Solutions For Boundary Value Research Articles

Ultralow‐frequency magnetohydrodynamics in boundary‐constrained geomagnetic flux coordinates

A new method is described for constructing geomagnetic flux coordinates, constrained by magnetospheric and ionospheric boundary surfaces. The technique is especially useful for computing boundary value solutions in a geometry defined by a realistic, three‐dimensional geomagnetic field with boundary conditions specified on surfaces oriented arbitrarily relative to the background magnetic field. An adaptable algorithm for calculating the metric tensor for a general class of geomagnetic flux coordinate systems is presented. Application to ultralow‐frequency wave propagation in the magnetosphere is illustrated by solving the equations of linear, one‐fluid magnetohydrodynamics for the driven field line resonance in a dipolar geomagnetic field bounded by a spherical ionosphere. A novel diagnostic for energy flow in the driven resonance problem shows that MHD wave power flows “radially” inward as a compressional wave from the boundary driver to the resonant flux surface, is diverted azimuthally at the resonant surface, and becomes field‐aligned in the resonant surface by coupling to the magnetically guided Alfvén wave at locations where the shear mode is in azimuthal phase quadrature with the compressional driver.

Optical Diffraction in Close Proximity to Plane Apertures. I. Boundary-Value Solutions for Circular Apertures and Slits.

In this paper the classical Rayleigh-Sommerfeld and Kirchhoff boundary-value diffraction integrals are solved in closed form for circular apertures and slits illuminated by normally incident plane waves. The mathematical expressions obtained involve no simplifying approximations and are free of singularities, except in the aperture plane itself. Their use for numerical computations was straightforward and provided new insight into the nature of diffraction in the near zone where the Fresnel approximation does not apply. The Rayleigh-Sommerfeld integrals were found to be very similar to each other, so that polarization effects appear to be negligibly small. On the other hand, they differ substantially at sub-wavelength differences from the aperture plane and do not correctly describe the diffracted field as an analytical continuation of the incident geometrical field.

The Nonlinearly Elastic Equilibrium of a Compound Rectangular Plate with Friction Between the Layers

We consider a spatial boundary-value problem in the physically nonlinear theory of elasticity for a multilayered rectangular plate with friction between the layers. The nonlinear relationships between stresses and small strains are assumed to have Kauderer's form. The nonlinear boundary-value solution is constructed as series in powers of a physical dimensionless parameter. The physically nonlinear problem posed is reduced to a sequence of linear inhomogeneous problems. The effect of physical nonlinearity on the elastic equilibrium of a two-layer thick plate with friction between the layers is studied.

Exact relativistic treatment of stationary counterrotating dust disks: Boundary value problems and solutions

This is the first in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which describe counter-rotating disks of dust. These disks can serve as models for certain galaxies and accretion disks in astrophysics. We review the Newtonian theory for disks using Riemann-Hilbert methods which can be extended to some extent to the relativistic case where they lead to modular functions on Riemann surfaces. In the case of compact surfaces these are Korotkin's finite gap solutions which we will discuss in this paper. On the axis we establish for general genus relations between the metric functions and hence the multipoles which are enforced by the underlying hyperelliptic Riemann surface. Generalizing these results to the whole spacetime we are able in principle to study the classes of boundary value problems which can be solved on a given Riemann surface. We investigate the cases of genus 1 and 2 of the Riemann surface in detail and construct the explicit solution for a family of disks with constant angular velocity and constant relative energy density which was announced in a previous Physical Review Letter.

Comparable solutions to nth-order linear difference equations

We consider the n th -order linear difference equation l ny(t) = L ny(t) + q(t)y (t + [ n 2 ]) = 0 , for t ϵ [ a, b] where q( t) is a real-valued function defined on [ a, b]. We define the (formal) adjoint operator l ∗ n of l n by l ∗ nz(t) = L ∗ nz(t) + (−1) nq(t)z (t + [ n 2 ]) , for t ϵ [ a, b]. We compare boundary value solutions of l n y( t) = 0 to similar solutions of the adjoint equation l ∗ nz(t) = 0 .

Scattering of a plane wave by an infinite grating of circular dielectric cylinders at oblique incidence: E-polarization

The boundary value solution to the problem of scattering by an infinite grating of circular dielectric cylinders is treated for an obliquely incident vertically polarized plane electromagnetic wave. The generalized representations of the scattered electric and magnetic fields as well as the internal fields within the dielectric circular cylinders are acquired in terms of special functions and series. The solution is an exact one and based on the separation-of-variables technique and the addition theorem for cylindrical waves. The scattering coefficients for the electric and magnetic fields are expressed in terms of two systems of simultaneous linear equations of infinite order in coupled form. It has been demonstrated that the solutions to the infinitely long dielectric cylinder at oblique incidence, and to the infinite grating of circular dielectric cylinders at non-oblique incidence can be obtained from the most generalized solution presented here.

Axisymmetric creeping flow past a porous prolate spheroidal particle using the Brinkman model

A boundary-value solution to axisymmetric creeping flow past and through a porous prolate spheroidal particle is presented. The Brinkman model for the flow inside the porous medium and the Stokes model for the free-flow region in their stream function formulation are used. As boundary conditions, continuity of velocity, pressure and tangential stresses across the interface are employed. A mainly analytical procedure for calculation the required eigenvalues and eigenfunctions for the porous region part of the solution is proposed. The coefficients of the convergent series expansions of the general solutions for the stream functions, and thus for the velocity, pressure, vorticity and stress fields, both for the flow inside and outside the porous particle, can be calculated to any desired degree of accuracy as the solution of a truncated algebraic system of linear equations, once the eigenvalues to the Brinkman equation for a given focal distance and permeability have been computed. The drag force experienced by the porous particle is then given as a function of only one of these coefficients. Streamline-pattern and drag-force dependence on permeability and focal distance are presented and discussed.

Diffusion-mediated permanence problem for a heterogeneous Lotka–Volterra competition model

We analyse the dynamics of a prototype model for competing species with diffusion coefficients (d1d2) in a heterogeneous environment Ω. When diffusion is switched off, at each pointx∊ Ω we have a pair of ODE's: thekinetic. If for somex∊ Ω kinetic has a unique stable coexistence state, we show that there existsuch that for everythe RD-model ispersistent, in the sense that it has a compact global attractor within the interior of the positive cone and has a stable coexistence state. The same result is true if there existxu,xv∊ Ω such that the semitrivial coexistence states (u, 0) and (0,v) of thekineticare globally asymptotically stable atx=xuandx=xv, respectively. More generally, our main result shows that, for most kinetic patterns, stable coexistence of xspopulations can be found for some range of the diffusion coefficients.Singular perturbation techniques, monotone schemes, fixed point index, global analysis ofpersistence curves, global continuation and singularity theory are some of the technical tools employed to get the previous results, among others. These techniques give us necessary and/or sufficient conditions for the existence and uniqueness of coexistence states, conditions which can be explicitly evaluated by estimating some principal eigenvalues of certain elliptic operators whose coefficients are solutions of semilinear boundary value problems.We also discuss counterexamples to the necessity of the sufficient conditions through the analysis of the local bifurcations from the semitrivial coexistence states at the principal eigenvalues. An easy consequence of our analysis is the existence of models having exactly two coexistence states, one of them stable and the other one unstable. We find that there are also cases for which the model hasthree or morecoexistence states.

Analysis of millimeter wave phase shifters coupled to a fixed periodic structure

The propagation characteristics of an active plasma-induced millimeter wave phase shifter coupled to a fixed periodic structure are discussed. The numerical calculation is based on an improved boundary value solution. Like a normal dielectric slab waveguide with a plasma layer, the grating waveguide displays a phase shift with the increase of the plasma density. This phase shift due to the significant change of the field distribution has an impact on the modal attenuation coefficient. It results in the scanning of the radiation beam in the vicinity of the second Bragg, Especially, at both weak and strong plasma densities when the mode losses are small, the resonances caused by the periodic structure do not appear to be weakened. The results can be used to design electronically controllable millimeter wave scanning antennas.

On pairs of positive solutions for a singular boundary value

In this paper, we study the existence of multiple solutions of some singular boundary value problems where p can be equal to zero at t = 0 and t = T, the function g can be singular at t = 0, t = T and a h for u = 0. Such singularities generalize Emden-Fowler equations. We consider the case where the “slope” is larger than the first eigenvalue near 0 and near infinity. In between the slope is controlled from the existence of a strict upper solution. To prove our result we extend to the singular case the theory of lower and upper solutions a;nd its relation with the Leray-Schauder degree.

A failure of a numerical electromagnetics code to match a simple boundary condition

It is shown that it is impossible to satisfy the boundary condition for a perfectly conducting hemisphere-capped dipole without creating a discontinuity in the first derivative of the scattered field on the boundary. Since solutions of Maxwell's equations are analytic in a source-free region, all field derivatives must be continuous in the region where the source-free equations are satisfied. The basis functions used by the generalized multipole technique (GMT) are solutions of Maxwell's source-free equations in a region which includes the scattering surface. Therefore, a GMT solution to the dipole problem can exist only in the sense that the boundary value error approaches zero as the number of basis functions approaches infinity. This in itself is not surprising, but the difficulty of matching the boundary condition at the discontinuity affects the convergence of the technique. For the method-of-moments (MOM) technique, where a source current exists on the scattering surface, it is not clear if a perfect boundary value solution to the dipole problem can be theoretically realized or not. >

Asymptotic modeling of thin asymmetric laminates. Boundary-value problems and solution methods

We have investigated the internal dynamic stress-strain state of elastic laminates with perfectly joined anisotropic layers in the long-wave approximation. In contrast to familiar cases, in this treatment we allow an asymmetric arrangement of plies across the thickness and the most general anisotropy. We derive the generalized governing equations for plates by an asymptotic method based on three-dimensional dynamic elasticity. We discuss the similarities and differences between the derived two-dimensional theory and earlier models of S. A. Ambarisumyan and R. M. Christensen. We show that the asymptotic accuracy of the equations depends on the type of anisotropy: (i) general anisotropy; (ii) monodinic anisotropy or orthotropy of the layers.

Analysis and experiments of the 3mm-band higher order modes waveguide Y-junction circulators

Based on the boundary value solution of electromagnetic fields, a calculated results of the 3mm-band higher order modes waveguide Y-junction cirulators, which are given with the frequency dependence of the eigenvalues of the junction scattering matrix, are presented for the first time. It has been verified exactly by the experiments. Using the higher order modes. the bandwidth of the circulators can be expanded.

Boundary-value solution to electromagnetic scattering by a rectangular groove in a ground plane

The diffraction of a plane electromagnetic wave by a rectangular groove corrugated in a perfectly conducting ground plane is rigorously formulated in terms of a Fourier-type integral for two orthogonal polarizations. I successfully expand the angular spectrum of the diffracted field in a series of Bessel functions. Furthermore, the Watson transform is adopted to compute the far-zone diffracted field. Results indicate that the groove width is inversely related to the beamwidth, while the groove depth is responsible for the power distribution among the mainlobe and the sidelobes. The solution remains computationally stable for a wide range of the parameters.

Transverse magnetic scattering by parallel conducting elliptic cylinders

A boundary value solution to the problem of transverse magnetic multiple scattering by M parallel perfectly conducting elliptic cylinders is presented. The solution is an exact one and based on the separation-of-variables technique and the addition theorem for Mathieu functions. It is expressed in terms of a system of simultaneous linear equations of infinite order, which is then truncated for numerical computations. Representative numerical results for the scattered field by two cylinders are then generated, for some selected sizes and orientations parameters, and presented.

Closed-form solutions for boundary value coupled differential and difference systems

In this paper we study existence conditions and explicit closed-form solutions of boundary value problems for coupled differential and difference systems. Our algebraic approach is based on the concept of rectangular co-solution of an algebraic matrix equation of polynomial type. This avoids the increase of the dimension of the problem and permits to find an explicit expression of the general solution of the problem.

Random-volume scattering: Boundary effects, cross sections, and enhanced backscattering.

Assuming a scalar wave, the Bethe-Salpeter (BS) equation for a system of random medium with rough boundaries is first reviewed, together with the scattering matrices involved. Emphasis is placed on the optical condition of each scattering matrix, as well as that of a random layer with pronounced boundary effect as one scatterer. Their optical expressions are obtained in terms of the cross sections along with the respective optical conditions. The enhanced backscattering can be understood as a natural consequence of requiring coordinate-interchange invariance of the second-order Green's function, and the BS equation is rewritten as an equation for the function of the four coordinates involved, so that the invariance is immediately clear. With the solution, specific expressions of cross sections are obtained for a random layer to the approximation of using the boundary-value solution of the diffusion equation. Nevertheless, the angle distribution in the enhanced backscattering holds sufficient accuracy as long as the optical width of the layer is long enough, although not quite for the background term. Another method of using asymptotic evaluation of the cross sections under the diffusion condition is also discussed. A numerical example is shown for the enhanced backscattering.

Plane wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric

The scattering properties of a dielectric-coated nonconfocal conducting elliptic cylinder are investigated. The theoretical treatment of such a problem is based on the boundary value solution, which is an exact treatment of the problem. Only the transverse magnetic (TM) case is considered, although the transverse electric (TE) case can be treated in the same way. It is shown that this solution is much more general than all previous solutions because it can handle a variety of scattering geometries. Numerical results are presented in graphical form for the echo width pattern of various geometries.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Boundary effects in second sound near Tλ

Abstract We investigate the effect of non-ideal boundary conditions applied to a second sound resonator near the lambda point. Appropriate boundary conditions for He-II are applied to a second sound resonator with a heater and detector at either end. We find the analytic boundary value solution for the homogeneous (zero gravity) differential equation. We show that the standing wave approximation is accurate to a few tenths of a percent at 10 −8 K from the transition.

Boundary conditions of the diffusion equation and applications.

A particular expression of the mutual coherence function of a wave is first derived for a system of random layers with rough boundaries, starting from the unified Bethe-Salpeter equation that involves the random medium and boundaries on exactly the same footing. An effective scattering matrix of the medium is then introduced, which is the only medium-dependent quantity involved in the expression and is obtained as a boundary-value solution of the diffusion equation. The diffusion approximation is based on an eigenfunction expansion by using a set of eigenfunctions of the medium scattering cross section that can be even rotationally variant, and the first term is good enough in the diffusion region. Here each expansion coefficient is obtained as the solution of an integral equation with terms of the boundaries, and, for the first term, the equation can be converted to a diffusion equation with a source term, subjected to boundary conditions determined by the boundary scattering cross sections. Here the condition at each boundary is valid even when the averaged refractive indices of the two media differ from each other by a large amount, as contrasted to the condition of no reflection used so far when obtaining boundary-value solutions of diffusion equations. Specific expressions are obtained for both backscattered and transmitted waves through a random layer, along with numerical examples. The boundary condition is finally generalized to meet the case of a rough boundary between two random media of different kinds, consistent with power conservation.