Cauchy Principal Value Research Articles

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.

The inverse gamma-difference distribution and its first moment in the Cauchy principal value sense

The inverse gamma-difference distribution and its first moment in the Cauchy principal value sense

The inverse gamma-difference distribution and its first moment in the Cauchy principal value sense

The inverse gamma-difference distribution and its first moment in the Cauchy principal value sense

A note on the discrete Cauchy‐Kovalevskaya extension

In this paper, we exploit the umbral calculus framework to reformulate the so‐called discrete Cauchy‐Kovalevskaya extension in the scope of hypercomplex variables. The key idea is to consider not only formal power series representation for the underlying solution, but also integral representations for the Chebyshev polynomials of first and second kind by means of its Cauchy principal values. It turns out that the resulting integral representation associated to our toy problem is a space‐time Fourier type inversion formula. Moreover, with the aid of some Laplace transform identities involving the generalized Mittag‐Leffler function, we are able to establish a link with a Cauchy problem of differential‐difference type.

Chaotic backward shift operator on Chebyshev polynomials

We obtain the representation of the backward shift operator on Chebyshev polynomials involving a principal value (PV) integral. Twice the backward shift on the space of square-summable sequences l2 displays chaotic dynamics, thus we provide an explicit form of a chaotic operator on L2 (−1, 1, (1−x2)–1/2) using Cauchy’s PV integral. We explicitly calculate the periodic points of the operator and provide examples of unbounded trajectories, as well as chaotic ones. Histograms and recurrence plots of shifts of random Chebyshev expansions display interesting behaviour over fractal measures.

Quaternion Boundary Element Method for Coupled Exterior and Interior Magnetostatic Fields

In this paper, a quaternion boundary element method (BEM) is proposed to solve the magnetostatic problem. The present quaternion-valued BEM is developed by discretizing the quaternion-valued boundary integral equation (BIE). The quaternion-valued BIE can be seen as an extension of the generalized complex variable BIE in 3-D space. In other words, quaternion algebra is an extension of the complex variable in 3-D space. To derive quaternion-valued BIEs, the quaternion-valued Stokes’ theorem is utilized. The quaternion-valued BIEs are noted for singularity, which exists in the sense of the Cauchy principal value (CPV). An analytical scheme is developed to evaluate the CPV by introducing a simple quaternion-valued harmonic function. For the domain points close to the boundary, some sorts of analogous, nearly singular, so-called “numerical boundary layer” phenomena appear and are remedied by using a similar analytic evaluation. The quaternion BEM features the oriented surface element, combining the unit outward normal vector with the ordinary surface element. It is noted that all derivations are done in quaternion algebra. In addition, quaternion algebra is more flexible than vector algebra in solving some 3-D problems from the point view of algebraic space. In deriving BIEs for exterior fields, the conditions at infinity for the quaternion-valued functions are carefully examined. Later, a magnetic sphere in a uniform magnetic field is considered. This problem is a magnetostatic problem of coupled exterior and interior magnetostatic fields. Finally, we apply the present approach to solve the magnetostatic problem. By comparing with exact solutions, the validity of the present approach is checked.

Accurate and Efficient Nyström Volume Integral Equation Method for Electromagnetic Scattering of 3-D Metamaterials in Layered Media

In this paper, we develop an accurate and efficient Nyström volume integral equation (VIE) method for the Maxwell equations to compute the electromagnetic scattering of three-dimensional metamaterials in layered media. The VIE for meta-atom scatterers in a layered medium is derived using dyadic Green's functions for layered media where Cauchy Principal Values are evaluated accurately using a finite-size exclusion volume with the help of some correction integrals of removable singularities. Several desingularization techniques are also introduced for the VIE, including interpolated quadrature formulas with tensor-product quadrature nodes for self-integration terms of typical meta-atoms, a regularization scheme for closely packed meta-atoms, and adaptive integration approaches for high gradient components of the dyadic Green's functions. The resulting Nyström method demonstrates fast numerical convergence with high accuracy and efficiency with only a small number of collocation points in calculating the scattered fields and reflection coefficients of metamaterials of cubic, spherical, and cylindrical shapes.

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 ratio S/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.

On the Solutions of a $$2+1$$ 2 + 1 -Dimensional Model for Epitaxial Growth with Axial Symmetry

In this paper, we study the evolution equation derived by Xu and Xiang (SIAM J Appl Math 69(5):1393–1414, 2009) to describe heteroepitaxial growth in $$2+1$$ dimensions with elastic forces on vicinal surfaces is in the radial case and uniform mobility. This equation is strongly nonlinear and contains two elliptic integrals and defined via Cauchy principal value. We will first derive a formally equivalent parabolic evolution equation (i.e., full equivalence when sufficient regularity is assumed), and the main aim is to prove existence, uniqueness and regularity of strong solutions. We will extensively use techniques from the theory of evolution equations governed by maximal monotone operators in Banach spaces.

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.

A plane-wave singularity subtraction technique for the classical Dirichlet and Neumann combined field integral equations

This paper presents expressions for the classical combined field integral equations for the solution of Dirichlet and Neumann exterior Helmholtz problems on the plane, in terms of smooth (continuously differentiable) integrands. These expressions are obtained by means of a singularity subtraction technique based on pointwise plane-wave expansions of the unknown density function. In particular, a novel regularization of the hypersingular operator is obtained, which, unlike regularizations based on Maue's integration-by-parts formula, does not give rise to involved Cauchy principal value integrals. Moreover, the expressions for the combined field integral operators and layer potentials presented in this contribution can be numerically evaluated at target points that are arbitrarily close to the boundary without severely compromising their accuracy. A variety of numerical examples in two spatial dimensions that consider three different Nyström discretizations for smooth domains and domains with corners—one of which is based on direct application of the trapezoidal rule—demonstrates the effectiveness of the proposed higher-order singularity subtraction approach.

Numerical computation of hypersingular integrals on the real semiaxis

In this paper we propose some different strategies to approximate hypersingular integrals∫=0+∞G(x)(x−t)p+1dx,where p is a positive integer, t > 0 and the integral is understood in the Hadamard finite part sense. Hadamard Finite Part integrals (shortly FP integrals), regarded as pth derivative of Cauchy principal value integrals, are of interest in the solution of hypersingular BIE, which model many different kind of Physical and Engineering problems (see [1] and the references therein, [2], [3, 4]).The procedure we employ here is based on a simple tool like the “truncated” Gaussian rule (see [5]), conveniently modified to remove numerical cancellation. We will consider functions G having different decays at infinity. The method is shown to be numerically stable and convergent and some error estimates in suitable Zygmund-type spaces are proved. Finally, some numerical tests which confirm the efficiency of the proposed procedures are presented.

Uniform approximation to Cauchy principal value integrals with logarithmic singularity

An approximation of Clenshaw–Curtis type is given for Cauchy principal value integrals of logarithmically singular functions I(f;c)=−∫−11f(x) (log|x−c|)∕(x−c)dx (c∈(−1,1)) with a given function f. Using a polynomial pN of degree N interpolating f at the Chebyshev nodes we obtain an approximation I(pN;c)≅I(f;c). We expand pN in terms of Chebyshev polynomials with O(NlogN) computations by using the fast Fourier transform. Our method is efficient for smooth functions f, for which pN converges to f fast as N grows, and so simple to implement. This is achieved by exploiting three-term inhomogeneous recurrence relations in three stages to evaluate I(pN;c). For f(z) analytic on the interval [−1,1] in the complex plane z, the error of the approximation I(pN;c) is shown to be bounded uniformly. Using numerical examples we demonstrate the performance of the present method.

Generalized Cauchy Theorem in Clifford Analysis and Boundary Value Problems for Regular Functions

In this paper, we establish the generalized Cauchy theorems on the para-sphere and the generalized Cauchy integral formulae on the strong para-sphere in Clifford analysis. As applications, the generalized Cauchy theorems and the generalized Cauchy integral formulae on the closed smooth surface and the cylindroid with crooked tips are respectively obtained. And these directly result in the Painleve theorem and the generalization of the Sochocki–Plemelj formula for the difference of boundary values in Clifford analysis. Then, by using these results the Riemann jump boundary value problems and Dirichlet boundary value problems for regular functions in Clifford analysis are discussed. Some singular integral equations are also solved and the inversion formula for Cauchy principal value is obtained by the results based on these boundary value problems solved.

Cauchy Principal Value Contour Integral with Applications

Cauchy principal value is a standard method applied in mathematical applications by which an improper, and possibly divergent, integral is measured in a balanced way around singularities or at infinity. On the other hand, entropy prediction of systems behavior from a thermodynamic perspective commonly involves contour integrals. With the aim of facilitating the calculus of such integrals in this entropic scenario, we revisit the generalization of Cauchy principal value to complex contour integral, formalize its definition and—by using residue theory techniques—provide an useful way to evaluate them.

Fundamental solutions of nonlocal Hörmander’s operators II

Consider the following nonlocal integro-differential operator: for $\alpha\in(0,2)$: \[\mathcal{L}^{(\alpha)}_{\sigma,b}f(x):=\mbox{p.v.}\int_{|z|<\delta}\frac{f(x+\sigma(x)z)-f(x)}{|z|^{d+\alpha}}\,\mathrm{d}z+b(x)\cdot\nabla f(x)+{\mathscr{L}}f(x),\] where $\sigma:\mathbb{R}^{d}\to\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ and $b:\mathbb{R}^{d}\to\mathbb{R}^{d}$ are smooth functions and have bounded partial derivatives of all orders greater than $1$, $\delta$ is a small positive number, p.v. stands for the Cauchy principal value and ${\mathscr{L}}$ is a bounded linear operator in Sobolev spaces. Let $B_{1}(x):=\sigma(x)$ and $B_{j+1}(x):=b(x)\cdot\nabla{B}_{j}(x)-\nabla{b(x)}\cdot B_{j}(x)$ for $j\in\mathbb{N}$. Suppose $B_{j}\in C_{b}^{\infty}(\mathbb{R}^{d};\mathbb{R}^{d}\otimes\mathbb{R}^{d})$ for each $j\in\mathbb{N}$. Under the following uniform Hörmander’s type condition: for some $j_{0}\in\mathbb{N}$, \[\inf_{x\in\mathbb{R}^{d}}\inf_{|u|=1}\sum_{j=1}^{j_{0}}|uB_{j}(x)|^{2}>0,\] by using Bismut’s approach to the Malliavin calculus with jumps, we prove the existence of fundamental solutions to operator $\mathcal{L}^{(\alpha)}_{\sigma,b}$. In particular, we answer a question proposed by Nualart [Sankhyā A 73 (2011) 46–49] and Varadhan [Sankhyā A 73 (2011) 50–51].

Riemann boundary value problems on the positive real axis

In this paper, some Riemann boundary value problems on the positive real axis are presented. Firstly, we introduce the concepts of principle part and order at infinity and zero point for the holomorphic function on the complex plane cut along the positive real axis, which are more extensive than those in the classic sense. Then, the behaviors of Cauchy-type integral and Cauchy principal value integral on the positive real axis at infinity and zero point are discussed, respectively. Based on those, Riemann boundary value problems for sectionally holomorphic functions with the positive real axis as their jump curve are established explicitly, which are different from Riemann problems on the finite curves and more complicated than those on the whole real axis. Finally, some boundary value problems for matrix-valued functions are also constructed, which play a very important role in the asymptotic analysis for orthogonal polynomials on the positive real axis.

Robust multiscale field-only formulation of electromagnetic scattering

We present a boundary integral formulation of electromagnetic scattering by homogeneous bodies that are characterized by linear constitutive equations in the frequency domain. By working with the Cartesian components of the electric, E and magnetic, H fields and with the scalar functions (r*E) and (r*H), the problem is cast as solving a set of scalar Helmholtz equations for the field components that are coupled by the usual electromagnetic boundary conditions at material boundaries. This facilitates a direct solution for E and H rather than working with surface currents as intermediate quantities in existing methods. Consequently, our formulation is free of the well-known numerical instability that occurs in the zero frequency or long wavelength limit in traditional surface integral solutions of Maxwell's equations and our numerical results converge uniformly to the static results in the long wavelength limit. Furthermore, we use a formulation of the scalar Helmholtz equation that is expressed as classically convergent integrals and does not require the evaluation of principal value integrals or any knowledge of the solid angle. Therefore, standard quadrature and higher order surface elements can readily be used to improve numerical precision. In addition, near and far field values can be calculated with equal precision and multiscale problems in which the scatterers possess characteristic length scales that are both large and small relative to the wavelength can be easily accommodated. From this we obtain results for the scattering and transmission of electromagnetic waves at dielectric boundaries that are valid for any ratio of the local surface curvature to the wave number. This is a generalization of the familiar Fresnel formula and Snell's law, valid at planar dielectric boundaries, for the scattering and transmission of electromagnetic waves at surfaces of arbitrary curvature.

Erratum: “The Cauchy principal value and the Hadamard finite part integral as values of absolutely convergent integrals” [J. Math. Phys. 57, 033502 (2016)

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Simple Method for Evaluating Singular Integrals

In this paper, we develop a method for evaluating one dimensional singular integrals (weakly, strongly, and hyper-singular) that converge in the sense of Cauchy principal value and Hadamard finite part integrals. A proof of convergence of this method is also provided.