Numerical quadratures for singular and hypersingular integrals

We present a procedure for the design of high-order quadrature rules for the numerical evaluation of singular and hypersingular integrals; such integrals are frequently encountered in solution of integral equations of potential theory in two dimensions. Unlike integrals of both smooth and weakly singular functions, hypersingular integrals are pseudo-differential operators, being limits of certain integrals; as a result, standard quadrature formulae fail for hypersingular integrals. On the other hand, such expressions are often encountered in mathematical physics (see, for example, [1]), and it is desirable to have simple and efficient “quadrature” formulae for them. The algorithm we present constructs high-order “quadratures” for the evaluation of hypersingular integrals. The additional advantage of the scheme is the fact that each of the quadratures it produces can be used simultaneously for the efficient evaluation of hypersingular integrals, Hilbert transforms, and integrals involving both smooth and logarithmically singular functions; this results in significantly simplified implementations. The performance of the procedure is illustrated with several numerical examples.     

A special property of the Lobatto weights

It is shown that if the abscissae of the Lobatto quadrature formula are known to accuracy ? the weights can be determined to accuracy O(?²).     

Fehlerabschätzungen für ein numerisches Verfahren zur Auflösung linearer Integralgleichungen mit schwachsingulären Kernen

Zusammenfassung Es werden Fehlerabschätzungen für eine Quadraturformel-methode zur LösungFredholmscher Integralgleichungen zweiter Art mit schwachsingulären Kernen gegeben. Im ersten Teil werden Konvergenzaussagen und Abschätzungen für beliebige schwachsinguläre Kerne im Anschluß an die Methoden vonKantorowitsch undAkilow diskutiert. Im zweiten Teil werden die Abschätzungen für periodische Kerne mit logarithmischen Singularitäten verschärft. Es zeigt sich, daß das Verfahren die Ordnungh ³ besitzt, wobeih die Schrittweite in der zugrundegelegten Rechteckformel ist. Ein Zahlenbeispiel, das der Theorie der Außenraumprobleme für dieHelmholtzsche Schwingungsgleichung entnommen ist, zeigt, daß diese Fehlerordnung realistisch ist.

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Summary Error estimates for approximate solutions ofFredholm integral equations with weakly singular kernels are derived. The approximation scheme is based on the rectangular formula for numerical quadrature. The first part of the paper discusses convergence questions and error estimates for arbitrary kernels, by using the general theory ofKantorowitsch andAkilow. In the second part the estimates are improved for the special case of periodic kernels with logarithmic singularities. It is shown that the approximation is of orderh ³ whereh denotes the step-size in the quadrature formula. The paper concludes with a numerical example, pertaining to the exteriorDirichlet problem for the reduced wave equation.

     

Algorithm 32 automatic computation of integrals with singular integrand,over a finite or an infinite range

A numerical quadrature algorithm is developed, for integrands which may exhibit some kind of singular behaviour within the finite of infinite integration range. Using the automatical FORTRAN IV integration program, one should provide the abscissae the function is not “smooth” at. The quadrature formula has been obtained by applying the trapezoidal rule after transformation of the integrand. Standing severe tests which were based on the test functions of Casaletto et al. and on Kahaner's sample set, the integration scheme turned out to be of a remarkable reliability, efficiency and accuracy.     

Error bound for product quadrature rules inL 1-weighted norm

The authors study the error of the product quadrature rules to compute the integral $\int_{ - 1}^1 {f\left( x \right) u\left( x \right)dx} $ . Estimates inL ₁-weighted norm are established whenu is a weight with algebraic and/or logarithmic singularities and the quadrature points are classical Jacobi zeros. Upper bounds for the generalized functions of second kind are also given.     

Gauss-Legendre and Chebyshev quadratures for singular integrals

Exact expressions are presented for efficient computation of the weights in Gauss-Legendre and Chebyshev quadratures for selected singular integrands. The singularities may be of Cauchy type, logarithmic type or algebraic-logarithmic end-point branching points. We provide Fortran 90 routines for computing the weights for both the Gauss-Legendre and the Chebyshev (Fejér-1) meshes whose size can be set by the user.

New program summary

Program title: SINGQUADCatalogue identifier: AEBR_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEBR_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 4128No. of bytes in distributed program, including test data, etc.: 25 815Distribution format: tar.gzProgramming language: Fortran 90Computer: Any with a Fortran 90 compilerOperating system: Linux, Windows, MacRAM: Depending on the complexity of the problemClassification: 4.11Nature of problem: Program provides Gauss-Legendre and Chebyshev (Fejér-1) weights for various singular integrands.Solution method: The weights are obtained from the condition that the quadrature of order N must be exact for a polynomial of degree?(N−1). The weights are expressed as moments of the singular kernels associated with Legendre or Chebyshev polynomials. These moments are obtained in analytic form amenable for computation.Additional comments: If the NAGWare f95 compiler is used, the option, “-kind = byte”, must be included in the compile command lines of the Makefile.Running time: The test run supplied with the distribution takes a couple of seconds to execute.     

Compact Numerical Quadrature Formulas for Hypersingular Integrals and Integral Equations

In the first part of this work, we derive compact numerical quadrature formulas for finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f(x)=g(x)|x?t| ^?? , ?? being real. Depending on the value of ??, these integrals are defined either in the regular sense or in the sense of Hadamard finite part. Assuming that g??C ^??[a,b], or g??C ^??(a,b) but can have arbitrary algebraic singularities at x=a and/or x=b, and letting h=(b?a)/n, n an integer, we derive asymptotic expansions for ${T}^{*}_{n}[f]=h\sum_{1\leq j\leq n-1,\ x_{j}\neq t}f(x_{j})$ , where x _j =a+jh and t??{x ₁,??,x _n?1}. These asymptotic expansions are based on some recent generalizations of the Euler?CMaclaurin expansion due to the author (A.?Sidi, Euler?CMaclaurin expansions for integrals with arbitrary algebraic endpoint singularities, in Math. Comput., 2012), and are used to construct our quadrature formulas, whose accuracies are then increased at will by applying to them the Richardson extrapolation process. We pay particular attention to the case in which ??=?2 and f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ , which arises in the context of periodic hypersingular integral equations. For this case, we propose the remarkably simple and compact quadrature formula $\widehat{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)-\pi^{2} g(t)h^{-1}$ , and show that $\widehat{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h??0 ???>0, and that it is exact for a class of singular integrals involving trigonometric polynomials of degree at most n. We show how $\widehat{Q}_{n}[f]$ can be used for solving hypersingular integral equations in an efficient manner. In the second part of this work, we derive the Euler?CMaclaurin expansion for integrals $I[f]=\int^{b}_{a} f(x)dx$ , where f(x)=g(x)(x?t) ^?? , with g(x) as before and ??=?1,?3,?5,??, from which suitable quadrature formulas can be obtained. We revisit the case of ??=?1, for which the known quadrature formula $\widetilde{Q}_{n}[f]=h\sum^{n}_{j=1}f(t+jh-h/2)$ satisfies $\widetilde{Q}_{n}[f]-I[f]=O(h^{\mu})$ as h??0 ???>0, when f(x) is T-periodic with T=b?a and $f\in C^{\infty}(-\infty,\infty)\setminus\{t+kT\}^{\infty}_{k=-\infty}$ . We show that this formula too is exact for a class of singular integrals involving trigonometric polynomials of degree at most n?1. We provide numerical examples involving periodic integrands that confirm the theoretical results.     

An algorithm for the numerical evaluation of the Randles-Sevcik function

An accurate numerical method for the evaluation of the Randles-Sevcik function χ(x) is given. The method is an improvement over the trapezoidal quadrature approach of Lether and Wenston. We obtain significant computational economy by adding correction terms to the quadrature sum, which compensates for the effect of the singularities of the integrand. For intermediate values of x, this method is shown to be more suited than other series methods. Practical error bounds are given for the discretization and truncation errors.     

Gauss-turán quadratures of chebyshev type and error formulae

This paper begins with an investigation of two special forms of the Gauss-Turán quadrature of Chebyshev-type of precision 6n?1. Then the remainder formulas of these quadratures are developed and sharp error bounds for the functions inC ^q [?1, 1] are shown, whereq is a positive integer. Most importantly this study proves that these reslts can be extended in order to yield sharp error estimates for all such quadratures of higher precision.     

Hermite and gauss type optimal quadratures for analytic functions

For integrals \(\int\limits_{ - 1}^\iota {w(x)f(x)dx} \) with a non-negative weight functionw(x) and analyticf, we develop Hermite and Gauss type optimal quadratures over the Hilbert spaceH ²(C_r) of functions analytic in a circle. Our development of these optimal quadratures in very similar to that of classical Hermite and Gauss quadratures; the role played by fundamental polynomials in the classical theory is replaced here by certain «fundamental rational functions». We then show, by arguments similar to those used in the classical case, that a Gauss type optimal quadrature has positive weights an abscissas lying in (?¹, 1).     

Derivation and implementation of an algorithm for singular integrals

A numerical construction of extended Gaussian quadrature rules for weight functions with algebraic and logarithmic singularities is presented. A computer program is described and numerical examples are given.     

Transient analysis of the dynamic stress intensity factors using SGBEM for frequency-domain elastodynamics

In this paper, a two-dimensional symmetric-Galerkin boundary integral formulation for elastodynamic fracture analysis in the frequency domain is described. The numerical implementation is carried out with quadratic elements, allowing the use of an improved quarter-point element for accurately determining frequency responses of the dynamic stress intensity factors (DSIFs). To deal with singular and hypersingular integrals, the formulation is decomposed into two parts: the first part is identical to that for elastostatics while the second part contains at most logarithmic singularities. The treatment of the elastostatic singular and hypersingular singular integrals employs an exterior limit to the boundary, while the weakly singular integrals in the second part are handled by Gauss quadrature. Time histories (transient responses) of the DSIFs can be obtained in a post-processing step by applying the standard fast Fourier transform (FFT) and algorithm to the frequency responses of these DSIFs. Several test examples are presented for the calculation of the DSIFs due to two types of impact loading: Heaviside step loading and blast loading. The results suggest that the combination of the symmetric-Galerkin boundary element method and standard FFT algorithms in determining transient responses of the DSIFs is a robust and effective technique.     

Characterizing the measures on the unit circle with exact quadrature formulas in the space of polynomials

In the present paper we characterize the measures on the unit circle for which there exists a quadrature formula with a fixed number of nodes and weights and such that it exactly integrates all the polynomials with complex coefficients. As an application we obtain quadrature rules for polynomial modifications of the Bernstein measures on [−1,1], having a fixed number of nodes and quadrature coefficients and such that they exactly integrate all the polynomials with real coefficients.     

Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method

New Gaussian integration schemes are presented for the efficient and accurate evaluation of weak form integrals in the extended finite element method. For discontinuous functions, we construct Gauss-like quadrature rules over arbitrarily-shaped elements in two dimensions without the need for partitioning the finite element. A point elimination algorithm is used in the construction of the quadratures, which ensures that the final quadratures have minimal number of Gauss points. For weakly singular integrands, we apply a polar transformation that eliminates the singularity so that the integration can be performed efficiently and accurately. Numerical examples in elastic fracture using the extended finite element method are presented to illustrate the performance of the new integration techniques.     

Weibull renewal equation solution

The existing solution methods for the Weibull Renewal Equation suffer from a lack of sufficient accuracy due to the singularity at the origin for some parameter values of the weibull density. The proposed method of solution provides accuracy to any desired degree of precision for all parameter values particularly in the singular range. The method utilizes a cubic spline approximation of the unknown renewal function and applies the Galerkin technique of integral equation solution. Gaussian quadratures are used to evaluate integrals. The singular nature of the integrand is handled by the Gauss-Jacobi quadrature. Results are compared with those obtained by simulation.     

On polynomial solutions of the linear stationary control system

It is known that the controllable system x′ = Bx + Du, where the x is the n-dimensional vector, can be transferred from an arbitrary initial state x(0) = x ⁰ to an arbitrary finite state x(T) = x ^T by the control function u(t) in the form of the polynomial in degrees t. In this work, the minimum degree of the polynomial is revised: it is equal to 2p + 1, where the number (p ? 1) is a minimum number of matrices in the controllability matrix (Kalman criterion), whose rank is equal to n. A simpler and a more natural algorithm is obtained, which first brings to the discovery of coefficients of a certain polynomial from the system of algebraic equations with the Wronskian and then, with the aid of differentiation, to the construction of functions of state and control.     

Lanczos <Emphasis Type="Italic">τ</Emphasis>-method regularization algorithm and its algebraic-programming implementation

An algebraic algorithm is developed for computing an algebraic polynomial y _n of order n ∈ N in computer algebra systems. This polynomial is the optimal approximation of the solution y = y(x), x ∈ [a,b], to a system of linear differential equations with polynomial coefficients and initial conditions at a regular singular zero point of this equation in a space C_{[ a,b ]}^k C_{\left[ {a,b} \right]}^k .     

On the convergence of some quadrature rules for Cauchy principal-value and finite-part integrals

In this paper sufficient conditions are derived to ensure the convergence of the Elliott and Hunter types of quadrature rules for the evaluation of weighted Cauchy principal-value integrals of the form: The simultaneous convergence in the interval (?1, 1) of both quadratures was established for a class of Hölder-continuous functionsf(f∈H _μ ). Corrections of some previous statements on the subject of convergence of such quadratures are also included. Moreover, a simple derivation of the Hunter and Elliott types of quadrature rules for the evaluation of the derivative of thep-th-order of the abovestated integral was given and sufficient conditions for the convergence of the Hunter-type quadrature were obtained. Thus, the convergence of this integral was ensured for functionsf such thatf ^(p) ∈H _μ .     

Singularity crossing phenomena in DAEs: A two-phase fluid flow application case study

This paper analyzes the existence of smooth trajectories through singular points of differential algebraic equations, or DAEs, arising from traveling wave solutions of a degenerate convection-diffusion model. The DAE system can be written in the quasilinear form A(x)x′ = b(x). In this setting, singularities are displayed when the matrix A(x) undergoes a rank change. The singular hypersurface may be smoothly crossed by trajectories in a finite time if x* is a geometric singularity satisfying certain directional conditions. The basis of our analysis is a two-phase fluid flow model in one spatial dimension with dissipative mechanism involved.