Infinite System Of Algebraic Equations Research Articles

On the Solvability of an Infinite System of Algebraic Equations with Monotone and Concave Nonlinearity

On the Solvability of an Infinite System of Algebraic Equations with Monotone and Concave Nonlinearity

THE SOLVABILITY OF AN INFINITE SYSTEM OF NONLINEAR ALGEBRAIC EQUATIONS WITH TOEPLITZ MATRIX

The work is focused on studying of the existence, uniqueness, and various qualitative properties of the constructive solution of an infinite system of algebraic equations with a concave nonlinearity property, which are generated by Toeplitz matrices. In addition to its independent mathematical interest, such systems have a significant application in several branches of mathematical physics and mathematical biology. Those particularly appear in discrete problems within radiative transfer theory, kinetic theory of gases, dynamic theory of p-adic strings, and the mathematical theory of epidemic propagation. We establish the existence of a positive solution for the system in the class of bounded sequences, as well as provide an iterative method to approximate to the solution. We also study the asymptotic behavior of the solution at infinity and the uniqueness of the nontrivial solution with non-negative elements in the class of bounded sequences. The last section of the paper provides examples of applications of the corresponding Toeplitz matrix and the function that describes the nonlinearity.

Elastic-plastic problem for a perforated plate under transversal shear

The problem of solving the problem of the transverse shear of a plate along the edges of the holes and weakened by a two-periodic system of rectilinear through cracks with plastic end zones, collinear to the abscissa and ordinate axes of unequal length, is considered. General representations of solutions are constructed that describe a class of problems with a doubly periodic distribution of moments outside circular holes and straight-line cracks with end zones of plastic deformations. Satisfying the boundary conditions, the solution of the problem of the plate shear theory is reduced to two infinite systems of algebraic equations and two singular integral equations. Then each singular integral equation is reduced to a finite system of linear algebraic equations.

EXACT SOLUTION OF CONTACT PROBLEMS IN A FINITE-WIDTH BAND ON A MULTILAYER MEDIUM

In this paper, for the first time, an exact solution of contact problems for rigid or deformable stamps in a strip of finite width is obtained. It is assumed that the strip is located on a multilayer base of finite thickness. The universal modeling method previously developed by the authors is used. With its help, the solutions of complex boundary value problems for systems of partial differential equations are reduced, using the Galerkin transform, to solving individual differential equations, among which the Helmholtz equations are the simplest. In an earlier work of the authors published, when studying the problem for a deformable stamp in a strip, we had to limit ourselves to an asymptotic solution that is valid only for strips of large relative width. The solution for a strip of any finite size was constrained by the impossibility of constructing an exact solution to the contact problem for a rigid stamp in a strip of any finite width. As a result of the exact solution of the Wiener-Hopf integral equation in a finite-width band for the case of a multilayer medium, this contact problem was solved. The approach applied to the solution consists in constructing an exact operator equation of an infinite system of algebraic equations for large-width bands, using the operator formula of functions from matrices and investigating the constructed solution in the range of small relative bandwidth. The solution obtained in this way coincides with the solution obtained another method, namely, the singular integral method for the case of a small relative bandwidth. The constructed solution brings closer to the problems of research for materials of complex rheologies of contact problems with a deformable stamp, the description of cracks of a new type in limited bodies, modeling of nano particles, the study of tectonic plates of limited dimensions.

Теоремы существования и единственности для одной бесконечной системы нелинейных алгебраических уравнений

We study a infinite system of algebraic equations with monotone nonlinearities and with an infinite Toeplitz type matrix. This system has applications in discrete problems of the dynamical theory of 𝑝-adic open-closed strings, the kinetic theory of gases, and mathematical biology. Under certain restrictions on the nonlinearities and on the corresponding Toeplitz matrix, it is possible to prove existence and uniqueness theorems for a nontrivial solution in the class of bounded sequences. The main tool for proving the uniqueness theorem for a nontrivial solution is an auxiliary independent theorem on the asymptotic behavior of a nonnegative nontrivial and bounded solution on ±∞. At the end of the paper, specific applied examples of nonlinearities and the corresponding matrix are given to illustrate the importance of the results obtained.

Solution of a mixed nonaxisymmetric problem of the theory of elasticity for anisotropic bodies of revolution

The paper developed a technique for solving mixed nonaxisymmetric problems of the theory of elasticity for bounded bodies of revolution made of a transversely isotropic material under the action of surface forces specified according to a cyclic law. The technique involves the development of the energy method of boundary states, which is based on the concepts of spaces of internal and boundary states, conjugated by isomorphism, which makes it possible to establish a one-to-one correspondence between the elements of these spaces. The internal state includes the components of the tensor of stresses, deformations, and the displacement vector. The boundary state includes efforts and displacements at the boundary of the body. The isomorphism of the state spaces is proved, which allows finding the internal state to be reduced to the study of the boundary state isomorphic to it. The basis is formed on the basis of the general solution of the boundary value problem of elastostatics for a transversely isotropic body of revolution. Orthogonalization of state spaces is carried out, where the internal energy of elastic deformation is used as scalar products in the space of internal states; in the space of boundary states, the work of external forces is used. Finally, finding the desired state is reduced to solving an infinite system of algebraic equations for the Fourier coefficients. The solution of the problem with mixed boundary conditions for a circular in plan cylinder of transversely isotropic coarse dark gray siltstone with anisotropy axis coinciding with the geometric axis of symmetry is presented. The solution is analytical and the characteristics of the stress-strain state have a polynomial form. Explicit and indirect signs of convergence of problem solutions and graphical visualization of the results are presented.

Axisymmetric torsion problem by a rigid disc of an elastic half-space weakened by an annular crack

This investigation is devoted to an axisymmetric torsion of a circular disc subjected to a half-space, containing an annular crack. The stress–strain state and the stress intensity factors pertaining to the inner and outer crack edges are investigated. The proposed problem is solved by applying the Hankel integral transformation method. The mixed boundary-value problem is reduced to a system of coupled dual and triple integral equations. We developed a new method to solve this system. By using the Gegenbauer formulas, we obtain a system of infinite algebraic equations for getting the unknown functions. The solution is analysed and discussed. Numerical computations are carried out and the results are presented in the form of tables and graphs. Some special cases are considered.

An Analytical Approach to Obtaining the Stress Field of a Cylindrical Shell with a Circular Hole under Tension

The problem of a cylindrical shell with a circular hole under uniaxial tension is considered. The main obstacle of solving this problem is the necessity to find such coefficients in the expansion of the solution into a sum of basis functions, for which this solution satisfies the boundary conditions. The study of the classical works led to understanding that none of the so far proposed approaches can be considered successfully, and the results of these approaches differ, so it is not clear, which results can be used as a basis. In the present paper, a new analytical approach to studying this issue is proposed. It allows expanding the range of applicability of the solution and gives the opportunity for the analytical study of the stress state. The idea consists in expanding each of the basis functions in a Fourier series by dividing the variables, which allows obtaining explicitly an infinite system of algebraic equations for finding coefficients. One of the important steps of this research is that the authors were able to prove which exact equation is a linear combination of the others and exclude, which made it possible to compose a reduced system for finding unknown coefficients. The proposed approach, in contrast to most classical works, does not impose mathematical restrictions on the values of the main parameter characterizing the cylindrical shell. The existing restrictions are of mechanical nature, as larger cutouts require another model. Moreover, the numerical results obtained by the new method are presented in a fairly complete manner and they are compared with the results of the classical works.

ELASTOPLASTIC PROBLEM FOR PERFORATED PLATE IN TRANSVERSE SHEAR

A solution is given to the problem of transverse shear of a thin plate clamped along the edges of the holes and weakened by a doubly periodic system of rectilinear through cracks with plastic end zones collinear to the abscissa and ordinate axes of unequal length. General representations of solutions are constructed that describe the class of problems with a doubly periodic stress distribution outside circular holes and rectilinear cracks with end zones of plastic deformations. Satisfying the boundary conditions, the solution of the problem of the theory of shear plates is reduced to two infinite systems of algebraic equations and two singular integral equations. Then each singular integral equation is reduced to a finite system of linear algebraic equations. Keywords: perforated thin plate, straight cracks with end zones, transverse bending, plastic deformation zones.

Acoustic field effect on encapsulated spherical drop located in liquid filled cylindrical vessel

The acoustical problem dealing with the interaction of a plane wave propagating along the liquid-filled cylindrical cavity with an elastic spherical shell filled, in turn, with the compressible liquid is solved. An axisymmetric case is studied. The boundary problem for simultaneous solution of the wave equations describing the motion in the fluid filling the vessel as well as in the fluid contained within the shell together with equations of motion of the thin elastic shell is formulated. The problem statement contains the boundary conditions for both fluid–shell interfaces, the condition at the vessel surface and far-field radiation conditions. The incident wave propagating along the vessel axis is expanded into a normal mode series. In order to meet all boundary conditions, the variable separation method along with the spherical wave function representations in terms of the cylindrical ones and vice versa are used. This approach yields exact solution of the boundary problem and enables one to analyze the scattered fields in the low and medium frequency ranges. The problem is reduced to an infinite system of algebraic equations that is solved by the truncation method. The goal of the study is to determine the interaction features between the shell and the vessel surfaces. Obtained numerical results describe scattered field of pressure and velocity in the neighborhood of the shell as functions of frequency, the internal liquid characteristics, and the elastic characteristics of the shell. It is ascertained, that peculiarities of the resonance behavior of the liquid-filled shell can be essentially amplified by the acoustical interaction with the vessel surface.

Estimating the influence of double-sided rounded screens on the acoustic field around a linear sound source

This paper reports a study into the acoustic field of transport flow around noise protection screens located on both sides of the sound source. Most research on noise protection involving noise protection screens relates to the assessment of the effectiveness of screens located on one side of the noise source. The influence of the second screen on the effectiveness of the first one has been investigated only experimentally. Therefore, it is a relevant task to assess the mutual impact of the two screens between which the linear sound source is located. A problem was stated in such a way that has made it possible to derive an analytical solution and find a sound field around a linear sound source. In this case, the sound source was limited on both sides by acoustically rigid screens with finite thickness. The screens' cross-sections were shaped as part of a ring with arbitrary angles and the same radius. The problem was solved by the method of partial domains. This method has made it possible to obtain an infinite system of algebraic equations that were solved by the method of reduction. Such an approach to solving a problem allows a given solution to be applied for different cases of the mutual location of screens, source, and territory protected from noise. The study results help estimate a field between the screens, the dependence of increasing sound pressure on the road on the geometric size of the screen and the width of the road. In addition, the solution resulted in the ability to assess the impact of one screen on the efficiency of another in the frequency range of up to 1,000 Hz. It has been shown that the mutual impact of screens could reduce the screen efficiency by 2 times. The study reported here could make it possible to more accurately calculate the levels of the sound field from traffic flows when using noise protection screens, which is often performed in practice when designing new and reconstructing existing highways.

Revealing the effect of rounded noise protection screens with finite sound insulation on an acoustic field around linear sound sources

This paper reports studying the reduction of traffic noise by rounded noise protection screens with finite sound insulation, that is, those that can pass sound. Almost all models of acoustic screens, which are examined by analytical methods, are either direct or such that disregard the passage of sound through the screen, that is, it is assumed that the screen sound insulation is non-finite. This approach made it possible to solve the problem for a simplified model analytically but made it impossible to analyze the required sound insulation of noise protection screens. In the current paper, the problem of investigating an acoustic field around the screen whose sound insulation is finite has been stated, that is, it was taken into consideration that a sound wave propagates through the body of the screen. In addition, a given problem considers a rounded screen, rather than vertical, which is also used in different countries. Such a problem was solved by the method of partial domains. This method has made it possible to strictly analytically build a solution to the problem by simplifying it to solving an infinite system of algebraic equations, which was solved by the method of reduction. The screen model was set by the values of the density and speed of sound in the screen material. This approach has made it possible to change the acoustic impedance of the screen material and thereby change the sound insulation of the screen. That has made it possible to quantify the effect of screen sound insulation on its effectiveness. It has been shown that the efficiency of noise protection screens with finite sound insulation is approaching the efficiency of acoustically rigid screens, provided that the screen's natural sound insulation is 13–15 dB greater than the estimated efficiency of the rigid screen. The study results could make it possible to more accurately assess the effectiveness of noise protection screens. Determining the screen acoustic efficiency would make it possible to set requirements for its sound insulation characteristics. That could make it possible to select the designs of noise protection screens with minimal physical parameters, such as thickness, weight, etc.

Determination of acoustic radiation acting on a spherical rigid particle placed in an elastic tube filled with liquid

Acoustic radiation force effect upon a rigid spherical particle placed in the thin elastic tube is studied. The problem of determination of the acoustic radiation forces acting on an obstacle in an ideal liquid is formulated with respect to the Lagrange coordinate system. Thus, the radiation pressure is defined as time-averaged value of the acoustic pressure over the obstacle surface. This approach is adequate if, at determining of the acoustic pressure in a fluid, the deviation of the pressure from the harmonic law in time domain is taken into account in the obstacle vicinity. An action of the acoustic radiation force on the rigid spherical particle placed in the thin tube with elastic wall is studied here for the case of the incident plane sound wave propagating along the tube axis. Model is developed to describe the response of the system consisting of the compliant infinite thin circular cylindrical tube filled with the ideal compressible liquid and rigid spherical body which is immovable and located on the tube axis under the plane wave propagating along the tube axis. The problem of the hydrodynamic characteristics determination is reduced to the solution of the infinite system of algebraic equations that can be solved by the reduction method. The formula for the acoustic radiation force calculation is derived to characterize the force acting upon rigid spherical particle in the thin compliant elastic cylindrical tube.

МАТЕМАТИЧНЕ МОДЕЛЮВАННЯ КОНТАКТНОЇ ВЗАЄМОДІЇ ПОПЕРЕДНЬО НАПРУЖЕНОГО КІЛЬЦЕВОГО ШТАМПА ТА ПРУЖНОГО ПІВПРОСТОРУ З ПОЧАТКОВИМИ НАПРУЖЕНЯМИ

Стаття присвячена математичному моделюванню контактної взаємодії попередньо напруженого кільцевого штампа та пружного півпростору з початковими (залишковими) напруженнями. Представлено розв’язок для кільцевого пружного штампа, що враховує вплив початкових напружень. Задачу розв’язано у випадку рівних коренів визначального рівняння. Дослідження представлено в загальному вигляді для теорії великих початкових деформацій і двох варіантів теорії малих початкових деформацій у рамках лінеаризованої теорії пружності при довільній структурі пружного потенціалу. Припускається, що початкові стани пружного кільцевого штампа та пружного півпростору однорідні та рівні. Дослідження проводиться в координатах початкового деформованого стану, які пов'язані з лагранжевими координатами (природного стану). Крім того, вплив кільцевого штампа викликає невеликі збурення відповідних величин основного напружено-деформованого стану. Також передбачається, що пружний кільцевий штамп та пружний півпростір виготовлені з різних ізотропних, трансверсально-ізотропних або композиційних матеріалів. Наведені загальні розв’язки основних диференціальних рівнянь лінеаризованої теорії пружності у випадку осесиметричної деформації для скінченної кільцевої області. У результаті, розв’язки поставленої задачі представлені у вигляді нескінченних рядів, коефіцієнти яких визначаються з нескінченної системи алгебраїчних рівнянь. Проведено дослідження впливу початкових (залишкових) напружень у півпросторі та кільцевому штампі на розподіл контактних характеристик в області контакту. У випадку рівних коренів та потенціалу Бартенєва-Хазановича наведено результати чисельного аналізу, що подані у вигляді графіків, які ілюструють достатньо значний вплив початкових напружень. Отже, вплив початкових напружень на напружено-деформований стан пружного кільцевого штампа, що втискається у пружний півпростір, полягає у тому, що: у випадку стиснення початкові напруження в півпросторі призводять до зменшення напружень у пружному штампі, а у випадку розтягу – до їх збільшення, а для переміщень – навпаки.

Contact problems of multilayer slabs interaction on an elastic foundation

A mathematical model and a method to assess the internal force factors in multilayer strip – slab on elastic foundation under various static loads were developed in the paper. A detailed review of well-known publications on assessing the stress-strain state and dynamic behavior of various structures interacting with the base is given. A closed system of integro-differential equations describing the strain process in multilayer slabs on an elastic foundation was derived. The problem under consideration was reduced (using the Chebyshev polynomial) to the solution of infinite systems of algebraic equations. The regularities of infinite systems of algebraic equations were proved and the corresponding estimates were obtained. A number of terms of the Chebyshev polynomial for obtaining a solution to the problem with the required accuracy was determined. The efficiency of the method was shown on the example of test problems.

Plane contact problem on the interaction of a pre-stressed strip with an infinite inhomogeneous stringer

The article is devoted to the study of problems of contact interaction of an infinite elastic inhomogeneous stringer with a prestressed strip clamped along one edge. As a result of the research, we have obtained a resolving system of recurrent systems of integro-differential equations. In general, the studies were carried out for the theory of large initial and various versions of the theory of small initial deformations within the framework of the linearized theory of elasticity with an elastic potential of an arbitrary structure. Integral integer differential equations are obtained using the integral Fourier transform. Their solution is presented in the form of quasiregular infinite systems of algebraic equations. The article also investigates the influence of the initial (residual) stresses in strips on the distribution law of contact stresses along the line of contact with an infinite stringer. The system is solved in a closed form using the Fourier transform. The stress expressions are represented by Fourier integrals with a fairly simple structure. The influence of the initial stress on the distribution of contact stresses has been studied and mechanical effects have been found under the action of concentrated loads.

Partial Opening of a Semiinfinite Crack on the Boundary of an Elastic Strip and a Rigid Wall

We study the problems of compression and shear of an elastic strip containing a semiinfinite crack on the boundary of the strip and a rigid wall. The crack is opened in the middle and closed at the ends, where its faces are under the conditions of friction sliding. By the Wiener–Hopf method, we reduce the system of integral equations of the problem to an infinite system of algebraic equations. The stress intensity factor and the crack opening displacements are determined.

Abnormal Frequencies in a Semi-Infinite Cylindrical Vessel Filled with a Fluid and Dynamically Excited by a Spherical Oscillator

A semi-infinite circular cylindrical cavity filled with a compressible ideal fluid and containing a spherical body near its end is considered. The body surface radiates periodic pressure at given frequency and amplitude. The hydrodynamic characteristics of the system depending on the frequency of excitation and geometrical parameters are determined. The method of separation of variables, the translational addition theorems for spherical wave functions, and the expressions of spherical wave functions in terms of cylindrical ones are applied. This approach allows satisfying all the boundary conditions and finding the exact solution of the boundary-value problem. An infinite system of algebraic equations is solved. Its solution found by the truncation method is stated to converge. The determination of the pressure and velocity fields shows that the system has a number of excitation frequencies at which the acoustic characteristics exceed the amplitude of excitation by several orders of magnitude. These abnormal frequencies differ from the frequencies typical for an infinite cylindrical cavity with a spherical body. Even if the radius of the spherical oscillator is small and the abnormal phenomena in the infinite vessel are weak, they can appear strong in a semi-infinite vessel.

A flat contact problem the interaction two prestressed stripes with an infinite stringer

The article is devoted to the research of problems of contact interaction of infinite elastic stringer with two identical clamped along one edge of pre-stressed strips. In general, the research was carried out for the theory of great initial and different variants of the theory of small initial deformations within the framework of linearized theory of elasticity with the elastic potential having arbitrary structure. The integral integer-differential equations are obtained using the integral Fourier transform. Their solution is represented in the form of quasiregular infinite systems of algebraic equations. In the article alsaw was investigated the influence of the initial (residual) stresses in strips on the law of distribution of contact stresses along the line of contact with an infinite stringer. The system is solved in a closed forms using transformation of Fourier. Expressions of stresses are represented by Fourier integrals with a simple enough structure. Influence of initial stress on the distribution of contact stresses is study and discovered the mechanical effects under the influence of concentrated loads

Aeroelastic stability of a cylindrical shell of linearly varying thickness

For the first time, the aeroelastic stability equations of a composite cylindrical shell of linearly varying thickness are obtained on the basis of the bending theory of orthotropic shells for a shell subjected to axial forces and supersonic gas flow. The solution of the equations is assumed of the form of a trigonometric series in the axial coordinate. The problem is reduced to an infinite system of algebraic equations by the Bubnov-Galerkin method. The obtained characteristic equation is approximated by the Lagrange polynomial, whose stability is investigated with the use of the Routh-Hurwitz criterion. As a numerical example, the effect of the thickness gradient, structural damping and axial force on the critical velocity for a composite shell of linearly varying thickness in supersonic gas flow is shown. The refinement in the value of the critical velocity resulting from the use of the suggested model is about 35% as compared to the results for a shell of averaged constant thickness. This indicates the relevance of this model for aircraft weight optimization. The suggested approach expands the range of problems to be solved and allows for the analysis of the aeroelastic stability for orthotropic cylindrical shells of linearly varying thickness in supersonic gas flow.