Cauchy Integral Research Articles

Solution the Dirichlet Problem for Multiply Connected Domain Using Numerical Conformal Mapping

We present a method for construction of continuous approximate 2D Dirichlet problems solutions in an arbitrary multiply connected domain with a smooth boundary. The method is based on integral equations solution which is reduced to a linear system solution and does not require iterations. Unlike the Fredholm’s solution of the problem ours applies not a logarithsmic potential of a double layer but the properties of Cauchy integral boundary values. We search for the solution of the integral equation in the form of Fourier polynomial with the coefficients being the solution of a linear equation system. The continuous solution of Dirichlet problem is the real part of a Cauchy integral.

Difference of Weighted Composition Operators on the Space of Cauchy Integral Transforms

In this paper, we provide a complete function theoretic characterizations for boundedness and compactness of difference of weighted composition operators from the space of Cauchy integral transforms to logarithmic weighted-type spaces. Surprisingly, an interesting feature of these characterizations is that they are free from pseudo-hyperbolic distance between $\varphi(z)$ and $\psi(z)$, which is different from the previous characterizations of difference of weighted composition operators acting between different holomorphic function spaces.

Analysis of elastic symbols with the Cauchy integral and construction of asymptotic solutions

This paper deals with the elastic wave equation $(D_t^2 - L(x, D_{x'}, D_{x_n})) u(t, x', x_n)=0$ in the half-space $x_n>0$. In the constant coefficient case, it is known that the solution is represented by using the Cauchy integral $\int_c e^{ix_n\zeta} (I-L(\xi', \zeta))^{-1} d\zeta$. In this paper this representation is extended to the variable coefficient case, and an asymptotic solution with the similar Cauchy integral is constructed. In this case, the terms $\partial_x^\alpha \int_c e^{ix_n\zeta} (I-L(x,\xi',\zeta))^{-1} d\zeta$ appear in the inductive process. These do not become lower terms necessarily, and therefore the principal part of asymptotic solution is a little different from the form in the constant coefficient case.

On the computation of hybrid modes in planar layered waveguides with multiple anisotropic conductive sheets

A reliable method is presented for the computation of the propagation constants and the corresponding hybrid modal fields in multilayered planar dielectric stacks comprising multiple anisotropic conducting sheets. The appropriate dispersion functions (DFs) are formulated and evaluated using a numerically stable scattering matrix methodology. The roots of the DFs are computed by the Cauchy integration method with automatic differentiation of the DF.

Numerically Stable and Reliable Computation of Electromagnetic Modes in Multilayered Waveguides Using the Cauchy Integration Method With Automatic Differentiation

A robust and efficient method is presented for the computation of the electromagnetic modes supported by planar multilayer waveguides that may comprise lossy, active, plasmonic, and uniaxial media, including graphene sheets. Pole-free and numerically stable dispersion functions (DFs) are developed for various shielding configurations using the $S$ -matrix formulation. The modal propagation constants are computed by the Cauchy integration method on the four-sheeted Riemann surface, using the derivative of the DF for greater reliability. Since analytical derivatives of the S-parameters are difficult to obtain, automatic differentiation is employed, implemented by operator overloading in modern Fortran. The method is validated using various benchmark problems found in the literature.

A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions

The Baker-Akhiezer (BA) function theory was successfully developed in the mid 1970s. This theory brought very interesting and important results in the spectral theory of almost periodic operators and theory of completely integrable nonlinear equations such as Korteweg-de Vries equation, nonlinear Schr\"odinger equation, sine-Gordon equation, Kadomtsev-Petviashvili equation. Subsequently the theory was reproduced for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchies. However, extensions of the Baker-Akhiezer function for the Maxwell-Bloch (MB) system or for the Karpman-Kaup equations, which contain prescribed weight functions characterizing inhomogeneous broadening of the main frequency, are unknown. The main goal of the paper is to give a such of extension associated with the Maxwell-Bloch equations. Using different Riemann-Hilbert problems posed on the complex plane with a finite number of cuts we propose such a matrix function that has unit determinant and takes an explicit form through Cauchy integrals, hyperelliptic integrals and theta functions. The matrix BA function solves the AKNS equations (the Lax pair for MB system) and generates a quasi-periodic finite-gap solution to the Maxwell-Bloch equations. The suggested function will be useful in the study of the long time asymptotic behavior of solutions of different initial-boundary value problems for the MB equations using the Deift-Zhou method of steepest descent and for an investigation of rogue waves of the Maxwell-Bloch equations.

A New Cauchy Integral Formula in the Complex Clifford Analysis

In this paper, we construct an analogue of Bochner–Martinelli kernel based on theory of functions of several complex variables in complex Clifford analysis, which has generalized complex differential forms with Clifford basis vectors. Using these complex differential forms, we obtain the Stoke’s formula of complex Clifford functions which are defined on a domain \(\Omega \subset C^{n+1}\) and take values in a complex Clifford algebra \(Cl_{0,n}(C)\). Then, we give a Stoke’s formula which has a classical form and an analogue of Cauchy–Pompeiu formula which is represented by Bochner–Martinelli kernel, and establish an analogue of Cauchy integral formula in complex Clifford analysis. It is possible to promote these results to complex manifold’s corresponding results in the Clifford analysis using the representation by generalized complex differential forms.

The Approximate Solution of 2D Dirichlet Problem in Doubly Connected Domains

We propose a new method for constructing an approximate solution of the two-dimensional Laplace equation in an arbitrary doubly connected domain with smooth boundaries for Dirichlet boundary conditions. Using the fact that the solution of the Dirichlet problem in a doubly connected domain is represented as the sum of a solution of the Schwarz problem and a logarithmic function, we reduce the solution of the Schwartz problem to the Fredholm integral equation with respect to the boundary value of the conjugate harmonic function. The solution of the integral equation in its turn is reduced to solving a linear system with respect to the Fourier coefficients of the truncated expansion of the boundary value of the conjugate harmonic function. The unknown coefficient of the logarithmic component of the solution of the Dirichlet problem is determined from the following fact. The Cauchy integral over the boundary of the domain with a density that is the boundary value of the analytical in this domain function vanishes at points outside the domain. The resulting solution of the Dirichlet problem is the sum of the real part of the Cauchy integral in the given domain and the logarithmic function. In order to avoid singularities of the Cauchy integral at points near the boundary, the solution at these points is replaced by a linear function. The resulting numerical solution is continuous in the domain up to the boundaries. Three examples of the solution of the Dirichlet problem are given: one example demonstrates the solution with constant boundary conditions in the domain with a complicated boundary; the other examples provide a comparison of the approximate solution with the known exact solution in a noncircular domain.

Efficient approximation of free-surface Green function and OpenMP parallelization in frequency-domain wave–body interactions

The evaluation of Green function without translation has always been the most time-consuming part of the hydrodynamic analysis based on potential flow theory. This research investigates the new implementations of efficient approximations of Green function and the code parallelization with OpenMP library. First, algorithms of infinite-depth case are developed with economized Chebyshev polynomials based on modified region and subdomain partitions. Second, an improved Gauss–Laguerre method for the Cauchy integral is presented to evaluate the finite-depth case with the convergence acceleration by the reduction of fraction and singularity elimination by the infinite-depth case together with the exponential integral. For the analyzed container vessel, on the premise of good numerical accuracy, we conclude that the present method has nearly obtain the same efficiency as Chen (Hydrodynamics in offshore and naval applications—Part I, Bureau Veritas, Paris, 2004) for the infinite-depth case and almost 30% higher efficiency for the finite-depth case. Moreover, we also point out that the numerical efficiency is significantly enhanced again due to the OpenMP parallelization of the serial code.

The Cauchy Integral Formula, Quadrature Domains, and Riemann Mapping Theorems

It is well known that a domain in the plane is a quadrature domain with respect to area measure if and only if the function z extends meromorphically to the double, and it is a quadrature domain with respect to boundary arc length measure if and only if the complex unit tangent vector function T(z) extends meromorphically to the double. By applying the Cauchy integral formula to $$\bar{z}$$ , we will shed light on the density of area quadrature domains among smooth domains with real analytic boundary. By extending $$\bar{z}$$ and T(z) and applying the Cauchy integral formula to the Szegő kernel, we will obtain conformal mappings to nearby arc length quadrature domains and even domains that are like the unit disc in that they are simultaneously area and arc length quadrature domains. These “double quadrature domains” can be thought of as analogs of the unit disc in the multiply connected setting and the mappings so obtained as generalized Riemann mappings. The main theorems of this paper are not new, but the methods used in their proofs are new and more constructive than previous methods. The new computational methods give rise to numerical methods for computing generalized Riemann maps to nearby quadrature domains.

Integral relations associated with the semi-infinite Hilbert transform and applications to singular integral equations

Integral relations with the Cauchy kernel on a semi-axis for the Laguerre polynomials, the confluent hypergeometric function, and the cylindrical functions are derived. A part of these formulas is obtained by exploiting some properties of the Hermite polynomials, including their Hilbert and Fourier transforms and connections to the Laguerre polynomials. The relations discovered give rise to complete systems of new orthogonal functions. Free of singular integrals, exact and approximate solutions to the characteristic and complete singular integral equations in a semi-infinite interval are proposed. Another set of the Hilbert transforms in a semi-axis are deduced from integral relations with the Cauchy kernel in a finite segment for the Jacobi polynomials and the Jacobi functions of the second kind by letting some parameters involved go to infinity. These formulas lead to integral relations for the Bessel functions. Their application to a model problem of contact mechanics is given. A new quadrature formula for the Cauchy integral in a semi-axis based on an integral relation for the Laguerre polynomials and the confluent hypergeometric function is derived and tested numerically. Bounds for the remainder are found.

The General Scheme for Calculating Series with Positive Terms

The general theory of series has a long history and originates from the time of the emergence of differential and integral calculi. The three main questions in the theory of series are: the study of convergence, the estimation of the remainder term, and the convergence improvement [2,3,6]. There is no universal solution to these issues. Therefore, the search for new methods of solving these problems remains an urgent problem for today. Moreover, the problems of improving the convergence and estimating the remainder terms of the series have an increasing role not only in problems of mathematical physics, but also in engineering practice [5].One of the possible approaches to solving these problems is the use of the method of conjugation of two series, which allows us to establish some general criteria for the convergence of series with positive terms, and to obtain corresponding estimates of the remainder terms and to suggest ways of improving convergence [6]. The correspondence between the method of improving the convergence of a series and one or another criterion of convergence is based on a single transformation of the series proposed by Kummer [1,6,9]. This transformation is little known. However, using this transformation, it is possible to obtain, as particular cases, known criteria for the convergence of series: the d'Alembert, Cauchy, Raabe, Gauss, integral Cauchy, etc. [3,5]. It is also possible to obtain new practically useful criteria [8,10]. We note that, in the framework of this transformation, the Kummer scheme itself, and the Ermakov criterion, and other indications, which follow from the theory of conjugation of series [4, 8], actually fit themselves. For each criterion of convergence, which can be obtained from the Kummer transform, it is also possible to establish corresponding estimates of the remainder term of the series. If the series converges slowly or the estimates of the remainder are rough, for each chosen characteristic, appropriate ways of improving convergence can be selected [7, 8]. In particular, the well-known transformation of power series proposed by Euler is a simple consequence of the method for improving the convergence of series corresponding to the d'Alembert test [7, 10].The article is devoted to the discussion of the Kummer transformation and its applications. The presentation is based on the elementary theory of series, and also on some facts from the theory of special functions [4].

Estimating the division rate and kernel in the fragmentation equation

We consider the fragmentation equation∂f∂t(t,x)=−B(x)f(t,x)+∫y=xy=∞k(y,x)B(y)f(t,y)dy, and address the question of estimating the fragmentation parameters – i.e. the division rate B(x) and the fragmentation kernel k(y,x) – from measurements of the size distribution f(t,⋅) at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance Xue and Radford (2013) [26] for amyloid fibril breakage. Under the assumption of a polynomial division rate B(x)=αxγ and a self-similar fragmentation kernel k(y,x)=1yk0(xy), we use the asymptotic behavior proved in Escobedo et al. (2004) [11] to obtain uniqueness of the triplet (α,γ,k0) and a representation formula for k0. To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.

Generalized Cauchy integrals on the plane

The integrals with homogeneous-difference kernels are considered on a smooth contour. The boundary properties of the integrals are described in the H?lder space. An analogue of the known Sokhotski-Plemelj formula is obtained. Moreover, the differentiation formula of these integrals is also given.

On functions of quasi-Toeplitz matrices

Let be a complex-valued function, defined for , such that . Consider the semi-infinite Toeplitz matrix associated with the symbol such that . A quasi-Toeplitz matrix associated with the symbol is a matrix of the form where , , and is called a -matrix. Given a function and a -matrix , we provide conditions under which is well defined and is a -matrix. Moreover, we introduce a parametrization of -matrices and algorithms for the computation of . We treat the case where is given in terms of power series and the case where is defined in terms of a Cauchy integral. This analysis is also applied to finite matrices which can be written as the sum of a Toeplitz matrix and a low rank correction. Bibliography: 27 titles.

Optimal quadrature formulas with derivatives for Cauchy type singular integrals

In the present paper in L2(m)(0,1) space the optimal quadrature formulas with derivatives are constructed for approximate calculation of the Cauchy type singular integral. Explicit formulas for the optimal coefficients are obtained. Some numerical results are presented.

Riemann–Hilbert problem in the class of Cauchy type integrals with densities of grand Lebesgue Spaces

The present paper deals with a solution of the Riemann–Hilbert problem with piecewise continuous coefficients in the class of Cauchy type integrals with densities in grand Lebesgue spaces. Necessary and sufficient solvability condition is established. In solvability case the solutions in explicit form are constructed.

Debonded arc-shaped interface anticracks

We analytically investigate a debonded arc-shaped anticrack lying on the interface between a circular elastic inhomogeneity and an infinite matrix when subjected to uniform remote in-plane stresses. One side of the anticrack is perfectly bonded to either the inhomogeneity or the matrix, whereas its other side has become fully debonded. Through the introduction of two sectionally holomorphic functions, the problem is reduced to a non-homogeneous Riemann–Hilbert problem of vector form that can be solved through a decoupling procedure and through evaluation of the Cauchy integrals. Solutions to both the non-degenerate case of distinct eigenvalues and the degenerate case of identical eigenvalues are derived.

On the asymptotic expansion of certain plane singular integral operators

We discuss the problem of the asymptotic expansion for some operators in a general theory of pseudo-differential equations on manifolds with borders. Using the distribution theory one obtains certain explicit representations for these operators. These limit distributions are constructed with the help of the Fourier transform, the Dirac mass-function and its derivatives, and the well-known distribution related to the Cauchy type integral.

Boundary value formula for the Cauchy integral on elliptic curve

In this paper we consider a Cauchy integral on elliptic curve $$\Gamma $$ parameterized by equation $$\eta (t)=a \cos t+ib \sin t, a,b>0$$ . We drive a formula for the boundary values of the Cauchy integral when integral function is Holder continuous on $$\Gamma $$ . Hence we extend Hilbert transform to elliptic curves.