School of Mathematics and Statistics, Central South University, Changsha 410083, China
Author to whom correspondence should be addressed.
Received: 28 April 2019 / Accepted: 22 May 2019 / Published: 28 May 2019
In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, , for a smooth function . This method consists of evaluation of the modified moments by stable recurrence relation and Cauchy kernel is solved by steepest descent method that transforms the oscillatory integral into the sum of line integrals. Later theoretical analysis and high accuracy of the method is illustrated by some examples.
Boundary element method and finite element method are intensively eminent numerical approaches to evaluate partial differential equations (PDEs), which appear in variety of disciplines from engineering to astronomy and quantum mechanics [1,2,3,4,5]. Although these methods lead PDEs to Fredholm integral equations or Voltera integral equations, but these kind of integral equations posses integrals of oscillatory, Cauchy-singular, logarithmic singular, weak singular kernel functions. However, these classical methods are failed to approximate the integrals constitute kernel functions of highly oscillation and logarithmic singularity.
This paper aims at approximation of the integral
where is relatively smooth function. For integral (1) the developed strategy for logarithmic singularity is valid for . In particular, the highly oscillatory integral, has been computed by many methods such as asymptotic expansion, Filon method, Levin collocation method and numerical steepest descent method [6,7,8,9,10]. For instant, Dominguez et al.  for function with integrable singularities have proposed an error bound, calculated in Sobolev spaces , for composite Filon-Clenshaw-Curtis quadrature. Error bound depends on the derivative of and length of the interval M, for some defined as for
On the other hand, one methodology for numerical evaluation of integral is replacing by different kind of polynomials [12,13]. Another technique is based on analytic continuation of the integral if the integrand is analytic in the complex region . As far as for solution methods and properties of the solution for relative non-homogenous integrals have been discussed by using Brestain polynomials and Chebyshev polynoimals of all four kinds in [3,15].
For integral Clenshaw-Curtise rule is applied for numerical calculation. Wherein the convergence rate is independent of k but depends on the number of nodes of quadrature rule and function . Furthermore, Piessense and Branders  established the Clenshaw-Curtis quadrature rule, relies on the recurrence relation for They replaced the nonoscillatory and nonsingular part of the integrand by Chebyshev series. Chen  presented the numerical approximation of the integral with , and For analytic function the integral was rewritten in the form of sum of line integrals, wherein the integrands do not oscillate and decay exponentially. Moreover, Fang  established the Clenshaw-Curtis quadrature for for general function where steepest descent method is illustrated for analytic function . Recently, John  introduced the algorithm for integral approximation of Cauchy-singular, logarithmic-singular, Hadamard type and nearly singular integrals having integrable endpoints singularities i.e., . Composed Gauss-Jacobi quadrature consists of approximating the function by Jacobi polynomials of degree .
However, all these proposed method are inadequate to apply directly on integral (1) in the presence of oscillation and other singularities. This work presents Clenshaw-Curtis quadrature to get recurrence relation to compute the modified moments, that takes just operations. The initial Cauchy singular values for recurrence relation are obtained by the steepest descent method, as it prominently renowned to evaluate highly oscillatory integrals when the integrands are analytic in sufficiently large region.
The rest of the paper is organized as follows. Section 2 delineates the quadrature algorithm for integral (1). Numerical calculation of the modified moments with recurrence relation by using some Chebyshev properties is defined. Also steepest descent method is established for Cauchy singularity where later the obtained line integrals are further approximated by generalized Gauss quadrature. Section 3 alludes some error bounds derived in terms of Clenshaw-Curtis points and the rate of oscillation k. In Section 4, numerical examples are provided to demonstrate the efficiency and accuracy of the presented method.
2. Numerical Methods
In the computation of integral , the Clenshaw-Curtis quadrature approach is extensively adopted. The scheme is postulated on interpolating the function at Clenshaw-Curtis points set Writing the interpolation polynomial as basis of Chebyshev series
where is the Chebyshev polynomial of first kind of degree N and double prime denotes a sum whose first and last terms are halved, the coefficients
can be computed efficiently by FFT in operations [8,9]. This paper appertains to Clenshaw-Curtis quadrature, which depends on Hermite interpolating polynomial that allow us to get higher order accuracy
For any fixed t, we can elect felicitous N such that and rewrite Hermite interpolating polynomial of degree in terms of Chebyshev series
can be calculated in operations once if are known [13,21]. Finally Clenshaw-Curtis quadrature for integral is defined as
more specifically are called the modified moments. Efficiency of the Clenshaw-Curtis quadrature depends on the fast computation of the moments. In ensuing sub-section, we deduce the recurrence relation for .
Following proof asserts the results for case , and for the same technique can be used as well. Since the integrand is analytic in the half strip of the complex plane, by Cauchy’s Theorem, we have
with all the contours taken in clockwise direction (Figure 1).
Setting we obtain that
Similarly for , we get
From the statement of the theorem, ,
Let , then
Thus, we complete the proof with
From Proposition 2.2 numerical scheme for the line integrals can be evaluated by generalized Gauss-Laguerre quadrature rule, using command lagpts in Chebfun . Let be the nodes and weights of the weight function and let be the nodes and weights of the weight function The line integrals and can be approximated by
By the same argument and can also be approximated with generalized Gauss-Laguerre quadrature rule. Aforementioned theorem enlightens the another interesting fact that can also be computed by it if is an analytic function.
Computation of the moments is derived as, by using Chebyshev property (8)
For integrating by parts, we derive
We deduce the following recurrence relation by inserting (20) in (19)
It is worth to mention that can be computed in operations . For we obtain the as
Unfortunately, practical experiments demonstrate that the recurrence relation for is numerically unstable in the forward direction for , in this sense so-called Oliver’s algorithm is stable and used to rewrite the recurrence relation in the tridiagonal form .
3. Error Analysis
([9,13,14]) Suppose , for a non-negative integer m with , then
([9,14]) Let be a Lipschitz continous function on [−1,1] and let be the interpolation polynomial of at Clenshaw-Curtis points. Then it follows that
In particular, if is analytic with in an Bernstein ellipse with foci and major and minor semiaxis lengths summing to , then
if has an absolutely continuous st derivative and of bounded variation on [−1,1] for some , then for
(van der Corput Lemma ) Suppose that , then for each , it follows
Moreover, for some special cases we have
Suppose that , then it follows for all k that
For simplicity, here we prove the first identity in (3.29). Similar proof can be directly applied to the second identity in (3.29).
it leads to the desired result by Lemma 3.3. □
Suppose that , and define
From Lemma 2.1, we see that and , in addition, is a polynomial of degree at most N with for . Then the error on the Clenshaw-Curtis quadrature (6) can be estimated by
Suppose that and is bounded on , then the Clenshaw-Curtis quadrature (6) is convergent
In particular, if is analytic and in a Bernstein ellipse , , then the error term satisfies
If has an absolutely continuous st derivative and of bounded variation on [−1,1] for some , then for ()
The error bound for for integral can be estimated as
In addition, for , it follows
by Lemma 3.3, it implies
which yields (3.33) together with the estimate on in .
The identity (3.34) follows from Lemma 3.4 due to that for some . □
From the convergence rates Corollary 3.1 and Theorem 3.1, compared with that in , the new scheme is of much fast convergence rate. It is also illustrated by the numerical results (see Section 4).
4. Numerical Results
In this section, we will present several examples to illustrate the efficiency and accuracy of the proposed method. The exact values of an integral (36) are computed through Mathematica 11. Unless otherwise specifically stated, all the tested numerical examples are executed by using Matlab R2016a on a 4 GHz personal laptop with 8 GB of RAM.
Let us consider the integral
for , , Table 1 shows the results for relative error compared with results of integral  in Table 2.
Table 3, Table 4 and Table 5 represent results for relative error computed by Clenshaw-Curtis quadrature. As exact value we just have used that returned by the rule when a huge number of points is used.
Let the integral be
Table 6, Table 7 and Table 8 represent results for relative error computed by Clenshaw-Curtis quadrature. As exact value is calculated by using the rule for large number of points.
Clearly, Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8 illustrate the relative error of the Clenshaw-Curtis quadrature taken as We can see that for proposed Clenshaw-Curtis quadrature based on Hermite interpolation polynomial, with small value of points higher precision of the numerical results of integrals is obtained in operations. Furthermore these tables show that more accurate results can be obtained as k increases with fixed value of N. Conversely, more accurate approximation can be achieved as N increases but k is fixed. Moreover, Tables demonstrate that results successfully satisfy the analysis derived in Section 3.
The Authors have equally contributed to this paper.
This work was supported by National Science Foundation of China (No. 11771454) and the Mathematics and Interdisciplinary Sciences Project of Central South University.
Conflicts of Interest
The authors declare no conflict of interest.The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
Karczmarek, P.; Pylak, D.; Sheshko, M.A. Application of Jacobi polynomials to approximate solution of a singular integral equation with Cauchy kernel. Appl. Math. Comput.2006, 181, 694–707. [Google Scholar] [CrossRef]
Cuminato, N.A. On the uniform convergence of a perturbed collocation method for a class of Cauchy integral equations. Appl. Numer. Math.1995, 16, 439–455. [Google Scholar] [CrossRef]
Eshkuvatov, Z.K.; Long, N.N.; Abdulkawi, M. Approximate solution of singular integral equations of the first kind with Cauchy kernel. Appl. Math. Lett.2009, 22, 651–657. [Google Scholar] [CrossRef]
Ursell, F. Integral equations with a rapidly oscillating kernel. J. Lond. Math. Soc.1969, 1, 449–459. [Google Scholar] [CrossRef]
Polyanin, A.D.; Manzhirov, A.V. Handbook of Integral Equations; CRC Press: Boca Raton, FL, USA, 1998. [Google Scholar]
Olver, S. Numerical Approximation of Highly Oscillatory Integrals. Ph.D. Thesis, University of Cambridge, Cambridge, UK, 2008. [Google Scholar]
Iserles, A.; Norsett, S.P. On quadrature methods for highly oscillatory integrals and their implementation. BIT Numer. Math.2004, 44, 755–772. [Google Scholar] [CrossRef]
Trefethen, L.N. Is Gauss quadrature better than Clenshaw Curtis? SIAM Rev.2008, 50, 67–87. [Google Scholar] [CrossRef]
Trefethen, L.N. Chebyshev Polynomials and Series, Approximation theory and approximation practice. Soc. Ind. Appl. Math.2013, 128, 17–19. [Google Scholar]
Dominguez, V.; Graham, I.G.; Smyshlyaev, V.P. Stability and error estimates for Filon-Clenshaw-Curtis rules for highly oscillatory integrals. SIMA J. Numer. Anal.2011, 31, 1253–1280. [Google Scholar] [CrossRef]
Dominguez, V.; Graham, I.G.; Kim, T. Filon Clenshaw Curtis rules for highly oscillatory integrals with algebraic singularities and stationary points. SIMA J. Numer. Anal.2013, 51, 1542–1566. [Google Scholar] [CrossRef]
Wang, H.; Xiang, S. Uniform approximations to Cauchy principal value integrals of oscillatory functions. Appl. Math. Comput.2009, 215, 1886–1894. [Google Scholar] [CrossRef]
He, G.; Xiang, S. An improved algorithm for the evaluation of Cauchy principal value integrals of oscillatory functions and its application. J. Comput. Appl. Math.2015, 280, 1–13. [Google Scholar] [CrossRef]
Xiang, S.; Chen, X.; Wang, H. Error bounds for approximation in Chebyshev points. Numer. Math.2010, 116, 463–491. [Google Scholar] [CrossRef]
Setia, A. Numerical solution of various cases of Cauchy type singular integral equation. Appl. Math. Comput.2014, 230, 200–207. [Google Scholar] [CrossRef]
Dominguez, V. Filon Clenshaw Curtis rules for a class of highly oscillatory integrals with logarithmic singularities. J. Comput. Appl. Math.2014, 261, 299–319. [Google Scholar] [CrossRef]
Piessens, R.; Branders, M. On the computation of Fourier transforms of singular functions. J. Comput. Appl. Math.1992, 43, 159–169. [Google Scholar] [CrossRef]
Chen, R. Fast computation of a class of highly oscillatory integrals. Appl. Math. Comput.2014, 227, 494–501. [Google Scholar] [CrossRef]
Fang, C. Efficient methods for highly oscillatory integrals with weak and Cauchy singularities. Int. J. Comput. Math.2016, 93, 1597–1610. [Google Scholar] [CrossRef]
Tsalamengas, J.L. Gauss Jacobi quadratures for weakly, strongly, hyper and nearly singular integrals in boundary integral equation methods for domains with sharp edges and corners. J. Comput. Phys.2016, 325, 338–357. [Google Scholar] [CrossRef]
Liu, G.; Xiang, S. Clenshaw Curtis-type quadrature rule for hypersingular integrals with highly oscillatory kernels. Appl. Math. Comput.2019, 340, 251–267. [Google Scholar] [CrossRef]
Boyd, J.P. Chebyshev and Fourier Spectral Methods; Courier Corporation: The North Chelmsford, MA, USA, 2001. [Google Scholar]