Time-dependent Boundary Value Research Articles

A time-dependent boundary value problem of the flow of noninteracting particles

A method for solving the initial value problem of the flow of a rarefied gas is extended to solve the corresponding boundary value problem. The results are particularly suitable for describing realistic physical situations.

On the time dependent diffusion of macromolecules through transient open junctions and their subendothelial spread. 2. Long time model for interaction between leakage sites

In Part 1 of this study (Weinbaum et al., 1988) a short time model has been proposed to describe the initial time dependent leakage of macromolecules at short distances (5 microns or less) from the exit of a transient open junction which the authors have hypothesized as a characteristic feature of endothelial cells in the process of turnover (Weinbaum et al., 1985). This open junction pathway has also been proposed (Weinbaum et al., 1988) to be the primary ultrastructural correlate of the 20 nm diameter large pore suggested by Renkin et al. (1977) using the predictions of cylindrical pore theory. The short time model in (Weinbaum et al., 1988), however, has major limitations in that it neglects the interaction between leakage sites, macromolecular entry through other pathways, the finite thickness of the vessel wall and the curvature of the cell perimeter. The longer time model developed herein will attempt to describe each of these features and also present an improved model and analytic solution for the steady state flux and uptake. In the previous steady state model developed by Weinbaum et al. (1985) the effect of the resistance of the transient open junctions and the non-isotropic diffusion in the underlying tissue due to the internal elastic lamina (IEL) were both neglected. New solutions are first presented which describe the effect of these important model refinements on the steady state macromolecular permeability of the major arteries. Time dependent solutions are then presented to predict the transient longer time labeling following the introduction of tracer macromolecules of varying size. These solutions and the corresponding short time solutions in Weinbaum et al. (1988) are the first solutions to our knowledge to describe the difficult time-dependent boundary value problem to determine how the channel exit concentration and flux at a leaky junction vary with time. This is accomplished by casting the boundary value problem in the form of an integral equation for the unknown flux at the cleft exit and then solving this problem using a specially designed numerical technique. The theoretical predictions are used to interpret the behavior of the localized leaks to HRP and albumin that have been reported in Stemerman et al. (1986) and our own recent experiments (Lin et al., 1988).

On a four-dimensional integrated geodesy

In contrast to continuous global considerations of time dependent boundary value problems an attempt is made to define 4D-linear observation equations in the framework of integrated geodesy for discrete, more or less regional and local applications (deformation analysis) where time variations in position and in the gravity field have to be considered. The derivation is a strict analogue and extension of the 3D integrated approach. In addition the construction of time dependent covariance functions is discussed, which are necessary to solve for unknown displacements and changes in the gravity potential in the generalized least squares collocation model.

On Modelling a Flexible Robot Arm as a Distributed Parameter System with Nonhomogeneous, Time-Varying Boundary Conditions

The problem of end-point position control of a structuraily flexible robot arm can generally be embedeed as a nonhomogeneous, time-dependent boundary value problem. we suggestt an organic appproach which relies on an elegant transformation of a boundary value problem with homogeneous, time-varrying boundary conditions B.L.S to modifier problem with homogeneous and distributed input forces. for which modal control technique are applicable. The tranetomator leads to an infinite dimensional state space representation, which added trunation. is known to be completely controllible and conditionality completely observe. The advantage of using this transformation is that it can utilize the modal shapes and frequencies with are obtained for a given arm through finite element methods. We formulated the modified problem for a single link invisible aim and derived the state space model required for an actual controller design

Electric Fields in the Middle Atmosphere

All of the solar electromagnetic energy impinging on the earth must pass through the middle atmosphere in some form. This weakly conducting region between the cloud tops and the ionosphere has often been considered to be electrically passive. This paper will review what is known about the large scale electrodynamics of the middle atmosphere and present the results of some recent research which contradicts the historical view of the region. One key to understanding middle atmospheric electrodynamics is a good understanding of the temporally and spatially varying electrical conductivity. With the knowledge of the conductivity, the middle atmospheric effects of external electric generators can often be determined by solving a time dependent boundary value problem often referred to as "mapping" the electric field. This has been a useful technique for study of current generators both below and above the middle atmosphere. Furthermore, contrary to the classical picture, there may be sources of electric fields within the middle atmosphere. Certainly there are sources of variability in the conductivity which actively modify the expected signal from external generators. The many-fold increase in available electric parameter data from within the middle atmosphere has been a great stimulus to recent research. However, this review will conclude that these measurements have tended to raise more questions than answers. That is a clear indication that the field of middle atmospheric electrodynamics is ripe for breakthrough research which will only come as a result of a new and vigorous measurement program.

A technique for time-dependent boundary value problems in the kinetic theory of gases part II. Application to sound propagation

Following the procedure indicated in part I, a canonical solution of the matrix Riemann-Hilbert problem for sound propagation in a gas occupying a half space is presented. The closed form solution is computed numerically for three particular values of the sound frequency. The results compare favorably with a previously known numerical solution.

A technique for time-dependent boundary value problems in the kinetic theory of gases Part I. Basic analysis

The matrix Riemann-Hilbert problem arising in the time-dependent BGK model for half space problems is analyzed. The method used is similar to the one originally introduced by one of the authors for steady problems. A general procedure is given which leads to closed form solutions.

Determination of source parameters by wavefield extrapolation

The extrapolation of an observed (seismogram) wavefield backward in time produces an image, in both time and space, of the source. This concept is implemented through a finite difference solution of the two-dimensional acoustic wave equation. The seismograms themselves are used as time-dependent boundary values that drive the finite difference mesh, Numerical examples illustrate the determination of hypocentre location and origin time for point sources and of source extent, orientation and rupture velocity for finite sources in both homogeneous and heterogeneous media. Spatial resolution of source parameters is typically one-half wavelength of the dominant frequency generated by the source.

Some elementary solutions of the heat equation and related equations

An elementary connection is shown between certain time dependent boundary value problems for the heat equation and problems of steady state heat conduction (or diffusion) in a medium with a radially varying thermal conductivity. Solutions are given for growing ellipsoids, transient heating of wedges and transient bending of wedge shaped plates.

Modified Smoluchowski equation and a unified theory of the diffusion-controlled recombination

A unified theory of the diffusion-controlled recombination based on the modified Smoluchowski equation is developed which takes into account simultaneously the diffusion transport, the chemical reaction, and the physical interaction between reactants. The stationary solution of this equation suggests a method for calculating the physical quantities characterizing the reactivity of active particles in the homogeneous and geminate processes with no need of any information about the solution of the corresponding time-dependent boundary value problem. A correlation between escape probability and apparent rate constant is established. Some results of the general theory are used for the theoretical description of ion pair neutralization and reaction between free ions. The validity of the physical model used in the present paper is briefly discussed.

A theoretical study of the effects of various laryngeal configurations on the acoustics of phonation.

Simulation of glottal volume flow and vocal fold tissue movement was accomplished by numerical solution of a time-dependent boundary value problem, in which nonuniform, orthotropic, linear, incompressible vocal fold tissue media were surrounded by irregularly shaped boundaries, which were either fixed or subject to aerodynamic stresses. Spatial nonuniformity of the tissues was of the layered type, including a mucosal layer, a ligamental layer, and muscular layers. Orthotropy was required to stabilized the vocal folds longitudinally and to accomodate large variations in muscular stress. Incompressibility and vertical motions at the golttis played an important role in producing and sustaining phonation. A nominal configuration for male fundamental speaking pitches was selected, and the regulation of fundamental frequency, intensity, average volume flow, and vocal efficiency was investigated in terms of variations around this nominal configuration. Parameters which were varied consisted of geometrical factors such as length, thickness, and depth, factors for shaping the glottis, as well as tissue elasticities, tissue viscosities, and subglottal pressure. Since nonlinear stress-strain properties were not included, subglottal pressure did not produce a pronounced effect upon fundamental frequency under these somewhat edealized conditions F0 rasing correlated strongly with increased tension in the ligament, and somewhat with increasing tension in the vocalis. F0 lowering correlated with increase in vocal fold length when the tensions were held constant, but not with increase in vocal fold thickness. Vocal intensity and efficiency are shown to have local maxima as the configurational parameters are varied one at a time. It appears that oral acoustic power output and vocal efficiency can be maximized by proper adjustments of longitudinal tension of nonmuscular (mucosal and ligamental) tissue layers in relation to muscular layers. Quantitative verification of the "body-cover" theory is therefore suggested, and several further implications with regard to control of the human larynx are considered.

Solution of time dependent boundary value problems by the Boundary Operator Method

A new procedure, the Boundary Operator Method (BOM), for deriving normal mode response formulas in systems with time dependent boundary conditions is demonstrated. The method of analysis is applied to a finite, nonhomogeneous bar. This example illustrates the following advantages of the BOM: (1) The resulting formulas for the displacement and stress are valid for the entire history of the system and for Dirichlet, Neumann or Cauchy boundary conditions. (2) The uniqueness of the solution is unobscured since, unlike the Mindlin-Goodman method, the present solution technique does not require specific spatial functions which render the transformed boundary conditions homogeneous. (3) The evaluation of the solution is expedited by the absence of integrals containing auxiliary spatial functions. (4) The analysis employs standard mathematical techniques.

Hydrodynamic voltammetry at channel electrodes Part IV. Theory of chronopotentiometry for reversible electrode reactions

The theory of chronopotentiometric measurements at channel electrodes is developed for reversible electrode reactions, by rigorously solving the corresponding time-dependent boundary value problem, where the non-uniform accessibility of the channel electrode surface is taken into consideration. The theoretical equations of the transition time and the potential-time curve are derived as functions of ( I d / I ), where I denotes the applied current intensity and I d the limiting diffusion current obtained at steady-state. Finally, the time variation of the current density distribution at the electrode surface is given.

Hydrodynamic voltammetry at channel electrodes: Part IV. Theory of chronopotentiometry for reversible electrode reactions

The theory of chronopotentiometric measurements at channel electrodes is developed for reversible electrode reactions, by rigorously solving the corresponding time-dependent boundary value problem, where the non-uniform accessibility of the channel electrode surface is taken into consideration. The theoretical equations of the transition time and the potential-time curve are derived as functions of ( I d/ I), where I denotes the applied current intensity and I d the limiting diffusion current obtained at steady-state. Finally, the time variation of the current density distribution at the electrode surface is given.

Half-range orthogonality relations basic to the solution of time-dependent boundary value problems in the kinetic theory of gases

A half-range orthogonality relation concerning the elementary solutions of the time-dependent, linearized BGK model of the Boltzmann equation is established, and the required normalization integrals are evaluated. In addition, the half-space reflection matrixR(μ′→μ) is developed in order to simplify the evaluation of various surface quantities.

Transient analysis of phase-locked tracking systems in the presence of noise

This paper is concerned with the problem Of obtaining time-dependent solutions to a class of Fokker-Planck equations that arise in the analysis and synthesis of a variety of first-order synchronization systems employing the phase-lock principle. These include the classical sinusoidal phase-locked loop, squaring and Costas loops, data-aided loops, hybrid loops, various symbol synchronizer mechanizations, and tunnel-diode oscillators. By analyzing the spectral properties of the associated time-dependent Fokker-Planck boundary value problem, eigenfunction expansions of the reduced modulo- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2 \pi</tex> phase-error-transition probability-density function are developed for a class of first-order synchronization systems. Whereas previous work treated the steady-state probability density function, this approach yields a complete statistical description of the phase-error process reduced modulo- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2 \pi</tex> .

Theory of flooded porous electrodes I. Galvanostatic transients and generalised impedance

Summary o (a) A description of the homogeneous one-dimensional model of the porous electrode used in the present analysis is given (section 2), (b) The time-dependent boundary value problem is posed and a formal analytical solution obtained (section 3, section 5). (c) The earlier results of Tobias and of Guruvich and Bagotskii are critically examined (section 4, 6); under certain conditions a close agreement with the calculations of Grens and Tobias is established (section 6). (d) Several special cases of the above analysis discussed are: (i) t ≪τ dl , timeconstant for double-layer charging; (ii) the concentration polarisation is negligible and (iii) ohmic-activation control is predominant with mass transfer effects in the exterior taken into consideration (section 6). The equivalence of “surface diffusion” and “porous” models is indicated (section 6). (e) The equivalent circuit elements representing the porous matrix-electrolyte interface are derived and the computed values are plotted for various values of the basic parameters (section 7). Further reports on this problem will discuss the dependence of the rise times and the steady-state values on various parameters under different reactant-supply conditions.

Theory of flooded porous electrodes: I. Galvanostatic transients and generalised impedance

Summary o (a) A description of the homogeneous one-dimensional model of the porous electrode used in the present analysis is given (section 2), (b) The time-dependent boundary value problem is posed and a formal analytical solution obtained (section 3, section 5). (c) The earlier results of Tobias and of Guruvich and Bagotskii are critically examined (section 4, 6); under certain conditions a close agreement with the calculations of Grens and Tobias is established (section 6). (d) Several special cases of the above analysis discussed are: (i) t ≪τ dl , timeconstant for double-layer charging; (ii) the concentration polarisation is negligible and (iii) ohmic-activation control is predominant with mass transfer effects in the exterior taken into consideration (section 6). The equivalence of “surface diffusion” and “porous” models is indicated (section 6). (e) The equivalent circuit elements representing the porous matrix-electrolyte interface are derived and the computed values are plotted for various values of the basic parameters (section 7). Further reports on this problem will discuss the dependence of the rise times and the steady-state values on various parameters under different reactant-supply conditions.

Interior Value Problems of Mathematical Physics. Part I. Wave Propagation

The Laplace transform has been applied to solve a class of problems in wave propagation that are converse to the well-known time-dependent Boundary Value Problems (BVP). From a knowledge of time-varying conditions on a suitably restricted set of points interior to a region, it is possible to deduce time-varying boundary conditions. These standard situations in slab, cylindrical, and spherical geometry have been treated for the finite domain. The solution of an Interior Value Problem (IVP) in the semi-infinite domain explicitly demonstrates the formulation of an advanced solution in contrast to the retarded solution for a Boundary Value Problem. Corresponding problems in heat conduction will be discussed in a subsequent article.