Infinite System Of Linear Equations Research Articles

The sedimentation of a small particle through a fluid-filled pore of finite length

This paper investigates the axisymmetric sedimentation of a small slowly rotating and translating particle through a fluid-filled circular pore of finite length which communicates with two half-space chambers of fluid. In the quasi-steady Stokes approximation the particle is modelled by either a rotlet or Stokeslet, and potential-theoretic methods are used to reduce each problem to the solution of coupled infinite systems of linear equations. The numerical solutions of these sets of equations are used to compute approximations to the resistive torque and drag experienced by the particle.

On two mixed problems of antiplane strain of an elastic wedge with circular holes

Two problems are examined: 1) a wedge with a circular hole, clamped along the lower face, is subjected to the action of shear forces along the upper face, and 2) a rigid stamp acts on the upper face instead of shear forces. The circular hole is assumed to be load-free. Both problems reduce to a set of infinite systems of linear equations with a completely continuous operator l 2 under the condition that the circle does not touch the sides of the angle. These equations enable the method of reduction to be used. Formulas obtained, that relate the basis solutions of the Laplace equation in two different polar coordinate systems, are utilized in the solution. The method can be extended to the case of a wedge with several circular holes. The problem of the deformation of a wedge with a circular hole was first examined in one special in /1/, however, the infinite system obtained there remained uninvestigated.

On the Construction of Series Solutions to the First Biharmonic Boundary Value Problem on a Rectangle

This paper is concerned with the problem of finding a biharmonic function u on a rectangle $\Omega $ which vanishes on $\partial \Omega $ and has a preassigned normal derivative there. The solution u is developed in a series; however the family of functions that enters into this series development is not orthogonal and thus the determination of the coefficients requires the solution of an infinite system of linear equations. In implementation of this method of course the coefficients are obtained by solving a truncation of the original infinite system. The bulk of the paper is devoted to the estimation of the error that results from this truncation.

Torsion problems for bonded rods (Rods being bonded each other by an adhesive layer)

Torsion problems for two rods bonded each other by an adhesive layer are investigated. It is assumed in analyses that the mechanical behavior of the adhesive layer is approximated by that of shear springs, and hence the shear stress τθz in the adhesive layer can be proportional to the difference of the displacements υθ between the upper and the lower surface of the adhesive layer. Making use of the series expansion techniques, we reduce the problems to the infinite systems of linear equations which can be solved numerically. Then, the stress distributions are obtained for various values of mechanical properties of the rods and the adhesive layer. We also investigate the special case of these problems i. e. the torsion of a half space by an adhered rigid rod.

Torsion Problem for Bonded Rods : Rods Bonded to Each Other by an Adhesive Layer

Torsion problems for two rods bonded to each other by an adhesive layer are investigated. It is assumed in analyses that the mechanical behavior of the adhesive layer is approximated by that of shear springs, and hence the shear stress in the adhesive layer can be proportional to the difference of the shear displacements between the upper and the lower surface of the adhesive layer. Making use of the series expansion techniques, we reduce the problems to infinite system of linear equations which can be solved numerically. Consequently the stress distributions are obtained for various values of mechanical properties of the rods and the adhesive layer. We also investigate a special case of these problems i.e. the torsion of a half space by an adhered rigid rod.

The scattering of an H‐polarized plane wave from an axially slotted infinite cylinder: A dual series approach

The azimuthal current induced by an H‐polarized plane wave with an arbitrary angle of incidence on an infinite cylinder with an infinite axial slot is considered. A system of dual series equations is derived from the modal expansions of the tangential field components by enforcing the electromagnetic boundary conditions. This dual series system is then solved for the modal coefficients with techniques borrowed from the Riemann‐Hilbert problem of complex variable theory. The resulting infinite system of linear equations for the modal coefficients can be handled in an efficient manner. A comparison of the generalized dual series solution with a purely numerical method of moments solution based upon vector and scalar potentials is made. In contrast to this method of moments solution, it explicitly contains the behavior of the solution near the aperture rim and can generate the current values in a shadow region for small to large ratios of cylinder radius to wavelength without additional special considerations.

Self-consistent evaluation of nonuniform superlattice parameters by the harmonic method

Starting from the fact that in superlattice structures the spatial dependence of effective mass is present, the self-consistent procedure based on the representation of all periodic quantities in the form of complex Fourier series is given in the paper. It was shown that the determination of the allowed energy values reduces to the determination of the eigenvalue of an infinite symmetric matrix. The analysis of off-diagonal elements of this matrix leads to the important conclusion that the energy spectrum has band structure, if either the potential energy or the effective mass is spatially dependent. The solution of the Poisson equation (more exactly the first Maxwell field equation) reduces to the solution of an infinite system of linear equations where the Fourier coefficients of potential energy appear as unknown.

Compactness of the product topology and the solvability of infinite systems of infinite linear equations

If every finite subsystem of an infinite system of linear equations (say, over the field of real numbers) each with finitely many unknowns has a solution then the entire system has a solution. The situation is not so if the equations contain infinitely many unknowns. In this case, as shown below, the solvability of every finite subsystem implies the solva. bility of the entire system provided finite subsystems have solution with common upper and lower bounds and the coefficients of ever equation satisfy some boundedness or convergence conditions. The passage from the solvability of finite subsystem to the solvability of the entire system is achieved based on Tychnoff’s theorem stating that any product of compact topological spaces is compact in their product topology.

The effect of absorptive side walls on the propagation from a line source inside a slot

The effect of absorptive side walls of a slotted geometry on the sound propagation from a line source inside the slot is investigated analytically. The problem is modeled as a rectangular waveguide with a reflective baffle whose side walls have finite acoustic impedances. The source is located along the axis of the waveguide. The acoustic field inside the waveguide is represented by a series of mode functions with unknown coefficients and the field outside the waveguide is represented by an inverse cosine transform of an unknown function. The unknown coefficients and unknown function are determined from the boundary conditions inside the waveguide, radiation condition and the continuity of both pressure and normal velocity on the aperture plane. An infinite system of linear equations is truncated by a finite one and its error is evaluated using the edge conditions. Also a closed form representation of the transform function is given in an asymptotic form of an infinite integral. Limiting cases are compared with the earlier results by Howard [J. Math. Phys. 13, 482–490 (1972)] who treated the same geometry with perfectly reflective and absorptive walls.

Iterative determination of permittivity and conductivity profiles of a dielectric slab in the time domain

A numerical method is presented to compute one unknown constitutive parameter of an inhomogeoeous lossy dielectric slab from the reflected field in the time domain. The method is based upon a space-time discretization of the integral equation for the reflected field. In the inversion, especially those space-time points where the numerical computation of the electric-field strength in the slab is most accurate are taken into account. This is achieved by computing the unknown parameter iteratively. Alternately solving equations for an approximate direct-scattering problem and an approximate inverse-scattering problem yields successive approximations for the electric field in the slab and the unknown constitutive coefficient. Both problems lead to an infinite system of linear equations from which a finite subsystem is selected. General criteria for this selection are presented. Various profiles have been reconstructed numerically from the reflected field due to a sine-squared incident pulse.

The spectral properties of many-electron atomic Hamiltonians and the method of configuration interaction. III. Compactness proof associated with an infinite system of linear equations for n-electron atoms

An infinite system of linear equations is derived from the Schrödinger equation of an n-electron atomic system (n⩾3). The linear operator defined by this system of equations is then shown to be compact in a region of the complex energy plane which excludes the various bound state and multiparticle scattering cuts (i.e., the essential spectrum of the Hamiltonian of the n-electron atomic system). It is further shown that the method can be used to deal with the case of the diatomic molecule. The above result then permits one, both in the case of the n-electron atomic system and the diatomic molecule, to truncate the infinite system of equations in question with the assurance that as the size of the truncated system is increased, the energy eigenvalues computed from the truncated system will uniformly converge to those of the original infinite system.

The spectral properties of many-electron atomic Hamiltonians and the method of configuration interaction. II. Compactness proof associated with an infinite system of linear equations for two-electron atoms

The Schrödinger equation for a two-electron atomic system is reduced to an infinite system of linear equations. The linear operator defined by this system of equations is then shown to be compact in a region of the complex energy plane which excludes the various bound state and multiparticle scattering cuts (i.e., the eseential spectrum of the Hamiltonian of the two-electron atomic system). This permits one to truncate the infinite system of equations with the assurance that the N energy eigenvalues obtained from the N×N truncated system will uniformly approximate the lowest N energy eigenvalues of the original infinite system.

The spectral properties of many-electron atomic Hamiltonians and the method of configuration interaction. I. Proof of convergence of the configuration interaction method

In the method of configuration interaction for atomic structures, the Schrödinger equation for a many electron atom is reduced to an infinite system of linear equations. The eigenvalue problem associated with the finite system of equations obtained by truncating the infinite system is solved to obtain energy eigenvalues and eigenvectors. It is generally assumed that the truncation procedure is a convergent one in the sense that as one increases the size of the truncated equations, the number of eigenvalues and eigenvectors will increase and tend to those of the original infinite set. It is shown that the method of configuration interaction is a convergent procedure in the sense that given any point E in the spectrum of the Schrödinger Hamiltonian of the many-electron system, there exists an eigenvalue of the truncated matrix which is arbitrarily close to E for a sufficiently large size of the truncated matrix. Further, it is shown that the convergence referred to cannot be uniform.

Lösung von Randwertproblemen der Schwingungsgleichung für ein Modell eines Triebwerkeinlaufs

AbstractWe consider a simple model of a jet engine inlet consisting of two coaxial cylinders. The inner cylinder extends in both directions to infinity and represents the hub of a jet engine. The outer cylinder is assumed to be semiinfinite representing the engine's jacket. This arrangement is approached by a compressible and invisced gas moving at a velocity U less than the speed of sound. Two problems are formulated. In problem A a normal speed distribution is given in the compressor‐inlet plane, in problem B we are given a pressure distribution. We look for the induced pressure and velocity fields. Using the acoustical approximation of the fundamental equations the Wiener‐Hopf‐method is applied then leading, after a multiplicative and additive decomposition procedure, to an infinite system of linear equations. These involve the expansion coefficients of the solution with respect to the eigenfunctions of the concentric cylindrical duct. Applying arguements from perturbation theory it is possible to extract the essential information from this system of equations. Apart from the possible exception of a denumerable set of values of the distance L of the blade‐row from the inlet the problem has a unique solution.

The eigenvalue problem for infinite systems of linear equations

Given an infinite system of linear equationswhere the aij depend on a parameter λ, the eigenvalue problem is to determine values of λ for which xj (j = 1, 2, …) are not all zero. This problem (Taylor (3) and Vaughan (4)) can arise in the vibration of rectangular plates. Little theoretical work, however, appears to have been done concerning the existence and determination of the eigenvalues. The usual procedure (see (3) and (4)) is to consider a truncated or reduced system of N equations and find the values of λ for which the determinant of the N × N matrix [aij] vanishes. If a particular λ tends to a constant value as N is increased then this value is assumed to be an eigenvalue. The question therefore arises as to what happens if no limit exists. Can we assert that there are no eigenvalues? By constructing an appropriate example we show that the non-existence of a limit does not imply the non-existence of eigenvalues. In order to construct our example we first establish a result concerning the Legendre polynomials.

Diffraction of electromagnetic waves by two spheres

The solution method consists in the following. With each body we couple a local coordinate system within which the electromagnetic field is represented by the amplitudes of impressed-current waves and by unknown wave amplitudes determined by the presence of another body. We use the boundary conditions at the surface of each body in the local coordinate system to form an incomplete system of infinite linear equations in the unknown wave amplitudes. The wave-amplitude transformations complete this system of equations for transition from one local system to another. Solution of this system by the truncation method provides the solution to our problem.

Direct and conjugate Neumann-Ulam schemes for solving non-linear integral equations

ELEMENTARY unbiased estimates are constructed for a linear functional of the solution of a non-linear integral equation of fairly general type. The method of constructing the estimates, which is based on the “equivalence” of the initial equation to an infinite system of linear equations, makes it possible to transfer to the so-called conjugate Neumann-Ulam scheme, which can prove more advantageous when solving physical problems. We shall discuss the problem of estimating the functional I=∝ ϕhdμ 0 of the solution ϕ of a non-linear integral equation of fairly general type. In accordance with the analogue of the Neumann-Ulam scheme, constructed for this case in [1], the functional is estimated on trajectories of a branched Markov process, which proves to be connected in a natural way with the iterations of the non-linear equation. Below we analyze a different approach to the Neumann-Ulam scheme, based on the equivalence of the initial equation to an infinite system of linear equations. It leads to somewhat more general Markov processes, the most natural of which prove to be branched, and it also enables us to consider the question of the so-called conjugate Neumann-Ulam scheme. Throughout, only elementary estimates of the “absorption” type will be constructed, though it is obviously possible to construct different types of estimate. In addition, we make no explicit stipulation about the absolute convergence of integrals and series, about the σ-finiteness of measures, or about the measurability of functions etc.; such conditions carry over almost automatically from the linear case examined in [2].

On the dynamic contact problem for a half-plane reinforced by a finite elastic strip

The dynamic contact problem for a half-plane reinforced on its boundary by a finite elastic strip is considered. The solution of the problem reduces to solving an integral equation of the first kind, and then an infinite system of linear equations by using Chebyshev polynomials. It is proved that this infinite system of equations is quasi-completely regular. Moreover, a simple analytical expression, completely admissible for practical applications and differing by an arbitrarily small amount from the exact expression, is obtained for the kernel of the integral equation. In this case, for definite values of some physical parameter, the complete regularity of the appropriate infinite system of equations is proved in addition to the quasi-complete regularity and numerical results are obtained for the law of variation of the amplitude of the tangential contact stresses under the strip. The problem under consideration is related to problems of load transfer from stringers to elastic solids which are important for engineering practice. The case of an infinite or semi-infinite strip has been examined earlier [1].

Singular perturbation methods in acoustics: diffraction by a plate of finite thickness

Singular perturbation methods are used to determine the field diffracted by a semi-infinite rigid plate of thickness 2 a under irradiation by a plane acoustic wave at wavenumber k . Six terms of both an outer and an inner expansion in the small parameter ε = ka are calculated in closed form, yielding simple results for the far-field directivity pattern. The outer series is determined by certain eigensolutions, and by a sequence of straightforward Wiener-Hopf problems, while the inner terms are all obtained from a simple conformal mapping. Previous discussions of this problem (e.g. Jones 1953) involve the formulation of a modified Wiener-Hopf equation, and reduce the problem to that of inverting an infinite matrix with elements dependent upon ε. Jones has given a numerical inversion of the truncated 4 x 4 matrix in the limit ε→ 0. Here we obtain exact expressions A 2n+1 = 1 / 2(2n+1) { J n (n+½) - J n+1 (n+½)} for Jones’s variables, and prove that they satisfy his infinite system of linear equations. It is also shown that to O (ε 2 In 2 ε) the plate may be replaced by a duct longer than the plate by an amount L = ( a /π) In 2, in agreement with Jones’s numerical value of L = 0.22 a , together with the monopole field necessary to annul the effect of plane wave propagation down the duct. The principle used here to match the inner and outer asymptotic expansions is a slightly, though significantly, modified version of the one used extensively by Van Dyke (1964). In the present problem Van Dyke’s matching principle appears to hold, in that matching can be formally accomplished, but leads to erroneous results violating the reciprocal theorem . Accordingly, an appendix here gives a discussion and justification, in elementary terms, of the proposed modified asymptotic matching principle.

Solution to the problem of transient thermal contact between two different disks

In performing the Laplace transformation, the problem of representation by changing from a series expansion in one system of eigenfunctions to a series expansion in another system is reduced to an infinite system of linear equations. The Kramer rule is used for constructing the original function. This method is particularly suitable for the analysis of a regular heating mode.