Infinite System Of Algebraic Equations Research Articles

Self-adjointness and limit pointness for adjacency operators on a tree

Equivalence of self-adjointness and limit pointness for symmetric adjacency operators on a tree is proved. It is shown that the corresponding Green functions are completely characterized by a certain infinite system of algebraic equations.

Fourier Series Solutions to Poisson's Equation in Rectangularly Decomposable Regions

Fourier senes solutions to Poisson problems defined in rectangularly decomposable regions are found by matching solutions in a weak sense across the interfaces dividing subregions. The matching process produces an infinite system of algebraic equations for the unknown expansion coefficients. Complex analysis is used to investigate theoretically the conditioning of the coefficient matrix for both the C0 and C1 matching problems. The former is shown to be not well-defined whereas the latter is extremely well-conditioned. Furthermore, the analysis of the coefficient matrix for the weak C1 problem suggests a rapidly convergent iterative method for solving the truncated algebraic system for the expansion coefficients. Accurate approximations are obtained after a handful of iterations and with few degrees of freedom.

On the interaction of cylindrical and spherical elastic shells during vibrations in an incompressible fluid

Extensively studied at this time are the vibrations of individual cylindrical or spherical shells in an incompressible fluid. Vibrations of systems of parallel cylindrical shells or systems of spherical shells (bodies) are examined in [3, 6, 9]. Considered in this paper is the problem of the forced vibrations of cylindrical and spherical shells arranged in an unlimited volume of incompressible fluid so that it iS impossible to neglect their mutual influence. The solution of the problem relies on the possibility or reexpansion of the solution of the Laplace equation written in cylindrical coordinates into a system of spherical volume harmonic functions and vice-versa. Consequently, an infinite system of algebraic equations is obtained in the coefficients of the desired solution.

Calculation of nonlinear magnetization components in ferrites

In this paper ferrites are treated in a phenomenological description. The motion of the magnetization vector is calculated under the influence of a strong magnetic field H O and an additional microwave magnetic field h (t). The magnetic field h (t) and the magnetization m (t) are represented by their Fourier components. From the nonlinear equation of motion due to Landau and Lifshitz, an infinite system of algebraic equations is derived. Only tensor notation is used: the vectors need not to be split up into their spatial components. This leads to clear and comprehensive solutions which will be given for some cases.

Scattering of sound waves by a periodic array of slotted waveguides

The problem of scattering of plane acoustic waves obliquely incident upon a barrier consisting of superposed rectangular slotted waveguides is treated by taking a waveguide approach. The wave equation is solved for each interior region, bounded by parallel walls, and for the half-spaces on both sides of the scattering arrangement. The use of boundary conditions then leads to two infinite systems of algebraic equations, the solution of which generates the complex transmission and reflection coefficients. The scattering effects of comblike structures show sharp variations over the frequency spectrum. They also vary strongly at certain particular frequencies with the angle of incidence of the incoming wave. Two geometries of slotted waveguides are studied and their behavior, regarding the amplitude and phase of the transmitted sound, is compared with that of a simple grating of rectangular bars having similar dimensions. Theoretical results compare favorably with experimental values which are presented in another article forming the second part of this study.

Low-frequency scattering pattern of an arbitrary penetrable obstacle

We consider the scattering of low-frequency sound waves by an arbitrary penetrable obstacle whose density and compressibility are different from those of the surrounding infinite medium. We present solutions both inside and outside the scatterer. The physical properties of the scatterer are implicit in the transition conditions at its surface and in the interior value of the wave length. This leads to a two-parameter boundary value problem which we convert into an integral equation valid both inside and outside the obstacle. This integral equation is then recast into an infinite system of algebraic equations which are judicially truncated to yield the required solutions.

Stress analysis of multilayered plates around circular holes

In this paper, a technique to study the 3-dimensional stress state around a circular hole in laminated plates is developed. First, the 3-dimensional elasticity problem for a thick plate with a circular hole is formulated in a systematic fashion by using the z-component of the Galerkin vector and that of Muki's harmonic vector function. This problem was originally solved by Alblas[1]. The reasons for reconsidering it are to introduce a technique which may be used in solving the elasticity problem for a multilayered plate and to verify and extend the results given by Alblas. Among the additional results of particular interest, one may mention the significant effect of the Poisson's ratio on the behavior and the magnitude of the stresses. Secondly, the elasticity problem for a laminated thick plate, which consists of two bonded dissimilar layers and which contains a circular hole, is considered. The problem is formulated for arbitrary axisymmetric tractions on the hole surface. Through the expansion of the boundary conditions into Fourier series, the problem is reduced to an infinite system of algebraic equations which is solved by the method of reduction. Of particular interest in the problem are the stresses along the interface as they relate to the question of delamination failure of the composite plate. These stresses are calculated and are observed to become unbounded at the hole boundary. An approximate treatment of the singular behavior of the stress state is presented, and the stress intensity factors are calculated. It is also observed that, the results compare rather well with those obtained from the finite element method.

An axisymmetric Neumann potential problem for the circular annulus

In this note we consider an axisymmetric three-part mixed boundary value problem wherein a Neumann condition is prescribed over a plane circular annulus and over the remainder of the plane of the annulus Dirichlet conditions are imposed. The problem is reduced to a problem of solving an infinite system of algebraic equations. Expressions are derived for the quantities of interest and some numerical results are given.

An asymptotic approach to the problems of the theory of elasticity of bodies of finite dimensions

A method is developed to solve the equilibrium problems of elastic bodies of fixed dimensions, based on a separation of the boundary-layer part of the solution by considering the problem for a half-strip. A closed solution in quadratures is constructed for the half-strip with a free lateral face and with given normal displaced longitudinal boundaries, using both symmetric and antisymmetric loading. When the normal stresses on the front boundaries are specified, the problem reduces to an integral equation of the first kind in a semi-infinite interval, the inversion of which is obtained by reduction to an infinite system of algebraic equations. The approach considered for problems of bodies of finite dimensions is asymptotic with respect to the small parameter characterizing the body's thickness. Testing of the method on a plane problem for an elastic rectangle enables the range of variation of this parameter to be investigated, in which this procedure is fairly accurate. In the example considered, for the case of a rectangular area, the stresses are found and compared with the results obtained earlier by other methods. The nature of the influence of the boundary layer on the stress distribution inside the body is investiaged. Asymptotic methods, used for bodies of slab configuration, one of whose characteristic dimensions (thickness) is significantly less than the other two /1–4/, can obviously be classified into three types. The first of them /5,6/ is characterized by the application of joined asymptotic expansions to a certain class of solutions of the equations of the theory of elasticity, namely uniform solutions. The second type /7,8/ is distinguished by an asymptotic analysis of the equations of the theory of elasticity. From this it is clear that, to separate the boundary-layer part of the solution, it is enough, to a first approximation, to examine the two-dimensional problem and the problem of torsion for the half-strip, the lateral face of which is combined with the generating lateral surface of the plate at a given point. Finally, the third class includes the Vekua-Poniatovskii theory /9,10/, in which asymptotic methods are also developed /11/. In this sense this paper relates to the second of these methods.

On diffraction of SH-waves by cuts in nonhomogeneous solids

The diffraction of SH-waves by an infinite periodic system of cuts in an infinite medium possessing nonhomogeneity has been studied. Assuming that shear modulus and density vary, the problem of diffraction of SH-waves by the periodic system of cuts depends on the solution of dual series equations which ultimately reduces to the solution of an infinite system of algebraic equations.

The Linear Magnetoelastic Problem of Two Coplanar Griffith Cracks in a Soft Ferromagnetic Elastic Strip

Following a linear theory for soft ferromagnetic elastic solids, we consider the problem of determining the stress-intensity factors in an infinite strip of a soft ferromagnetic elastic material containing two coplanar Griffith cracks. We assume that the solid is a homogeneous and isotropic one and it is permeated by a uniform magnetostatic field normal to the cracks surfaces and that the cracks are opened by a constant internal pressure. By the use of Fourier transforms we reduce the problem to that of solving two simultaneous triple integral equations. These equations are reduced to a singular integral equation of the first kind. By expanding the solution into the form of the product of the series of Chebyshev polynomials of the first kind and their weight function, the singular integral equation is further reduced to the infinite system of algebraic equations for the determination of the unknown coefficients. Numerical calculations are carried out and the influence of the magnetic fields on the stress-intensity factors is graphically shown in detail.

An Asymmetric Mixed Boundary-value Problem of the Elastic Halfspace under Shear by an Annular Rigid Punch

The problem considered is equivalent to a mixed boundary value problem in the theory of elasticity where the displacement of the contact area is assumed to be translation of an annular rigid punch in a direction parallel to the surface of an elastic half-space. The solution is obtained by expressing the two stress components of the contact area as appropriate Fourier series and then reducing the problem to consideration of an infinite system of algebraic equations. Numerical results are presented for the distributions of displacements and stresses on the plane z=0.

The Effect of Shear on the Infinite Elastic Solid with an Annular Crack

Solution is presented, with in the classical theory of elasticity, for a class of asymmetric problems of an annular crack in an infinite elastic solid under shear. For the case when the components of the displacement on the crack face may be expressed in the form of appropriate Fourier series, the problem reduces to consideration of an infinite system of algebraic equations. Numerical results are presented for the stress intensity factors and the distribution of stresses and displacements on the crack face.

Theory of a waveguide tee

A rigorous approach is proposed to the solution of the wave diffraction problem at the joint of a waveguide tee in the E plane. The electrodynamic problem reduces the solution of an inhomogeneous system of infinite algebraic equations of the second kind in the unknown coefficients of the field in the principal arm of the tee. This system is solved by the reduction method using a computer; the convergence of the algorithm is investigated and new numerical results are offered.

An Asymmetric Mixed Boundary Value Problem of the Elastic Half-Space Subjected to Moment by an Annular Rigid Punch

The solution is presented, within the classical theory of elasticity, for an asymmetric mixed boundary-value problem of the elastic half-space subjected to moment by an annular rigid punch. In the analysis the problem is reduced solving an infinite system of algebraic equations under the assumption that the normal stress on the contact region may be expressed as appropriate Fourier series. Numerical results are illustrated for the distribution of displacements and stresses.

The Effect of Bending on the Infinite Elastic Solid with an Annular Crack

The problem considered is one of determining the stress field in the vicinity of an annular crack embedded in the infinite elastic solid under bending. The solution is obtained by assuming that the displacement component on the crack face may be expressed in the form of appropriate Fourier series and then the resulting system of integral equations to the problem of solving the infinite system of algebraic equations. The numerical results given in the paper include the stress intensity factors, and the stress and displacement distributions. The effect of the crack on the stress produced in the solid by an asymmetric normal loading is also considered.

Inclusions of arbitrary shape in an elastic medium

The problem of elasticity where a homogeneous linear solid of arbitrary shape and elastic constants is embedded in an infinite homogeneous isotropic medium is discussed in a manner which is a straightforward generalization of Eyges’ work on the Laplace equation. An exact integral equation is recast into an infinite system of algebraic equations and a systematic scheme of approximation is obtained by truncation. The lowest order approximation is discussed in detail, and its accuracy is shown to be quite good even for large differences in moduli between the two phases. Application to the effective moduli of composite materials is briefly discussed.

Plate in the form of an infinite strip on an elastic half-space: PMM vol. 40, n≗ 2, 1976, pp. 298–305

A method is given for solving the problem about the contact of a thin plate in the form of an infinite strip lying on an elastic frictionless half-space in a threedimensional formulation. This method reduces the problem to the solution of an infinite system of algebraic equations with a completely continuous form. The corresponding plane problem has been examined by a number of authors [1–6]. The results obtained in [3–6] should be considerd most complete. This same problem for a linearly deformable base of general type was examined in [7–9].

Heat transfer from two hot plates in laterally bounded uniform flow as an example of anN-part Wiener-Hopf-problem

By means of a specific example, namely the heat transfer from two hot plates due to laterally bounded conduction and convection at vanishingly small Prandtl number, it is shown that application of the Wiener-Hopf technique can be extended to the solution of certain linearN-part mixed boundary-value problems. In essence the method is a superposition ofN−1 ordinary Wiener-Hopf problems, the coupling of which leads to an infinite system of algebraic equations. But due to the exponentially small off-diagonal terms in the coefficient matrix, rapidly convergent solutions of this infinite system of equations can be obtained in most cases.

State of stress in a plat circular ring with a crack: PMM vol. 39, n≗ 6, 1975, pp. 1118–1128

The stress distribution in a circular isotropie ring with a crack on part of the concentric circle is investigated. A system of functional equations governing the coefficients of the complex Fourier series expansion of the stresses acting on the circle on which the crack is located is obtained. The solution of the mentioned system of equations is obtained by using a factorization method, which permitted reduction of the initial system of equations to two coupled infinite systems of algebraic equations. The possibility of using the method of truncation to solve these systems is proved. The singularity originating in the neighborhood of ends of the crack in the formulas governing the stresses is isolated. Stress intensity coefficients for the effect of a uniform load on the external contour are presented.