Cauchy Principal Value Integrals Research Articles

A boundary point interpolation method for acoustic problems with particular emphasis on the calculation of Cauchy principal value integrals

A boundary point interpolation method (BPIM) is presented in this paper for numerical solving acoustic problems. The BPIM is a boundary-type meshless method that combines the point interpolation technique for constructing approximate functions with boundary integral equations for reformulating boundary value problems. Since effective numerical integration techniques are crucial in the BPIM and boundary integral equation methods for calculating regular and singular integrals, two Clenshaw-Curtis integration techniques (CCITs) for regular and weakly singular integrals are derived in a unified manner with the same integration points on quadratic integration cells. In addition, a unilateral CCIT, a power series expansion technique, and an improved CCIT are proposed to effectively calculate strongly singular integrals or Cauchy principal value integrals in the BPIM. These integration techniques provide direct and efficient formulas for calculating boundary integrals on quadratic integration cells, and can be directly applied to the calculation of regular and singular integrals in the boundary element method and other meshless boundary integral equation methods. Explicit expressions of integration weights in all CCITs are derived, and the values of integration points and weights are listed. Numerical results are finally provided to validate the effectiveness of the present meshless method and integration techniques.

Riemann boundary value problems on the Archimedean spiral

This paper studies the Riemann boundary value problems on the Archimedean spiral. We characterize the functions which are integrable on the Archimedean spiral. We also study the asymptotic behaviors of Cauchy-type integral and Cauchy principal value integral on Archimedean spiral at the origin and infinity. At the end, we discuss the Riemann boundary value problems for sectionally holomorphic functions with the Archimedean spiral as their jump curve and obtain the singular integral representations of the solutions if solvable.

A Class of Quadrature Rules for Complex Cauchy Principal Value Integrals

This article is fully devoted to the numerical approximation of Cauchy-type integrals in the complex plane. A class of degree eight quadrature rules is formulated from a family of Gauss-type two-point rules based on the method of extrapolation. The basic rules are developed and then composite rules are constructed from the basic rules. The Error bounds for each rule are determined. The validity of the method has been demonstrated by the provision of numerical experiments and their results.

Gaussian type quadrature rules related to the oscillatory modification of the generalized Laguerre weight functions

In this paper we consider weighted integrals with respect to a modification of the generalized Laguerre weight functions wα(x)=xαe−x, α>−1, on (0,+∞) by the oscillatory factor exp(iζx), where ζ>0. For calculation of such integrals for analytic real-valued functions in D={z∈ℂ∣Rez≥0,Imz≥0} we construct polynomials orthogonal with respect to the linear functional L:P→ℂ, given by L[p]=∫0+∞p(x)wα(x)exp(iζx)dx, where P is a linear space of all algebraic polynomials, as well as the corresponding quadrature rules of Gaussian type. The existence of such orthogonal polynomials and the corresponding three-term recurrence relations are proved. A few numerical examples are presented, including computation of the Cauchy principal value integrals.

Efficient computation of oscillatory Bessel transforms with a singularity of Cauchy type

In this article, we present and analyze efficient methods for computing the Cauchy principal value integral of the oscillatory Bessel function ∫abf(x)x−cJm(ωx)dx, where 0≤a<c<b. For the two cases of a>0 and a=0, we establish new steepest descent integration paths, and employ the complex line integration technique to obtain affordable quadrature rules. According to the relation formula between the Meijer G-function and the Bessel function, a crucial explicit expression of the required integral ∫0+∞xjx−cJm(ωx)dx can be derived in terms of the Meijer G-function. Furthermore, for the proposed quadrature formulae, we perform rigorous error analysis in inverse powers of ω. Particularly, compared with a combination method given in the work (Kang et al., 2022), the proposed method takes less time and produces more accurate results. We verify error analysis of the proposed method by experimental results. Numerical examples are presented to illustrate the accuracy and efficiency of the proposed numerical methods.

Efficient numerical methods for Cauchy principal value integrals with highly oscillatory integrands

Efficient numerical methods for Cauchy principal value integrals with highly oscillatory integrands

Numerical methods for Cauchy principal value integrals of oscillatory Bessel functions

In this paper, we first propose different combination methods to compute the Cauchy principal value integrals of oscillatory Bessel functions. By special transformations, the considered integrals are converted to finite integrals and infinite integrals. Then, the finite integrals can be calculated through the Filon-type method, the Clenshaw–Curtis–Filon method and the Clenshaw–Curtis–Filon-type method, respectively. We compute the infinite integral through the numerical steepest descent method. Moreover, the error analysis with respect to frequency ω is given through theoretical analysis. Eventually, we present several numerical experiments which are in accord with our analysis. Particularly, the accuracy can be improved by either using more nodes or adding more derivatives interpolation at endpoints. The accuracy will increase drastically with the growth of frequency ω if both the number of nodes and interpolated multiplicity are fixed.

ОПЕРАТОРНИЙ МЕТОД В ЗАДАЧІ ПРО ДИФРАКЦІЮ Н-ПОЛЯРИЗОВАНОЇ ХВИЛІ НА ДВОХ ОДНАКОВИХ НАПІВНЕСКІНЧЕННИХ РЕШІТКАХ, РОЗТАШОВАНИХ В ОДНІЙ ПЛОЩИНІ

Purpose: Problem of the H-polarized plane wave diffraction by the structure, which consists of two semi-infinite strip gratings, is considered. The gratings are placed in the same plane. The gap between the gratings is arbitrary. The purpose of the paper is to develop the operator method to the structures, which scattered fields have both discrete and continuous spatial spectra. Design/methodology/approach: In the spectral domain, in the domain of the Fourier transform, the scattered field is expressed in terms of the unknown Fourier amplitude. The field reflected by the considered structure is represented as a sum of two fields of currents on the strips of semi-infinite gratings. The operator equations are obtained for the Fourier amplitudes. These equations use the operators of reflection of semi-infinite gratings, which are supposed to be known. The field scattered by a semi-infinite grating can be represented as a sum of plane and cylindrical waves. The reflection operator of a semi-infinite grating has singularities at the points, which correspond to the propagation constants of plane waves. Consequently, the unknown Fourier amplitudes of the fi eld scattered by the considered structure also have singularities. To eliminate these latter, the regularization procedure has been carried out. As a result of this procedure, the operator equations are reduced to the system of integral equations containing the integrals, which should be understood as the Cauchy principal value and Hadamar finite part integrals. The discretization has been carried out. As a result, the system of linear equations is obtained, which is solved with the use of the iterative procedure. Findings: The operator equations with respect to the Fourier amplitudes of the field scattered by the structure, which consists of two semi-infinite gratings, are obtained. The computational investigation of convergence has been made. The near and far scattered fields are investigated for different values of the grating parameters. Conclusions: The effective algorithm to study the fields scattered by the strip grating, which has both discrete and continuous spatial spectra, is proposed. The developed approach can be an effective instrument in solving a series of problems of antennas and microwave electronics. Key words: semi-infinite grating, operator method, singular integral, hypersingular integral, regularization procedure

Extended Error Expansion of Classical Midpoint Rectangle Rule for Cauchy Principal Value Integrals on an Interval

The classical composite midpoint rectangle rule for computing Cauchy principal value integrals on an interval is studied. By using a piecewise constant interpolant to approximate the density function, an extended error expansion and its corresponding superconvergence results are obtained. The superconvergence phenomenon shows that the convergence rate of the midpoint rectangle rule is higher than that of the general Riemann integral when the singular point coincides with some priori known points. Finally, several numerical examples are presented to demonstrate the accuracy and effectiveness of the theoretical analysis. This research is meaningful to improve the accuracy of the collocation method for singular integrals.

A least squares-based approach to evaluating strongly singular integrals

The aim of this paper is to tackle the numerical calculation of the so-called finite-part integrals, including Cauchy principal value integrals and any supersingular integral. The numerical procedure consists of replacing the density function f(t) by the polynomial that is the best fit in a least-squares sense for the values f(zk), where {zk} is a set of distinct points located on an ellipse that surrounds the interval [−1,1]. This method works in all mentioned cases, it does not depend on derivatives of f(t), and provides very accurate results when f is smooth enough. Some examples show the performance of this approach.

Rational Transformations for Evaluating Singular Integrals by the Gauss Quadrature Rule

In this work we introduce new rational transformations which are available for numerical evaluation of weakly singular integrals and Cauchy principal value integrals. The proposed rational transformations include parameters playing an important role in accelerating the accuracy of the Gauss quadrature rule used for the singular integrals. Results of some selected numerical examples show the efficiency of the proposed transformation method compared with some existing transformation methods.

Total integrals of Ablowitz‐Segur solutions for the inhomogeneous Painlevé II equation

AbstractIn this paper, we establish a formula determining the value of the Cauchy principal value integrals of the real and purely imaginary Ablowitz‐Segur solutions for the inhomogeneous second Painlevé equation. Our approach relies on the analysis of the corresponding Riemann‐Hilbert problem and the construction of an appropriate parametrix in a neighborhood of the origin. Obtained integral formulas are consistent with already known analogous results for the Ablowitz‐Segur solutions of the homogeneous Painlevé II equation.

Numerical Integration Method for Cauchy Principal-Value Integrals by Taylor Seires

Numerical Integration Method for Cauchy Principal-Value Integrals by Taylor Seires

Singularity cancellation method for time-domain boundary element formulation of elastodynamics: A direct approach

In the implementation of time-domain boundary element method for elasto-dynamic problems, there are two types of singularities: the wave front singularity arising when the product of wave velocity and time is equal to the distance between the source point and the field point, and the spatial singularity arising when the source point coincides with the field point. In this paper, the singularity of the first type in the integrand is eliminated by an analytical integration over time, Cauchy principal value and Hadamard finite part integral. Four types of singularities with different orders appear in the integrand after analytical time integration. In order to accurately calculate the integral, in which the integrand is piecewise continuous, the integral domain is subdivided into several patches based on the relation between the product of wave velocity and time and the distance. In singular patches, the integrands are separated into a regular part and a singular part. The regular part can be computed by traditional numerical integration method such as Gaussian integration, while the singular part can be analytically integrated. Using the proposed method, the spatial singular integrals can be calculated directly. Numerical examples using various kinds of elements are presented to verify the proposed method.

Non-normal type singular integral-differential equations by Riemann-Hilbert approach

This article deals with one class of singular integral-differential equations of non-normal type with convolution and Cauchy principal value integral in class {0}. By using Fourier transform, this classes of equations are transformed into a Riemann boundary value problem with nodal point, and we prove the existence of solutions and the Noether theory for the equations. For such equations, we propose one method different from classical one, and we obtain the general solutions and the conditions of solvability. All cases about the behaviors of the solution are considered at nodal points. Therefore, the theory of integral equations and the classical Riemann boundary value problems is extended further.

Asymptotic error expansions and splitting extrapolation algorithm for two classes of two-dimensional Cauchy principal-value integrals

This paper proposes numerical quadrature rules for two-dimensional Cauchy principal-value integrals of the forms ∫∫Ωf(x,y)(x−s)2+(y−t)2dydx and ∫∫Ωf(x,y)(x−s)(y−t)dydx. The derivation of these quadrature rules is based on the Euler–Maclaurin error expansion of a modified trapezoidal rule for one-dimensional Cauchy singular integrals. The corresponding error estimations are investigated, and the convergence rates O(hm2μ+hn2μ) are obtained for the proposed quadrature rules, where hm and hn are partition sizes in x and y directions, μ is a positive integer determined by integrand. To further improve accuracy, a splitting extrapolation algorithm is developed based on the asymptotic error expansions. Several numerical tests are performed to verify the effectiveness of the proposed methods.

Numerical evaluation of oscillatory-singular integrals

ABSTRACTIn this paper, we consider an asymptotic Filon-type method for computing the oscillatory Cauchy principal value integrals and end-point oscillatory-singular integrals. The method is constructed by using a Lagrange interpolating polynomial to approximate the non-oscillatory function so that it successfully avoids the singularity and solving the systems of equations. Numerical examples and theoretical results show the accuracy and efficiency of the proposed method.

Analysis of Vertical Dynamic Cross Interaction between Adjacent Footings for Railway Tracks through Soil

An efficient semi-analytical method is presented for the vertical analysis of adjacent railway tracks considering the dynamic cross interaction (DCI) effect through the semi-infinite soil medium. The interfaces between soil and tracks are discretized into a series of strip elements to solve the unknown contact pressure influenced by DCI effect. The Green function for each element under uniform harmonic excitation is derived and calculated by the piecewise integration and Cauchy principal value integral. The vertical impedance matrix is finally obtained by the superposition method. The accuracy and the effectiveness in high frequency range are verified by the convergence study, as well as the comparison study with the rigorous results from the mix-boundary-value method. The influence of the soil properties and the tracks distance on the DCI effect are discussed in detail in the parameter studies. It is shown that the DCI effect increases with the decrease of the distance ratio S/L.

A Simple Model for Vertical Dynamic Interactions among a Group of Strip Footings Rested on Homogeneous Half-Space

A simple model for vertical dynamic interactions among a group of strip footings rested on half-space is presented in this paper. An analytical method is presented to obtain the contact pressures and the impedance matrix for a group of surface strip footings. In order to conveniently solve the unknown contact pressures between the soil and footings, the soil-footing interfaces are discretized into a series of strip elements. The Green function for each element under uniform harmonic force is derived and calculated by the piecewise integration and Cauchy principal value integral. The influences of footing and soil parameters on contact pressures and vertical dynamic impedances of footing groups are discussed in detail. The SSSI effect between adjacent footings increases with the decrease of the distance ratioS/L. For three footings in a group, the middle footing experiences greater cross-interactional effect than the side ones. The present method has high accuracy, which is not only simple but also suitable for the high-frequency analysis.

Error expansion of trapezoidal rule for certain two-dimensional Cauchy principal value integrals

In this article, we discuss the classical composite trapezoidal rule for the computation of two dimensional singular integrals. The purpose is to obtain the convergence results O(h2) which is the same as the Riemann integral convergence rate at certain points of the classical composite trapezoidal rule. With the error functional of trapezoidal rule for computing two dimensional singular integrals, we get the superconvergence phenomenon when the special function in error functional is equal to zero. At last, some numerical examples are reported to illustrate our theoretical analysis which agree with it very well.