Integral Equation Analysis Research Articles

Stable Hydrodynamic Limit Fluctuations of a Critical Branching Particle System in a Random Medium

A particle system in Euclidean space is considered where the particles are subject to spatial motion according to a symmetric stable law and to a critical law in the domain of attraction of a stable law. The branching intensity may be position-dependent (varying medium) or be given by a realization of a random field (random medium). It is shown that under natural assumptions the hydrodynamic limit fluctuations around the macroscopic flow are the same as those given by the averaged medium, the limit being a generalized stable Ornstein-Uhlenbeck process. The convergence proof is based on an analysis of a nonlinear integral equation with random coefficients.

Local boundary integral equation analysis of 2‐dimensional potential problems using Fourier series

AbstractLocal boundary integral equation (boundary element) methods make it possible to determine the variables of interest in a small pre‐selected region of a solution domain, without the need to determine the unknowns over the rest of the boundary. A method of this type based on Fourier series is developed for two‐dimensional potential problems governed by Laplace's equation. Very accurate values of the function and its derivatives can be obtained.

Boundary integral equation analysis for radiation and scattering of elastic waves in three dimensions

Abstract In this paper, the boundary integral equation (BIE) method is employed to investigate the radiation and scattering of time‐harmonic elastic waves by obstacles of arbitrary shape embedded in an infinite medium. Based on the vector BIE, entirely free of Cauchy principal value integrals, an efficient numerical scheme using quadratic isoparametric boundary elements is proposed. Furthermore, the difficulty of non‐uniquess of a solution inherent with BIE formulations for exterior elastodynamic problems is studied numerically and analytically. The counterparts of the combined Helmholtz integral formulation method for elastodynamics together with the least‐square or Lagrange‐multiplier technique are derived and applied to overcome this difficulty successfully. In addition, the elastic‐wave fields radiated or scattered by either a spherical cavity or a rigid sphere in an infinite medium are calculated and the results are compared with the analytical solutions to demonstrate the accuracy and versatility of...

Integral equation analysis of radiating structures of revolution

A methods for solving radiation problems involving structures of revolution is presented that is based on the direct solutions of one-dimensional integral equations for the currents. The kernels of these equations are derived explicitly and their singularities are studied. Efficient methods for their computation are presented. The numerical solution is obtained by a method of moments technique using sinusoidal basis functions. Applications include the determination of the current densities, reflection coefficient, and far-field patterns for several waveguide-fed structures. Numerical checks and comparisons with experiment demonstrate the accuracy of the method. >

Numerical analysis and validation of the combined field surface integral equations for electromagnetic scattering by arbitrary shaped two-dimensional anisotropic objects

The numerical solution of coupled integral equations for arbitrarily shaped two-dimensional, homogeneous anisotropic scatterers is presented. The combined theoretical and numerical approach utilized in the solution of the integral equations is based on the combined field formulation and is specialized to both transverse electric (TE) and transverse magnetic (TM) polarizations. As opposed to the currently available methods for anisotropic scatterers, the present approach involves integration over the surface of the scatterer in order to determine the unknown surface electric and magnetic current distributions. The solution is facilitated by developing a numerical approach employing the method of moments. The various difficulties involved in the numerical effort are pointed out, and ways of overcoming them are discussed in detail. The results obtained for two canonical anisotropic structures, namely, a circular cylinder and a square cylinder, are given and validated by results obtained by alternative methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An integral equation analysis of inelastic shells

An integral equation analysis of inelastic shells

Information content analysis of aerosol remote-sensing experiments using singular function theory 2: Scattering measurements

We employ the singular function theory, which is the natural framework within which to discuss the analysis of first kind Fredholm integral equations, to analyze fully the information available from an aerosol aureole scattering experiment. This information is, of course, of two kinds: first, the number of pieces of information available for a given experimental error level and, second, the type (or location) of this information. To appreciate fully the latter, we apply this theory to the inversion of eleven synthetic data sets. These inversions are compared with those obtained previously from an extinction experiment.

Asymptotic analysis of a volterra integral equation

A uniformly valid aymptotic solution is obtained for a class of perturbed Volterra integral equations, in which a naive expansion breaks down as t → ∞. The procedure used is an adaption of the formal methodology presented in [1] for the construction of a uniform asymptotic solution to Volterra equations which possess a boundary layer near t = 0.

Complex space multipole expansion theory with application to scattering from dielectric bodies

Classical multipole theory valid for radiation from sources with small dimensions is extended to complex space. It is shown that by locating the multipole in complex space, the first terms in multipole series can be made more dominating than when restricting to real space. Expression for the complex location is derived to minimize the effect of the second-order multipole term. With an example, radiation from a phased cubic source it is seen that the radiating characteristic is an order of magnitude better for the dipole approximation when the dipole is in complex space instead of being at the center of the cube. Simultaneously, it is demonstrated that the validity of the dipole approximation is extended from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L \ll \lambda</tex> to about <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L \leq \lambda\/2</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> being the measure of the source. Complex location for the electric dipole approximating a dielectric spherical scatterer is found through integral equation analysis corresponding to a linear approximation of the inner field of the scatterer.

Information content analysis of aerosol remote-sensing experiments using singular function theory 1: Extinction measurements

An important early step in any planned remote-sensing experiment is an analysis of the information content of the equations which will ultimately be inverted. In this paper, we employ the singular function theory, which is the natural framework within which to discuss the analysis of first kind Fredholm integral equations. Using this theory, we are able to fully analyze the information available from an aerosol extinction experiment. It is important to appreciate that this information is actually of two forms: first, the number of pieces of information available for a given experimental error level, and second, the type (or location) of this information. To fully appreciate the latter, we apply this theory to the inversion of eleven synthetic data sets, including three multimodal model size distributions.

Boundary Integral Equation Analysis of Three-Dimensional Transient Heat Conduction in Composite Media

The problem of heat conduction in composite media is formulated in terms of integral equations by use of the tempered elementary solution of the unsteady diffusion operator. The integral equations are discretized in both time and space. Each homogeneous subre-gion and its surface and interfaces are represented by tetrahedral and triangular finite elements with quadratic variation of geometry and of unknown functions with respect to intrinsic coordinates. The time dependence of the unknown functions is assumed to be linear over each time step. Integration in the time domain is performed analytically, while integration with respect to spatial coordinates is developed by numerical quadrature formulas. As a result, a linear algebraic system of equations for the unknown in the nodal points is obtained. The procedure was checked with a number of test cases. The numerical experiments show that good accuracy may be obtained provided the parameters of the computation, in particular the number of Gaussian quadratur...

Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems.

Content.- 1 - General Introduction.- 2 - An Integral Equation Method for the Solution of Singular Slow Flow Problems.- 3 - Modified Integral Equation Solution of Viscous Flows Near Sharp Corners.- 4 - Solution of Nonlinear Elliptic Equations with Boundary Singularities by an Integral Equation Method.- 5 - Boundary Integral Equation Solution of Viscous Flows with Free Surfaces.- 6 - A Boundary Integral Equation Method for the Study of Slow Flow in Bearings with Arbitrary Geometrics.- 7 - General Conclusions.

Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems (Derek B. Ingham and Mark A. Kelmanson)

Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems (Derek B. Ingham and Mark A. Kelmanson)

Equations of the dynamics of sets of reachability in problems of optimization and control under conditions of uncertainty

An integral funnel equation is used to interpret the problem of control under conditions of uncertainty in terms of the dynamics of sets. Localization of this equation is obtained. The right-continuity of the bound of the set of reachability is proved, on the basis of which the dynamics of the set of reachability is reduced to an analysis of the local integral funnel equation at points of the bound of the sets of reachability. The local integral funnel equation reduces to a differential relation at the points of continuous differentiability of this bound, from which partial differential equations of the dynamics of the sets of reachability in the space of the positions, in the conjugate space and in the parametric form of the notation are obtained. A classification of these equations is given in accordance with the forms of representation of the sets and surfaces in Euclidean space. Bellman's well-known equation serves as a special case of the equation in the space of positions. A derivation of the maximum principle with a normalized conjugate system is presented for the boundary solutions. Its normalization eliminates the increase in the norm of the vector of conjugate variables and increases the time for the numerical calculations. The optimal control problem reduces to one of obtaining boundary solutions. A considerable number of papers (/1–5/, etc.) cover control under conditions of uncertainty.

Integral equations for some inverse problems of radiative heat exchange

We consider the derivation and analysis of integral equations for some inverse problems of radiative heat exchange.

Freezing in krypton monolayers adsorbed on graphite

The freezing of submonolayer krypton adsorbed on graphite is examined within the framework of recently proposed analysis of the nonlinear integral equation that relates the densities of the coexisting nonuniform surface liquid and solid phases. Density changes for freezing of the two-dimensional fluids were evaluated for various temperatures, liquid state densities, and values of the parameter characterizing the magnitude of the periodic part of the gas–solid interaction. It is concluded that the theory gives reasonable results which could be improved if more precise values for the structure factors of the fluid phase were available.

Analysis of integral equations attached to skin effect

The paper is a mathematical background of the paper of D. Mayer, B. Ulrych where the mathematical model of the skin effect is established and discussed. It is assumed that the currents passing through parallel conductors are under effect of a variable magnetic field. The phasors of the density of the current are solutions of $f(x)-jq \sum{^k_{i=1}}b_i \int_{Si} f(y) V(|y-x|)dy-c_p=h(x)$ for $x\in S_p, p=1,\dots,k$, $\int_{Si}f(x)dx=I_i, i=1,\dots, k$, where $j$ is the imaginary unit, $b_i,q,I_i$ are given constants, #h(x)$ is a given function and $f(x)$ is an unknown function and $c_i$ are unknown constants. The first and the second section of this paper are devoted to the problem of existence and unicity of a solution. The third section is devoted to a numerical method.

Application of the BEM and surface impedance method for the computation of the high frequency TM-eddy current losses in conductors with edges

The paper deals with the behaviour of the electromagnetic fields at the edges of long conductors subjected to time-harmonic transverse-magnetic (TM) fields and the computation of the losses in these conductors in the high frequency limit. The differences between the solutions obtained from a finite conductivity formulation in the high frequency limit and those based on a perfect conductor formulation are emphasized. The approximate computation of the losses through the use of a surface impedance is examined. Results obtained from a boundary integral equations analysis of rectangular crosssection cylinders-including strips are presented.

A study of the freezing transition in the Lennard-Jones system

The freezing of a Lennard-Jones fluid into a face-centered-cubic lattice is examined within the framework of a recently proposed analysis of the nonlinear integral equation that relates the densities of the coexisting liquid and solid phases. The temperature dependent densities of the liquid and solid phases along the coexistence line in the temperature–density plane are calculated and found to be in reasonable agreement with the results of computer simulations.

Boundary integral equation analysis of axisymmetric thermoelastic problems

The boundary integral equation (BIE) numerical method for linear elastic stress analysis is applied to axisymmetric problems which also involve temperature variations due to steady-state thermal conduction. The boundary of the two-dimensional solution domain is discretized into isoparametric quadratic line elements, which provide an excellent modelling capability. Satisfactory agreement is obtained with both exact analytical solutions for test problems and numerical results obtained by the finite difference and finite element methods. The BIE method is applied to the problem of a hollow cylinder with an external semi-circular groove, subjected to axial tensile loading and a temperature difference across the cylinder wall. The results emphasise the very high stress concentrations which can be obtained when the effects of thermal and mechanical loading are superimposed. An important practical advantage of the BIE method is the small amount of labour involved in preparing the mesh data.