Fundamental Integral Equation Research Articles

Inverse Spectral Problem for the Schrödinger Equation with an Additional Linear Potential

We consider the one-dimensional Schrodinger equation with an additional linear potential on the whole axis and construct a transformation operator with a condition at −∞. We obtain the fundamental integral Gelfand-Levitan equation on the half-axis (−∞,x) and prove the unique solvability of this fundamental equation.

Determining the radial pair distribution function from X-ray absorption spectra by use of the Landweber iteration method

The Landweber iteration approach is used to construct the radial pair distribution function (RPDF) from an X-ray absorption (EXAFS) spectrum. The physical motivation for the presented investigation is the possibility to also reconstruct asymmetric RPDFs from the EXAFS spectra. From the methodical point of view the shell fit analysis in the case of complicated spectra would be much more eased if the RPDF for the first shell(s) are computed precisely and independently. The RPDF, as a solution of the fundamental EXAFS integral equation, is examined for theoretical examples, and a detailed noise analysis is performed. As a real example the EXAFS spectrum of curium(III) hydrate is evaluated in a stable way without supplementary conditions by the proposed iteration, i.e. by a recursive application of the EXAFS kernel.

Prediction of Aerodynamic Characteristics of the Wing Stroke Mechanism in Lower Tip-Speed Ratio Condition

The author previously developed a wing stroke mechanism WSM as a human-friendly wind apparatus suitable for distributed power generation system in urban areas. The results of the operation in various kinds of wind condition and the result of the computer simulation show a clear tendency for the tip-speed ratio of the system to remain around one or two. The operation under very low tip-speed ratio condition with single or twin blade(s) may suggest that, different from normal horizontal axis wind turbine (HAWT), the azimuthal variation of the wake in the rotor can not be ignored. In addition, the influence of the interference between contra rotating twin blades in one rotor disc requires special calculation based on vortex theory. In this paper, the basic aerodynamic characteristics of the wind-stroke mechanism are investigated utilizing the lifting line theory. The fundamental integral equation is led and solved numerically using MATLAB and Simulink. Successive parametric research is undertaken according to parameters like aspect ratio sweepback angle taper-ratio and the asymmetry of the wing section of rotor blade(s). The wing stroke mechanism (WSM) has sweep back angle in order to get the directional stability. If the blade(s) with sweep back angle rotates, it makes a cone with a coning angle as is conventionally defined on HAWT. Considering again that the tip-speed ratio of the operating single or twin blade(s) is very small, it is preferable to apply the lifting line theory instead of common blade element momentum theory even though recently much improvement is established for that kind of complexity. The result shows that the wing stroke mechanism has a mixed character of the fixed wing of the airplane and the rotating blade(s) of HAWT.

Harmonic oscillations of a rigid punch on a porous elastic layer

The plane dynamic contact problem of the harmonic oscillations of a rigid punch on the free surface of an elastic layer of porous isotropic material with linear properties is considered. The Fourier transformation of the problem is reduced to a Fredholm integral equation of the first kind in the contact pressure. The properties of the kernel of the fundamental integral equation are investigated and a numerical method of solving it is constructed. Numerical results are compared with existing results in classical limiting cases.

An investigation into the validity of the Born approximation in reverberation calculations

Ocean reverberation models for active sonar applications assume the validity of the Born or single scatter approximation. In many areas of physics, a Born series does not converge due to a combination of the scattering strength and the spatial scale over which the scattering occurs. In these cases, the validity of the Born approximation is at best questionable. It is natural to attempt to examine the validity of the Born approximation and the convergence of the Born series in general for well-defined scattering problems in ocean acoustics. The problems examined in this study involve scattering from specific realizations of bottom roughness for 2-D waveguides. This study examines several idealized problems from the 1st Reverberation Workshop in November 2006 for the purpose of comparing a Born approximation solution to an exact solution provided by a 2-way coupled-mode integral equation approach. The coupled integral equations are solved exactly to within the limitations of finite differences and numerical quadrature. The exact solutions are then compared to those that solve the fundamental integral equation using the Born approximation and to solutions obtained with a finite-element model. [Work supported by ARL:UT IR&D.]

Modelling Rupture Dynamics of a Planar Fault in 3-D Half Space by Boundary Integral Equation Method: An Overview

In this article, we first reviewed the method of boundary integral equation (BIEM) for modelling rupture dynamics of a planar fault embedded in a 3-D elastic half space developed recently (ZHANG and CHEN, 2005a,b). By incorporating the half-space Green's function, we successfully extended the BIEM, which is a powerful tool to study earthquake rupture dynamics on complicated fault systems but limited to full-space model to date, to half-space model. In order to effectively compute the singular integrals in the kernels of the fundamental boundary integral equation, we proposed a regularization procedure consisting of the generalized Apsel-Luco correction and the Karami-Derakhshan algorithm to remove all the singularities, and developed an adaptive integration scheme to efficiently deal with those nonsingular while slowly convergent integrals. The new BIEM provides a powerful tool for investigating the physics of earthquake dynamics. We then applied the new BIEM to investigate the influences of geometrical and physical parameters, such as the dip angle (δ) and depth (h) of the fault, radius of the nucleation region (R asp), slip-weakening distance (D c ), and stress inside (T i ) and outside (T e ) the nucleation region, on the dynamic rupture processes on the fault embedded in a 3-D half space, and found that (1) overall pattern of the rupture depends on whether the fault runs up to the free surface or not, especially for strike-slip, (2) although final slip distribution is influenced by the dip angle of the fault, the dip angle plays a less important role in the major feature of the rupture progress, (3) different value of h, δ, R asp, T e , T i and D c may influence the balance of energy and thus the acceleration time of the rupture, but the final rupture speed is not controlled by these parameters.

Analysis of application of Langmuir isotherm to heterogeneous systems: High-pressure conditions

In this paper, an adsorption model that is based on the statistical mechanics approach was applied to study the saturation phenomena in adsorption to calculate the minimum pressure needed to attain the complete surface coverage on a physical adsorption. The fundamental integral equation, Θ T ( Q ) = ∫ N ( Q ) ⋅ Θ ( Q ) d Q , for the calculation of the coverage degree of the surface was developed for an exponential distribution function, N ( Q ) = ( m / R T ) exp ( − m Q / R T ) , and the representation of local adsorption sites is given by the Langmuir expression, Θ ( Q ) = b 0 C exp ( Q / R T ) / ( 1 + b 0 C exp ( Q / R T ) ) . At high values of the pressure C, a solution of the fundamental integral equation was obtained by imposing the condition b 0 C > 1 . The expression for the saturation condition, b 0 C > m / ( m + 1 ) , was obtained; that is, the saturation phenomenon is dependent on two parameters, correlated with the energetic heterogeneity and adsorption energy of the system. The pressure in the analysis of the cited expression shows that, for low m values (more heterogeneous systems), saturation is attained for b 0 C ≫ m , while for m ≅ 1 (more homogeneous systems), the saturation is attained for b 0 C ≫ 0.5 .

Transmitter Optimization and Theoretical Bounds for Dispersion-Limited Optical Fiber Links

In this paper, we present a novel mathematical investigation on the dispersion-limited optical communication channel. We tackle the problem using a comprehensive approach that is based on the optimization of the energy transfer from the input to the output of the channel. We solve the optimization problem deriving a fundamental integral equation, which we are able to solve analytically. We show that the dispersion-limited optical channel has a very interesting equivalence to the ideal lowpass filter. This equivalence allows us to derive new bounds on the maximum achievable bit rate on the channel with limited intersymbol interference (ISI). In particular, we demonstrate that by suitably increasing the memory of the modulator and using the optimal pulses derived in this paper, one can transmit with limited ISI over a channel with arbitrarily high dispersion.

A method for deriving stellar space densities

The fundamental integral equation of stellar statistics represents a direct, model-independent approach to calculating stellar densities. Many techniques exist for its solution, but some of these require assumptions, such as a Gaussian luminosity function or a specific form for the density function, that may be unrealistic. To solve the equation as an undeterdetermined total least squares system with Tikhanov regularization recognizes that the problem is ill-posed and generally ill-conditioned as well and oers decided advantages: it is unnecessary to assume a Gaussian luminosity function nor a specific form for the density function; discretization error in the kernel of the integral equation as well as the Poisson error in the star counts are accounted for; mean errors for the densities are calculated; the densities are constrained to be both continuous and positive. The greatest drawback to the method comes from the selection of the ridge parameter, but the drawback becomes surmountable. The method is first applied to three examples, general star counts, the distribution of K0 giants, and the distribution of M 2-M 4 dwarfs, and compared with densities calculated from methods such as Malmquist's and the ( m ,l og) table. Regularized total least squares competes well with these methods. Then the method is applied to a new data set from the AC2000.2 catalog to calculate the densities of M giants and supergiants in the directions of the north and south galactic poles. The densities decrease exponentially to near zero at 2000 pc, with half-density points near 550 pc: No evidence for asymmetry between the two hemispheres can be seen.

On stationary oscillations of galaxies

This paper describes a general investigation of stationary oscillations of galaxies. It begins with a linear analysis of modes of oscillation with continuous spectra of real frequencies. Such modes are gravitational analogues of the van Kampen modes of oscillation in plasmas. The characteristic value problem governing modes of the van Kampen type in a galaxy is solved with the aid of a modified version of the matrix method of Kalnajs in which the perturbation of the distribution function is expressed in terms of generalized functions. In general, there is no characteristic equation governing the frequencies in the continuous spectrum. However, isolated frequencies in the continuous spectrum do satisfy a characteristic equation, which, for stellar systems, is a counterpart of the dispersion relation proposed by Vlasov for plasma oscillations. The linear analysis also provides a characteristic equation for modes with a discrete spectrum of real and/or complex frequencies. The second part of the paper describes a perturbation theory for a stationary oscillation of a galaxy with a small but finite amplitude. Integrals of the stellar motion are constructed with the aid of canonical perturbation theory and used in conjunction with Jeans' theorem to specify the density of stars in the six-dimensional phase space. These oscillations are slightly non-linear counterparts of the modes of the van Kampen type, and they are stellar-dynamical counterparts of the non-linear plasma waves described by Bernstein, Greene and Kruskal in the 1950s. Fully non-linear models of stationary oscillations of galaxies can be constructed with the aid of Schwarzschild's numerical method for the solution of the fundamental integral equation describing the self-consistency of a stellar system.

Fundamental solution and boundary integral equation of 3-D coupled thermoelasticity with non-small temperature changes in Laplace transform domain

The coupled term in generalized thermoelasticity formulations is modified with non-small temperature change, where the absolute temperature is not replaced by the temperature of the body in its undeformed state, and is expressed as a linear function related to time. The term including ( ( d/ ds) u i,i ) in Laplace transform domain is treated with an approximate method. The applicability of the assumptions to a lot of engineering problems such as finite body subjected to sudden heating shock is explained. The new fundamental solutions and boundary integral equations of the generalized thermoelasticity with non-small temperature changes are derived.

Numerical equivalent inclusion method: a new computational method for analyzing stress fields in and around inclusions of various shapes

The present study has attempted to develop a new computational method for the elastic stress analysis of inclusions based on the equivalent inclusion method. The proposed method can avoid the complexity of mathematics required for the analysis of non-uniform eigenstrain distributions within inclusions having various shapes. The paper is focused on the formulation for two-dimensional case. The fundamental integral equation is shown first to have a kernel with the 1/ r-singularity. The two-dimensional equations are then discretized by using the triangle polar coordinates. It is shown that the adoption of this coordinate system can eliminate the singularity. Eigenstrain distributions within inclusions having various shapes were calculated by the present method in order to obtain stress distributions within them as well as those in the vicinity of the matrix–inclusion interfaces. The shapes of the inclusions described here are ellipse, circle, triangle and rectangle. The results obtained by the present method were compared and showed good agreements with those obtained by the theories and/or by the FEM analyses except for the sharp corner points of the triangular inclusions where the outside normal vectors can not be determined uniquely and thus the stress becomes singular.

Sommerfeld and Zenneck wave propagation for a finitely conducting one-dimensional rough surface

Starting with Zenneck and Sommerfeld wave propagation over a flat finitely conducting surface has been extensively studied by Wait (see IEEE Antennas Propagat. Mag., vol.40, p.7-24, 1998) and many other authors. We examine propagation over a finitely conducting rough surface, also studied by many people including Feinberg (1944), Bass and Fuks (1979), and Barrick (see Radio Sci., vol.6, p.517-26, and vol.6., p.527-33). This paper extends the multiple scattering theories based on Dyson and Bethe-Salpeter equations and their smoothing approximations. The theory developed here applies to rough surfaces with small root-mean-square (RMS) heights (/spl sigma/<0.1/spl lambda/). We limit ourselves to the one-dimensional (1-D) rough surface with finite conductivity excited by a magnetic line source, which is equivalent to the Sommerfeld dipole problem in two dimensions (x-z plane). With the presence of finite roughness, the total field decomposes into the coherent field and the incoherent field. The coherent (average) field is obtained by using Dyson's equation, a fundamental integral equation based on the modified perturbation method. Once the coherent field has been obtained, we determine the Sommerfeld pole, the effective surface impedance, and the Zenneck wave for rough surfaces of small RMS heights. The coherent field is written in terms of the Fourier transform, which is equivalent to the Sommerfeld integral. Numerical examples of the attenuation function are compared to Monte Carlo simulations and are shown to contrast the flat and rough surface cases. Next, we obtain the general expression for the incoherent mutual coherence functions and scattering cross section for rough conducting surfaces.

Henry's Law as a Limit for an Isotherm Model Based on a Statistical Mechanics Approach

The fundamental integral equation for isotherms derived from statistical mechanics was applied to the analysis of adsorption under low concentrations (or pressures). The model considers adsorption on an energetically heterogeneous surface characterized by an energy distribution function in the semiexponential formN(Q) = (m/RT)e−mQ/RTand by the individual adsorption of each site obeying the Langmuir isotherm. The expression derived guarantees a linear relationship of the covering degree,q,and the concentration,C,that is,q=HC(Henry's law), under the condition of low values of concentration. In this derivationH=mb0/(m− 1) is Henry's constant, whilem> 1 andb0are parameters to be adjusted to experimental data.

Local-Field Effects on Photon Drag in Multiple QuantumWells

A nonlocal analysis of the photon-drag effect due to intersubbandtransitions in a multiple-quantum-well system is presented. Startingwith a fundamental integral equation for local field in amultiple-quantum system, the prevailing local field in quantum wellsis studied. Then, the determined local-field is used to calculate thephoton-drag current along wells for different parameters: angle ofincidence, electron density, relaxation energy, and number of wells.Numerical results show that the local-field effects associated withstrong radiative interaction among quantum wells may lead to notablechanges on the photon-drag current in many ways.

Freundlich's Isotherm Extended by Statistical Mechanics

In this paper we obtain an asymptotic development compatible with Freundlich's isotherm, θ =KCm. That expression was derived underlying the theory of heterogeneous surface. We assume the distribution function of adsorption sites to be in the exponential form and the sites to follow the Langmuir energy homogeneous isotherm. The derived expression represents an analytical solution to the fundamental integral equation for the overall adsorption isotherm in its correct integration limit. From this expression we clearly identify the parametersKandm, the Freundlich parameters. We applied this expression to fit the experimental data for the adsorption of CO2into BPL-activated carbon.

A new off-boundary element method for analyzing plate and shallow shell bending

The paper has developed a new off-boundary element method for analyzing plate and shallow shell bending problems. The new method only needs a fundamental boundary integral equation, but locates the singular points within two auxiliary boundaries outside the domain. The calculation results of the new method have excellent accuracy. Theoretically, it is also proved that the new off-boundary element method is equal to the conventional off-boundary element method. However, the new method will simplify the calculation formulas and computer program significantly.

Importance of the Accuracy of Experimental Data in the Nonlinear Chromatographic Determination of Adsorption Energy Distributions

Adsorption energy distributions (AEDs) are calculated from the classical, fundamental integral equation of adsorption using adsorption isotherms and the expectation-maximization method of parameter estimation. The adsorption isotherms are calculated from nonlinear elution profiles obtained from gas chromatographic data using the characteristic points method of finite concentration chromatography. Porous layer open tubular capillary columns are used to support the adsorbent. The performance of these columns is compared to that of packed columns in terms of their ability to supply accurate isotherm data and AEDs. The effect of the finite column efficiency and the limited loading factor on the accuracy of the estimated energy distributions is presented. This accuracy decreases with decreasing efficiency, and approximately 5000 theoretical plates are needed when the loading factor, L[sub f], equals 0.56 for sampling of a unimodal Gaussian distribution. Increasing L[sub f] further increases the contribution of finite efficiency to the AED and causes a divergence at the low-energy endpoint if too high. This occurs as the retention time approaches the holdup time. Data are presented for diethyl ether adsorption on porous silica and its C-18-bonded derivative. 36 refs., 8 figs., 2 tabs.

Local-field effect on the linear optical intersubband absorption in multiple quantum wells.

Using a microscopic local-field formulation, the linear optical absorption of a multiple-quantum-well (MQW) structure associated with intersubband transitions is investigated. Taking as a starting point a fundamental self-consistent integral equation for the local field, the p-polarized electric-field distribution inside each quantum well in the MQW structure is studied. It is demonstrated that the local field is essentially determined from a set of linear algebraic equations. Within the infinite-barrier approximation with an effective well width, numerical calculations of the optical absorption spectra are presented for a GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As multiple quantum well for different parameters such as the two-dimensional (sheet) electron concentration, the angle of incidence, and the number of wells. The numerical results show that the local-field effect leads to a notable blueshift of the peak-position energy in the absorption spectra as the sheet density of the electrons is increased, and that for larger angles of incidence the electromagnetic interaction among quantum wells gives rise to a significant broadening of the absorption peak when the number of wells is sufficiently large.

Local-field study of the optical second-harmonic generation in a symmetric quantum-well structure.

A nonlocal analysis of the infrared second-harmonic generation associated with intersubband transitions in a symmetric semiconductor quantum-well structure is presented. Taking as a starting point a fundamental self-consistent integral equation for the local field, the p-polarized first-harmonic field inside the quantum well is studied. By using the infinite-barrier wave functions and taking into account only the two lowest subbands, analytical expressions are obtained for the local field. The result of the local-field calculation at the first-harmonic frequency, in turn, is used to calculate the p-polarized second-harmonic local field. The conversion efficiency of the second-harmonic generation from the quantum well is thus determined. Numerical calculations of the frequency and angular spectra of the second-harmonic intensities are presented for a symmetric GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As quantum well. The numerical results show that strong second-harmonic generation occurs in the vicinity of the resonance frequencies for both the first- and second-harmonic local field inside the quantum well. The influence of the dynamic local-field interaction of the electrons on the optical second-harmonic generation is investigated. It is demonstrated that the dynamic screening can lead to a notable upward shift of the locations of the resonant peaks in the frequency spectra of the second-harmonic conversion coefficient.