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SIAM J. on Applied Mathematics

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1999

Volume 59, Issue 6, pp. 1917-2300


Two Tandem Queues with General Renewal Input I: Diffusion Approximation and Integral Representations

Charles Knessl and Charles Tier

SIAM J. Appl. Math. 59, pp. 1917-1959 (43 pages) | Cited 2 times

Online Publication Date: July 26, 2006

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We consider two tandem queues with exponential servers. Arrivals to the first queue are governed by a general renewal process. If the arrivals were also exponentially distributed, this would be a simple example of a Jackson network. However, the structure of the model is much more complicated for general arrivals. We analyze the joint steady-state queue length distribution for this network, in the heavy traffic limit, where the arrival rate is only slightly less than the service rates. We formulate and solve the boundary value problem for the diffusion approximation to this model. We obtain simple integral representations for the (asymptotic) steady-state queue length distribution.

Two Tandem Queues with General Renewal Input II: Asymptotic Expansions for the Diffusion Model

Charles Knessl and Charles Tier

SIAM J. Appl. Math. 59, pp. 1960-1997 (38 pages) | Cited 3 times

Online Publication Date: July 26, 2006

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In Part I we formulated and solved a diffusion model for two tandem queues with exponential servers and general renewal arrivals. We thus obtained the heavy traffic diffusion approximation to the steady state joint queue length distribution for this network. Here we study asymptotic and numerical properties of the diffusion approximation. In particular, analytical expressions are obtained for the tail probabilities. Both the joint distribution of the two queues and the marginal distributionof the second queue are considered. We also give numerical illustrations of how this marginal is affected by changes in the arrival and service processes.

Influence of Surfactant on Rounded and Pointed Bubbles in Two-Dimensional Stokes Flow

Michael Siegel

SIAM J. Appl. Math. 59, pp. 1998-2027 (30 pages) | Cited 9 times

Online Publication Date: July 26, 2006

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A simple plane flow model is used to examine the effects of surfactant on bubbles evolving in slow viscous flow. General properties of the time-dependent evolution as well as exact solutions for the steady state shape of the interface and distribution of surfactant are obtained for a rather general class of far-field extensional flows. The steady solutions include a class for which "stagnant caps" of surfactant partially coat the bubble surface. The governing equations for these stagnant cap bubbles feature boundary data which switches across free boundary points representing the cap edges. These pointsare shown to correspond to singularities in the surfactant distribution, the location and strength of which are determined as part of the solution. Our steady bubble solutions comprise shapes with rounded as well as pointed ends, depending on the far-field flow conditions. Unlike the clean flow problem, we find in all cases an upper bound on the strain rate for which steady solutions exist. A possible connection with the phenomenon of tip streaming is suggested.

A Singular Field Method for the Solution of Maxwell's Equations in Polyhedral Domains

Anne-Sophie Bonnet-Ben Dhia, Christophe Hazard, and Stephanie Lohrengel

SIAM J. Appl. Math. 59, pp. 2028-2044 (17 pages) | Cited 51 times

Online Publication Date: July 26, 2006

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It is well known that in the case of a regular domain the solution of the time-harmonic Maxwell's equations allows a discretization by means of nodal finite elements: this is achieved by solving a regularized problem similar to the vector Helmholtz equation. The present paper deals with the same problem in the case of a nonconvex polyhedron. It is shown that a nodal finite element method does not approximate in general the solution to Maxwell's equations, but actually the solution to a neighboring variational problem involving a different function space. Indeed, the solution to Maxwell's equations presents singularities near the edges and corners of the domain that cannot be approximated by Lagrange finite elements.
A new method is proposed involving the decomposition of the solution field into a regular part that can be treated numerically by nodal finite elements and a singular part that has to be taken into account explicitly. This singular field method is presented in various situations such as electric and magnetic boundary conditions, inhomogeneous media, and regions with screens.

Chirality in the Maxwell Equations by the Dipole Approximation

H. Ammari, J. C. Nédélec, and K. Hamdache

SIAM J. Appl. Math. 59, pp. 2045-2059 (15 pages) | Cited 3 times

Online Publication Date: July 26, 2006

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In this paper, we show how chiral materials are realized by embedding chiral objects (helices) in an isotropic medium. More precisely, we derive the Drude--Born--Fedorov constitutive relations, governing the propagation of electromagnetic waves in chiral media, from the standard constitutive relations for a homogeneous, isotropic medium by embedding in the medium a large number of regularly spaced, ramdomly oriented helical conductors, each modeled as a dipole.

Viscosity Solutions and Convergence of Monotone Schemes for Synthetic Aperture Radar Shape-from-Shading Equations with Discontinuous Intensities

Daniel N. Ostrov

SIAM J. Appl. Math. 59, pp. 2060-2085 (26 pages) | Cited 8 times

Online Publication Date: July 26, 2006

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The shape-from-shading (SFS) equation relating u(y,r), the unknown (angular) height of a surface, to I(y,r), the known synthetic aperture radar (SAR) intensity data from the surface, is I = \frac{u_r^2}{\sqrt{1+u_r^2+u_y^2}}, where y and r are axial and radial cylindrical coordinates. Unlike the more common eikonal SFS equation which relates surface height in Cartesian coordinates to optical/photographic intensity data, the above radar equation can be transformed into Hamilton--Jacobi Cauchy form: ur+g(I,uy)=0. We explore the case where I is a discontinuous function, which occurs commonly in radar data. By considering sequences of continuous intensity functions that converge to I, we obtain corresponding sequences of viscosity solutions. We prove that these sequences must converge. We also establish conditions that guarantee that these sequences converge to a common limit, which we define as the solution to the radar equation. Finally, we establish and demonstrate that when this common limit exists, monotone numerical schemes must converge to this solution as the mesh size decreases.

Second-Order Phase Field Asymptotics for Unequal Conductivities

Robert F. Almgren

SIAM J. Appl. Math. 59, pp. 2086-2107 (22 pages) | Cited 55 times

Online Publication Date: July 26, 2006

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We extend Karma and Rappel's improved asymptotic analysis of the phase field model to different diffusivities in solid and liquid. We consider both second-order "classical" asymptotics, in which the interface thickness is taken much smaller than the capillary length, and the new "isothermal" asymptotics, in which the two lengths are considered comparable. In the first case, if the phase field model is required to be gradient flow for an entropy functional, then for unequal diffusivities it is impossible to construct a phase equation with finite kinetics which converges with second-order accuracy to a Gibbs--Thomson equilibrium condition with infinitely fast kinetics. In the second case, some error terms are pushed to higher orders, and it is easy to eliminate the remaining errors with finite phase kinetics.

Maximizing Band Gaps in Two-Dimensional Photonic Crystals

David C. Dobson and Steven J. Cox

SIAM J. Appl. Math. 59, pp. 2108-2120 (13 pages) | Cited 40 times

Online Publication Date: July 26, 2006

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Photonic crystals are periodic structures composed of dielectric materials and designed to exhibit band gaps, i.e., ranges of frequencies in which electromagnetic waves cannot propagate, or other interesting spectral behavior. Structures with large band gaps are of great interest for many important applications. In this paper, the problem of designing structures that exhibit maximal band gaps is considered. Admissible structures are constrained to be composed of "mixtures" of two given dielectric materials. The optimal design problem is formulated, existence of a solution is proved, a simple optimization algorithm is described, and several numerical examples are presented.

Mathematical Model of the Graded-Band-Gap Semiconductor Structure with High Internal Quantum Efficiency

Ivan Gavrilyuk, Nataliya Rossokhataya, Victor Rossokhaty, and Vladimir Makarov

SIAM J. Appl. Math. 59, pp. 2121-2138 (18 pages) | Cited 1 time

Online Publication Date: July 26, 2006

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Graded A3B5-semiconductors are prospective materials for devices with a wide range of applications because of their peculiarities regarding many physical effects. One of these peculiarities, connected with the effect of photon recycling in semiconductors with high internal quantum efficiency, may essentially influence the parameters of devices operating at high levels of excitation (e.g., power diodes and transistors, light-emitting diode lasers). In this paper, a one-dimensional mathematical model of the diode structures based on $Al_{\chi}Ga_{1-\chi}As$ graded-gap semiconductors with high efficiency of spontaneous radiative recombination is considered. This model supposes that the base region of structure is neutral and leads to an initial-boundary value problem of a nonlinear integrodifferential equation. An existence-uniqueness result in the Sobolev spaces is proved. The scheme of the method of lines is constructed to find approximate solution. The convergence theorem is proved. The results of numerical experiments and their physical interpretation are presented.

Solitary-Wave Solutions of the Benjamin Equation

Jerry L. Bona, John P. Albert, and Juan Mario Restrepo

SIAM J. Appl. Math. 59, pp. 2139-2161 (23 pages) | Cited 5 times

Online Publication Date: July 26, 2006

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Considered here is a model equation put forward by Benjamin that governs approximately the evolution of waves on the interface of a two-fluid system in which surface-tension effects cannot be ignored. Our principal focus is the traveling-wave solutions called solitary waves, and three aspects will be investigated. A constructive proof of the existence of these waves together with a proof of their stability is developed. Continuation methods are used to generate a scheme capable of numerically approximating these solitary waves. The computer-generated approximations reveal detailed aspects of the structure of these waves. They are symmetric about their crests, but unlike the classical Korteweg--de Vries solitary waves, they feature a finite number of oscillations. The derivation of the equation is also revisited to get an idea of whether or not these oscillatory waves might actually occur in a natural setting.

The Whitham Equations for Optical Communications: Mathematical Theory of NRZ

Yuji Kodama

SIAM J. Appl. Math. 59, pp. 2162-2192 (31 pages) | Cited 22 times

Online Publication Date: July 26, 2006

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We present a model of optical communication system for high-bit-rate data transmission in the nonreturn-to-zero (NRZ) format over transoceanic distance. The system operates in a small group velocity dispersion regime, and the model equation is given by the well-known Whitham equations describing the slow modulation of multiphase wavetrains of the (defocusing) nonlinear Schrödinger (NLS) equation. The model equation is of hyperbolic type, and NRZ pulse with certain initial phase modulation develops a shock. We then show how one can obtain a global solution by choosing an appropriate Riemann surface on which the Whitham equation is defined. We also discuss the effect of third order dispersion by using an integrable hierarchy of the NLS equation, and we give a condition to avoid a shock formation.

Weakly Connected Quasi-periodic Oscillators, FM Interactions, and Multiplexing in the Brain

Eugene M. Izhikevich

SIAM J. Appl. Math. 59, pp. 2193-2223 (31 pages) | Cited 14 times

Online Publication Date: July 26, 2006

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We prove that weakly connected networks of quasi-periodic (multifrequency) oscillators can be transformed into a phase model by a continuous change of variables. The phase model has the same form as the one for periodic oscillators with the exception that each phase variable is a vector. When the oscillators have mutually nonresonant frequency (rotation) vectors, the phase model uncouples. This implies that such oscillators do not interact even though there might be physical connections between them. When the frequency vectors have mutual low-order resonances, the oscillators interact via phase deviations. This mechanism resembles that of the FM radio, with a shared feature---multiplexing of signals. Possible applications to neuroscience are discussed.

Cone Beam Local Tomography

Alexander Katsevich

SIAM J. Appl. Math. 59, pp. 2224-2246 (23 pages) | Cited 11 times

Online Publication Date: July 26, 2006

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In this paper we study three-dimensional cone beam local tomography. We analyze the local tomography function $f_{\Lambda}^c$, which was proposed earlier in [A.K. Louis and P. Maass, IEEE Trans. Medical Imaging, 12 (1993), pp. 764--769]. Let f be an unknown density distribution inside an object being scanned. We find a relationship between the wave fronts of $f_{\Lambda}^c$ and f and compute the principal symbol of the operator which maps f into $f_{\Lambda}^c$. Our results prove the fact, which was first noted in Louis and Maass, that one can recover most of the singularities of f knowing $f_{\Lambda}^c$. It is shown that these are precisely the singularities of f that are visible from the data. A simple and efficient algorithm for finding values of jumps of f knowing local cone beam data is proposed. The nature of artifacts inherent in cone beam local tomography is studied.

A Lattice Cellular Automata Model for Ion Diffusion in the Brain-Cell Microenvironment and Determination of Tortuosity and Volume Fraction

Longxiang Dai and Robert M. Miura

SIAM J. Appl. Math. 59, pp. 2247-2273 (27 pages) | Cited 3 times

Online Publication Date: July 26, 2006

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In the brain-cell microenvironment, the movement of ions is by diffusion when there is not any electrical activity in either the cells or the externally applied electric field. In this complex medium, the primary constraints on long-range diffusion are due to the geometrical properties of the medium, especially tortuosity and volume fraction, which are lumped parameters that incorporate local geometrical properties such as connectivity and pore size. In this paper, we study the effects of these geometrical properties in mimicking the experimental situation in the brain. We build a lattice cellular automata model for ion diffusion within the brain-cell microenvironment and perform numerical simulations using the corresponding lattice Boltzmann equation. In this model, particle injection mimics extracellular ion injection from a microelectrode in experiments. As an application of the model, we combine the results from the simulations with porous media theory to compute tortuosities and volume fractions for various regular and irregular porous media. Porous media theory previously had been combined with diffusion experiments in brain tissue to determine tortuosity and volume fraction. As in the case of the diffusion experiments, porous media theory gives a good approximation to the numerical simulations. We conclude that the lattice Boltzmann equation can accurately describe ion diffusion in the extracellular space of brain tissue.

On Some Breakup and Singularity Formation Mechanisms for Inviscid Liquid Jets

J. J. L. Velázquez and M. A. Fontelos

SIAM J. Appl. Math. 59, pp. 2274-2300 (27 pages) | Cited 2 times

Online Publication Date: July 26, 2006

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The goal of this paper is to describe several mechanisms of singularity formation and breaking for the well-known one-dimensional model of inviscid jets. We describe a procedure, based on the use of the so-called hodograph transformation, that allows us to construct a large wealth of solutions to this problem yielding very different singularity formation mechanisms.
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