Pair Of Integral Equations Research Articles

On solving paired integral equations by means of $p$-analytic functions

On solving paired integral equations by means of $p$-analytic functions.

Numerical simulation of electromagnetic wave diffraction on a finite number of slits in a flat screen

The article presents an algorithm for modeling diffraction on slits made on a flat screen made of actual materials. As a result, the boundary conditions of Schukin-Leontovich lead to the mixed boundary value problem. According to the method of Y.V. Gandel and V.D. Dushkin the solution of the problem can be reduced to a system of paired integral equations with singular and logarithmic singularities. The author has created and debugged a PC program that performs the numerical solution of the arising problems, and a series of computational experiments has been performed. This work has been carried at School of Mathematics and Computer Sciences of V. N. Karazin Kharkiv National University within the state budget themes: "Modeling of the dynamics of folding systems with the method of identifying problem situations".

Mechanical model for a thermoelectric thin film bonded to an elastic infinite substrate

A thermoelectric thin film bonded to an elastic infinite substrate is modeled with consideration of the bending stiffness of the film. Through the stress balance and the strain compatibility between the film and the substrate, a pair of integral equations is derived. The results show that the interfacial tensile and shear stresses exhibit singularity at the end of the thin film. For this, the concept of stress intensity factor is used to describe the singularity behavior of the film/substrate interface. The effects of the electric current density, the thermal conductivity of the film, the film to substrate stiffness ratio and the half-length to thickness ratio of the film on the stress intensity factors are examined. Comparison between beam model and membrane model for thermoelectric thin film analysis is made and the results suggest that consideration of film stiffness is essential for correct evaluation of the thermal stress level in the thermoelectric film.

Water wave scattering by a pair of thin vertical barriers with submerged gaps

The scattering of obliquely incident water waves by two thin vertical barriers with gaps at different depths has been studied assuming linear theory. Using Havelock’s expansion of water wave potential, the problem is reduced to two pairs of integral equations of the first kind, one pair involving a horizontal component of velocity across the gaps and the other pair involving the difference of potentials across each wall. These two pairs of integral equations can be solved approximately by employing a Galerkin single-term approximation technique to obtain numerical estimates for the reflection and transmission coefficients. These estimates for the reflection and transmission coefficients thus obtained are seen to satisfy the energy identity. The reflection coefficient is plotted against wave number in a number of figures for different values of various parameters involved in the problem. It is observed that the reflection coefficient vanishes at discrete frequencies when the vertical barriers are identical. For nonidentical vertical barriers the reflection coefficient never vanishes, though at some wave number it becomes close to zero. The results for a single barrier and fully submerged two barriers, and for a single barrier with a narrow gap, are also recovered as special cases.

Criterion for connections of solutions of the general convolution type paired integral equations in banach algebras with weight

Existence and connection between the solutions of the abstract general convolution type paired integral equations with arbitrary right-hand side and equal to Dirac’s delta function are considered. The theorem - criterion for such connection is formulated and proved. Procedure is a free from the theory of Cauchy integral and Hölder requirement

A note on efficient performance evaluation of the Cumulative Sum chart and the Sequential Probability Ratio Test

We establish a simple connection between certain in‐control characteristics of the Cumulative Sum (CUSUM) Run Length and their out‐of‐control counterparts. The connection is in the form of paired integral (renewal) equations. The derivation exploits Wald's likelihood ratio identity and the well‐known fact that the CUSUM chart is equivalent to repetitive application of Wald's Sequential Probability Ratio Test (SPRT). The characteristics considered include the entire Run Length distribution and all of the corresponding moments, starting from the zero‐state average run length. A particular practical benefit of our result is that it enables the in‐control and out‐of‐control characteristics of the CUSUM Run Length to be computed concurrently. Moreover, owing to the equivalence of the CUSUM chart to a sequence of SPRTs, the Average Sample Number and Operating Characteristic functions of an SPRT under the null and under the alternative can all be computed simultaneously as well. This would double up the efficiency of any numerical method that one may choose to devise to carry out the actual computations. Copyright © 2016 John Wiley & Sons, Ltd.

A strip cut in a composite elastic wedge

Integral equations of new three-dimensional problems concerning a cut in the plane of symmetry of a composite elastic wedge are obtained. The wedge consists of three wedge-shaped layers with a common apex, connected by a sliding clamp: in the middle there is a compressible layer with a cut, to the sides of which two identical incompressible layers are pressed. For the case of a strip cut emerging on the edge of the composite wedge, using the method of paired integral equations, the problem is reduced to Fredholm integral equations of the second kind. An expression is obtained for the stress intensity factor.

Two Parallel Queues with Infinite Servers and Join the Shortest Queue Discipline

We study the stationary distribution of a system of two parallel M/M/∞ queues managed by the Join the Shortest Queue load balancing policy; one motivation for characterizing the efficiency of that policy is its potential application to resource allocation issues in cloud computing. For the general system with distinct service rates, we first show that the tail of each marginal queue length distribution exhibits a much faster decay than that of a Poisson distribution. Second, the determination of the joint stationary distribution is shown to reduce to the resolution of a pair of linear integral equations. In the case when service rates are identical (“symmetric case”), that pair of integral equations simplifies to a single Fredholm integral equation of the first kind whose solution is explicitly given in terms of Legendre polynomials; this enables us to entirely determine the stationary distribution of the system. We provide, in particular, asymptotics for the second moment and the tail of the queue length distribution.

Forced oscillations of the elastic strip with a longitudinal crack

The problem of forced oscillations of an elastic strip containing a longitudinal crack of a nite length is considered. The diraction problem is reduced to the system of paired integral equations. The system of integral equations is reduced to the system of linear algebraic equations by using the Galerkin method. Singularities of integrand functions, through which coecients of the system matrix are calculated, are determined.

Functional of the Diffusion Path of a Two-State Markov-Chain Model for Option Pricing

This paper studies the functional of the path of a diffusion in which volatility switches between two states: high and low. For this two-state Markov-chain model, we derive a closed-form expression for the distribution function for the time spent in the high volatility state by guessing the form of, and calculating the coefficients in, the solution to a pair of integral equations.

Influence of initial stresses on the stressed state of a composite with a periodic system of parallel coaxial normal tensile cracks

Within the framework of the linearized mechanics of deformable solids, we have investigated the axisymmetric problem of fracture of a prestressed composite material with a periodic system of parallel coaxial normal tensile cracks. Using the representations of the general solutions of linearized equilibrium equations via the harmonic potential functions and the technique of Hankel integral transformations, we have reduced the problem to a system of paired integral equations and then to a resolving second-kind Fredholm integral equation. We have obtained expressions for the stress intensity factors near the crack contours and for the case of a layered composite with isotropic layers. We have also analyzed the dependence of stress intensity factors on the initial stresses, the mechanical characteristics of components of the material, and the geometric parameters of the problem.

The radio-transmitting properties of a carbon nanotube vibrator located on the boundary of a dielectric

The quantum and mechanical properties of a carbon nanotube are described in a model with the use of a macroscopic parameter, viz., surface impedance. Solving the boundary problem for the impedance vibrator is reduced to solving paired integral equations for the vibrator current. The paired integral equations are solved by the Galerkin method with Chebyshev’s basis. The influence of the substrate on the amplitude and frequency characteristics of the antenna, a carbon nanotube, is investigated.

Axial compression of circular cylindrical bar with coaxial subsurface cylindrical crack of finite length

A fracture stability of a circular cylindrical bar with a coaxial surface cylindrical crack subjected to an axial compression is considered. A state of subcritical initial strain is assumed. A non-classical fracture criterion is based on a local stability loss near the defect. The theory of integral Fourier transforms and series expansions are used to reduce these problems to a system of paired integral equations and then to a system of linear algebraic equations with respect to the contraction parameter.

Problems of cuts in a composite elastic wedge

Problems of strip and elliptical cuts (tensile cracks) in the middle of a three-layer elastic wedge are investigated in a three-dimensional formulation. Free or rigid clamping conditions or the stress-free condition are stipulated on the outer surfaces of the composite wedge. The problems are assumed to be symmetrical about the plane of the cut. The wedge-shaped layer containing the cut is incompressible and hinged along both faces with two other layers. The integral equations of the problems with respect to the opening of the cut are derived. Inverse operators are obtained for the operators occurring in the kernels of these equations. The relation between problems on cuts and the corresponding contact problems for a composite wedge of half the aperture angle is used. The method of paired integral equations is used for the case of a strip cut emerging from the edge of the wedge. The problems are reduced to Fredholm integral equations of the second kind in certain auxiliary functions, in terms of the values of which the normal stress intensity factors are expressed. A regular asymptotic solution is constructed for the case of an elliptic cut.

Layered soft ferromagnetic materials with moving interfacial crack in uniform magnetic field

Within the framework of linear magnetoelasticity, the plane strain problem of a layered medium composed of three dissimilar soft ferromagnetic materials with a Yoffe-type moving crack at the interface is considered. The medium is subjected to a uniform in-plane magnetic field and the crack is opened by internal normal and shearing tractions. By using the Fourier integral transform technique, the current mixed boundary problem is reduced to a pair of integral equations of the second kind with Cauchy-type singularities. To determine the dynamic magnetoelastic stress intensity factors, the singular integral equations are solved numerically by means of the Jacobi polynomial expansion method. Results are presented graphically to show the dependence of the DSIFs on the velocity of the moving crack, the magnetic field, the magnetic properties of the materials and the geometric parameter of the problem.

Moving interfacial crack between two dissimilar soft ferromagnetic materials in uniform magnetic field

This paper presents the dynamic magnetoelastic stress intensity factors of a Yoffe-type moving crack at the interface between two dissimilar soft ferromagnetic elastic half-planes. The solids are subjected to a uniform in-plane magnetic field and the crack is opened by internal normal and shear tractions. The problem is considered within the framework of linear magnetoelasticity. By application of the Fourier integral transform, the mixed boundary problem is reduced to a pair of integral equations of the second kind with Cauchy-type singularities. The singular integral equations are solved by means of a Jacobi polynomial expansion method. For a particular case, closed-form solutions are obtained. It is shown that the magnetoelastic stress intensity factors depend on the moving velocity of the crack, the magnetic field and the magnetoelastic properties of the materials.

Non-local theory solution of two collinear mode-I cracks in piezoelectric materials

In this paper, the basic solution of two collinear cracks in a piezoelectric material plane subjected to a uniform tension loading is investigated by means of the non-local theory. Through the Fourier transform, the problem is solved with the help of two pairs of integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the integral equations, the jumps of displacements across the crack surfaces are directly expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the interaction of two cracks, the materials constants and the lattice parameter on the stress field and the electric displacement field near crack tips. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularities are present at crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to using the maximum stress as a fracture criterion in piezoelectric materials.

Integral formulation for a circular cylindrical cavity in infinite solid and a finite length coaxial cylindrical crack compressed axially

A method for solving problems of fracture of an infinite solid with a circular cylindrical cavity and a coaxial cylindrical crack near the surface under an uniform axial compression is proposed using a non-classical criterial approach associated with a mechanism of a local stability loss near the defect. The theory of integral Fourier transforms and series expansions are used to reduce these problems to a system of paired integral equations and then to a system of linear algebraic equations with respect to the contraction parameter.

Paired Sum Equations and Singular Integral Equations in Electromagnetic Scattering by Open Conical Structures

The use of the Kontorovich–Lebedev integral transform together with the semi-inversion method is an effective approach to the solution of electrodynamic boundary value problems with open conical geometry. It permits one to construct a numerical-analytic algorithm for electromagnetic scattering by a cone with periodically arranged longitudinal slits (one ribbon per period) and to obtain a closed-form solution [1, 2]. For many-element structures (several ribbons of different widths per period), the problem is more complicated, and this approach encounters serious difficulties. The aim of the present paper is to construct a new method for solving boundary value problems for oneand many-element open conic structures on the basis of the technique of singular integral equations [3, 4].

Analysis on the cohesive stress at half infinite crack tip

The nonlinear fracture behavior of quasi-brittle materials is closely related with the cohesive force distribution of fracture process zone at crack tip. Based on fracture character of quasi-brittle materials, a mechanical analysis model of half infinite crack with cohesive stress is presented. A pair of integral equations is established according to the superposition principle of crack opening displacement in solids, and the fictitious adhesive stress is unknown function. The properties of integral equations are analyzed, and the series function expression of cohesive stress is certified. By means of the data of actual crack opening displacement, two approaches to gain the cohesive stress distribution are proposed through resolving algebra equation. They are the integral transformation method for continuous displacement of actual crack opening, and the least square method for the discrete data of crack opening displacement. The calculation examples of two approaches and associated discussions are given.