems into hypersingular integral equations via the complex variable function method. Obayis et al. [10] developed an automatic quadrature scheme (AQS) based on Clenshaw-Curtis Chebyshev quadrature methods for the evaluation of HSIs of the form where , , and are the weight functions and is a smooth or continuous. Error estimation for the developed AQS for was obtained in the class of functions .
In this note, we have developed the automatic quadrature scheme based on Chebyshev Gauss quadrature formula for the semibounded solutions of the hypersingular integrals of the form where and are the weights and is a smooth function. The proposed automatic quadrature schemes constructed cover the different weights functions that have not been considered by Obayis et al. [10]. Unlike the work of Obayis et al. [10] which takes advantage of Clenshaw-Curtis Chebyshev quadrature methods, the construction of our AQSs is based on Chebyshev Gauss quadrature methods. The method proposed by Hui and Shia [7] works only for even number of knots points whereas AQS for our problems works well for all and converges very fast to the exact solution.
Present in Section 2 are the mathematical concepts of the third and fourth kinds of Chebyshev polynomials and singular integrals. In Section 3, the construction of automatic quadrature schemes for semibounded solutions of hypersingular integrals is given. Rate of convergence of the suggested method is discussed in Section 4. Numerical examples and results are provided in Section 5.
2. Mathematical Concepts of Third and Fourth Kinds of Chebyshev Polynomials and Singular Integrals
The Chebyshev polynomials and are defined as follows.
Definition 1 (Mason and Handscomb [11]). The chebyshev polynomial of the third kind on is a polynomial of degree in defined by
Definition 2 (Mason and Handscomb [11]). The chebyshev polynomial of the fourth kind on is a polynomial of degree in defined by
Both polynomials satisfy the recurrence relation with the same starting value
The third and fourth kinds of Chebyshev polynomials are orthogonal on with the weights and , respectively. They also have continuous orthogonality given in [11], respectively, as
The third kind and fourth kind of Chebyshev polynomials have discrete orthogonality, respectively, as
The four kinds of Chebyshev polynomials are connected via the relations and the differentiation formulae
The Hilbert type integral transform is as follows.
Lemma 3 (Mason and Handscomb [11]). Consider the following for any , .
Lemma 4 (Mason and Handscomb [11]). Consider the following for any , .
The estimations for some integrals are established.
Lemma 5 (Israilov [12]). Let and , , and for all
where and .
The following estimations are true:(i),
(ii), ,(iii), .
3. Automatic Quadrature Schemes for Semibounded Hypersingular Integrals
To construct AQSs for the semibounded HSIs (4), the density function is approximated with the truncated series of third and forth kinds of Chebyshev polynomials:
Substituting (20) into (4), we have where
By applying the orthogonality relations (10), we obtain the coefficients , as
For the discrete coefficients , we choose the collocation points, , at the zeros of ; namely, and, using the orthogonality conditions (11) and relations (12) and (14), respectively, as well as applying composite trapezium rule, we arrive at
4. Error Estimation for the Semibounded Hypersingular Integrals
Let us introduce the classes of functions.(i) is a class of function satisfying Hölder condition on the interval with the index and constant .(ii).(iii) is a class of continuous functions on the interval with the norm (iv).
Let and let the maximum norm of the error be defined as
It is known in [11] that for any
As a consequence of (29), we arrive at the following estimations: where
Lemma 6. The Lipschitz constants , , of and are where
Proof. Let be the Lipschitz constant of the third kind Chebyshev polynomial ; that is,
where
which gives sequence of numbers:
These sequences can be generated by
which leads to
In a similar way we can get
Let be the Lipschitz constant of the derivative of third kind Chebyshev polynomial ; that is,
where
This Lipschitz constants generate the following sequence of numbers:
and its recurrence relation is
where
These recurrence relations lead to
In a similar way we obtain
Main results are given in the following Theorem 7.
Theorem 7. Let for and let the series of Chebyshev polynomials be defined by (20); then the AQS (21) has an error bound where is defined by (32) and
Proof of Theorem 7. In view of error norms (27), we can compute the error bound of AQS (21) for as follows:
Taylor series expansion of around is given by
where
Due to (51)-(52), we have
where
It is easy to check that
For the estimation of , we divide the interval into three subintervals , , and :
where
For , according to Lemma 5, we obtain
In the similar way, for , we have
Utilizing (30) and (31) into (59) and (60), we arrive at
For the error of ,
It follows from (29) and Lemma 6 that
where
Substituting (63) into (62) yields
Using Lemma 5 gives
By choosing for , we obtain
By substituting (61) and (67) into (57), we arrive at
Due to (31) and (56),
Since, from the hypothesis in Lemma 5,
we have
Substituting (69)–(72) into (54) yields
Choosing the statement of Theorem 7 follows.
5. Numerical Results and Discussion
In this session, we evaluate the hypersingular integrals (4) for different choices of , and compare with the exact solution.
Example 1. Consider the HSIs of the form where . The exact solution is .
The numerical results in Table 1 show that the AQS gives a very good result for the quadratic function for the case and . This is further illustrated in Figure 1(a) where the graphs of the exact and AQS solutions are superimposed for the same choices of singular point values. Details of the accuracy of the proposed AQS is observed from the absolute error in Figure 1(b).
(a)
(b)
Example 2. Consider the HSIs of the form where . The exact solution is .
The numerical results in Table 2 show that the AQS (21) gives a very good result for the quadratic function for the case and . Figure 2(a) indicates that the exact and AQS solutions are nearly the same. Further details of the accuracy of the absolute error of the proposed AQS are demonstrated in Figure 2(b).
(a)
(b)
Example 3. Consider the HSIs of the form where . The exact solution is .
The numerical results in Table 3 show that the AQS (21) is highly accurate for the rational function for the case and . Figure 3(a) reveals that there js good agreement between the exact solution and AQS solution which can be seen from absolute error in Figure 3(b) when . The absolute error is improved in Figure 4(b) when with the corresponding exact and AQS solution in Figure 4(a). Best result is obtained in Figures 5(a) and 5(b) when .
(a)
(b)
(a)
(b)
(a)
(b)
Example 4. Consider the HSIs of the form where . The exact solution is .
The numerical results in Table 4 show that the AQS (21) is highly accurate for the rational function for the case and . Figure 6(a) shows that there is good agreement between the exact solution and AQS solution which can be seen from absolute error in Figure 6(b) when . The absolute error is reduced in Figure 7(b) when with the corresponding exact and AQS solution in Figure 7(a). The best result is obtained and illustrated in Figures 8(a) and 8(b) when .
(a)
(b)
(a)
(b)
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(b)
6. Conclusion
In this paper, we have constructed the automatic quadrature schemes for the semibounded weighted Hadamard type hypersingular integrals using the truncated series of the third and fourth kinds of Chebyshev polynomials. The unknown coefficients are found by applying the discrete orthogonality condition and the choice of collocation points on . The error estimations for the semibounded solutions of the hypersingular integrals are obtained. Numerical examples have shown the accuracy of the developed methods and agreed with theoretical findings.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank Professor Dr. Fudziah Ismail (Universiti Putra Malaysia) for providing financial support for this research paper. Fundamental Research Grant Scheme (FRGS) code is 010-10-12-1138FR.