The superconvergence of the composite midpoint rule for the finite-part integral

The composite midpoint rule is probably the simplest one among the Newton-Cotes rules for Riemann integral. However, this rule is divergent in general for Hadamard finite-part integral. In this paper, we turn this rule to a useful one and, apply it to evaluate Hadamard finite-part integral as well as to solve the relevant integral equation. The key point is based on the investigation of its pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate of the midpoint rule is higher than what is globally possible. We show that the superconvergence rate of the composite midpoint rule occurs at the midpoint of each subinterval and obtain the corresponding superconvergence error estimate. By applying the midpoint rule to approximate the finite-part integral and by choosing the superconvergence points as the collocation points, we obtain a collocation scheme for solving the finite-part integral equation. More interesting is that the inverse of the coefficient matrix of the resulting linear system has an explicit expression, by which an optimal error estimate is established. Some numerical examples are provided to validate the theoretical analysis.     

Numerical evaluation for Cauchy type singular integrals on the interval

The singular integral (SI) with the Cauchy kernel is considered. New quadrature formulas (QFs) based on the modification of discrete vortex method to approximate SI are constructed. Convergence of QFs and error bounds are shown in the classes of functions H^α([−1,1]) and C¹([−1,1]). Numerical examples are shown to validate the QFs constructed.     

On the evaluation of Cauchy principal value integrals of oscillatory functions

The problem of the numerical evaluation of Cauchy principal value integrals of oscillatory functions , where −1<τ<1, has been discussed. Based on analytic continuation, if f is analytic in a sufficiently large complex region G containing [−1, 1], the integrals can be transformed into the problems of integrating two integrals on [0,+∞) with the integrand that does not oscillate, and that decays exponentially fast, which can be efficiently computed by using the Gauss-Laguerre quadrature rule. The validity of the method has been demonstrated in the provision of two numerical experiments and their results.     

The evaluation of Cauchy principal value integrals involving unknown poles

A method is described for the evaluation of a Cauchy principal value integral of the formf ₀ ^p f(t)dt, wheref is analytic in the interval [0,p] except at a simple pole at an unknown point in (0,p), with an unknown residue. The method is based on the trapezoidal rule.     

Superconvergence and ultraconvergence of Newton-Cotes rules for supersingular integrals

In this article, the general (composite) Newton-Cotes rules for evaluating Hadamard finite-part integrals with third-order singularity (which is also called “supersingular integrals”) are investigated and the emphasis is placed on their pointwise superconvergence and ultraconvergence. The main error of the general Newton-Cotes rules is derived, which is shown to be determined by a certain function . Based on the error expansion, the corresponding modified quadrature rules are also proposed. At last, some numerical experiments are carried out to validate the theoretical analysis.     

On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals

Summary In a previous paper the authors proposed a modified Gaussian rule _* ^m (wf;t)to compute the integral (wf;t) in the Cauchy principal value sense associated with the weightw, and they proved the convergence in closed sets contained in the integration interval. The main purpose of the present work is to prove uniform convergence of the sequence { _* ^m (wf;t)} on the whole integration interval and to give estimates for the remainder term. The same results are shown for particular subsequences of the Gaussian rules _m (wf;t) for the evaluation of Cauchy principal value integrals. A result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made.     

Interpolatory product quadratures for Cauchy principal value integrals with Freud weights

Summary. We prove convergence results and error estimates for interpolatory product quadrature formulas for Cauchy principal value integrals on the real line with Freud–type weight functions. The formulas are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration. As a by–product, we obtain new bounds for the derivative of the functions of the second kind for these weight functions. Received July 15, 1997 / Revised version received August 25, 1998     

Application of Cauchy integrals and singular integral equations in scattered data problems

A new approach is suggested for constructing “smooth” surfaces which contain discontinuities in functional values or derivatives at prescribed locations. The approach is based on solving singular integral equations with Cauchy-type kernels. It is applicable in the interpolation or approximation of scattered data. We investigate, in particular, several two-dimensional biharmonic and triharmonic problems which have smooth solutions except on given line segments, across which different types of discontinuities occur. We also discuss some issues concerning the application of the approach in practical problems.     

Superconvergence of the composite Simpson’s rule for a certain finite-part integral and its applications

The superconvergence phenomenon of the composite Simpson’s rule for the finite-part integral with a third-order singularity is studied. The superconvergence points are located and the superconvergence estimate is obtained. Some applications of the superconvergence result, including the evaluation of the finite-part integrals and the solution of a certain finite-part integral equation, are also discussed and two algorithms are suggested. Numerical experiments are presented to confirm the superconvergence analysis and to show the efficiency of the algorithms.     

Convergence of Gauss type product formulas for the evaluation of two-dimensional Cauchy principal value integrals

The authors consider product rules of Gauss type for the numerical approximation of certain two-dimensional Cauchy principal value integrals with respect to generalized smooth Jacobi weight functions.Convergence results for these rules are given, and some asymptotic estimates of the remainder are established.Work sponsored by Italian Research Council and by the Ministero della Pubblica Istruzione of Italy.     

Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals

Summary. We show that, if (), the error term of every modified positive interpolatory quadrature rule for Cauchy principal value integrals of the type , , fulfills uniformly for all , and hence it is of optimal order of magnitude in the classes (). Here, is a weight function with the property . We give explicit upper bounds for the Peano-type error constants of such rules. This improves and completes earlier results by Criscuolo and Mastroianni (Calcolo 22 (1985), 391–441 and Numer. Math. 54 (1989), 445–461) and Ioakimidis (Math. Comp. 44 (1985), 191–198). For the special case of the Gaussian rule, we show that the restriction can be dropped. The results are based on a new representation of the Peano kernels of these formulae via the Peano kernels of the underlying classical quadrature formulae. This representation may also be useful in connection with some different problems. Received November 21, 1994     

The superconvergence of Newton–Cotes rules for the Hadamard finite-part integral on an interval

We study the general (composite) Newton–Cotes rules for the computation of Hadamard finite-part integral with the second-order singularity and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite) Newton–Cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points. Several numerical examples are provided to validate the theoretical analysis. The work of J. Wu was partially supported by the National Natural Science Foundation of China (No. 10671025) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. CityU 102507). The work of W. Sun was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (No. City U 102507) and the National Natural Science Foundation of China (No. 10671077).     

Convergence of product formulas for the numerical evaluation of certain two-dimensional Cauchy principal value integrals

Summary We consider product rules of interpolatory type for the numerical approximation of certain two-dimensional Cauchy principal value integrals. We present convergence results which generalize those known in the one-dimensional case.Work sponsored by the Ministero della Pubblica Istruzione of Italy     

The numerical evaluation of cauchy principal value integrals with non-standard weight functions

The authors develop an algorithm for the numerical evaluation of the finite Hilbert transform, with respect to non-standard weight functions, by a product quadrature rule. In particular, this algorithm allows us to deal with the weight functions with algebraic and/or logarithmic singularities in the interval [−1, 1], by using the Chebyshev points as quadrature nodes. The practical application of the rule is shown to be straightforward and to yield satisfactory numerical results. Convergence theorems are also given, when the nodes are the zeros of certain classical Jacobi polynomials and the weight is defined as a generalized Ditzian-Totik weight. This work was supported by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica (first author) and by the Italian Research Council (second author).     

Computation of poisson type integrals

We consider problems of computing the Poisson integral when the point at which the integral is evaluated approaches the ball surface. Techniques are proposed enabling one to improve the computation efficiency.     

Efficient quadrature for highly oscillatory integrals involving critical points

This paper based on the Levin collocation method and Levin-type method together with composite two-point Gauss–Legendre quadrature presents efficient quadrature for integral transformations of highly oscillatory functions with critical points. The effectiveness and accuracy of the quadrature are tested.     

A modified Gaussian quadrature rule for integrals involving poles of any order

By applying the theory of completely symmetric functions we derive a Gaussian quadrature rule which generalizes that due to McNamee. A feature of this generalization is the inclusion of an explicit correction term taking account of the presence of poles (of any order) of the integrand close to the integration-interval. A numerical example is provided to illustrate the formulae.     

Definitions,properties and applications of finite-part integrals

By using standard calculus, we discuss some definitions and properties of finite-part integrals, point out the essence of this concept and show how these integrals naturally arise in some integral equation applications. A couple of new examples are also described.