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SIAM J. on Applied Mathematics

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1996

Volume 56, Issue 6, pp. 1523-1819


A Front Dynamics Approach to Curvature-Dependent Flow

D. W. Schwendman

SIAM J. Appl. Math. 56, pp. 1523-1538 (16 pages) | Cited 2 times

Online Publication Date: July 05, 2006

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A front dynamics approach is developed to study the evolution of planar curves whose normal speed depends on curvature. The formulation is similar to Whitham’s shock dynamics theory for the propagation of shock wave in gases but assumes a different propagation rule. Equations that describe the motion of the front are obtained, and these are evolution equations for the normal direction and local arc length of the front. The solution of these equations leads to the front positions using an appropriate integration along rays. A similarity solution of the equations is found for the evolution of an initial corner. Free-boundary problems for the motion of a junction connecting front segments are discussed. A numerical method is presented to calculate the evolution of any number of front segments. The segments can be closed or open, connected to wall boundaries or not, or connected to other segments at 3-segment junctions. Several sample problems are considered to illustrate the method. An extension of the method for curvature-dependent motion under a constant area constraint is also discussed.

The Conservation Law $\partial _y u + \partial _x \sqrt {1 - u^2 } = 0$ and Deformations of Fibre-Reinforced Materials

Rustum Choksi

SIAM J. Appl. Math. 56, pp. 1539-1560 (22 pages)

Online Publication Date: July 05, 2006

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The conservation law $\partial _y u + \partial _x \sqrt {1 - u^2 } = 0$ is found to govern planar deformations of incompressible materials containing a continuous linear distribution of inextensible fibres. Kinematically feasible deformations are discussed, with emphasis on admissibility and the resolution of nonuniqueness. Many of the aspects of hyperbolic conservation laws have direct consequences in the kinematics of these materials, thus providing an illustrative guide to the theory. Alternatively, the study of this conservation law is geometrically motivated by questions on the structure of the set of points above a continuous function curve whose minimum distance to the curve is achieved in several places.

Band-Gap Structure of Spectra of Periodic Dielectric and Acoustic Media. II. Two-Dimensional Photonic Crystals

A. Figotin and P. Kuchment

SIAM J. Appl. Math. 56, pp. 1561-1620 (60 pages) | Cited 40 times

Online Publication Date: July 05, 2006

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We consider the two-component dielectric medium consisting of a periodic array of parallel air columns of square cross section embedded into a lossless optically dense host material with the dielectric constant $\zeta > 1$. We show that if$\zeta $ is large enough and the relative distance $\delta $ between the air columns is such that $\zeta \delta \gg 1$ and $\zeta \delta ^2 \ll 1$, then the corresponding Maxwell operator has a series of gaps in the spectrum. We also provide some analytic formulas that enable one to detect location of bands and gaps in the spectrum. In particular, the typical wavelength exhibiting a photonic band gap is $2\pi L\sqrt {\zeta \delta } $ where $L$ is the distance between the axes of adjacent air columns. We also give some estimates on the space distribution of electric field energy for different eigenmodes.

Stability of Cellular States of the Kuramoto–Sivashinsky Equation

John N. Elgin and Xuesong Wu

SIAM J. Appl. Math. 56, pp. 1621-1638 (18 pages) | Cited 3 times

Online Publication Date: July 05, 2006

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This paper is concerned with the instability property of a particular type of solution to the Kuramoto–Sivashinsky equation, namely, the symmetric, time-independent cellular states. Attention is focused on the dimension of their unstable manifolds. We show that as a control parameter varies, the dimension changes in an ordered way governed by certain properties of an associated ordinary differential equation. Further change in dimension is shown to be related to Hopf bifurcations from the cellular states. An asymptotic analysis explains the onset of these bifurcations and predicts approximate parameter values at which they occur.

Numerical Solution of Transport Equations for Bacterial Chemotaxis: Effect of Discretization of Directional Motion

Benjamin J. Brosilow, Roseanne M. Ford, Sten Sarman, and Peter T. Cummings

SIAM J. Appl. Math. 56, pp. 1639-1663 (25 pages) | Cited 8 times

Online Publication Date: July 05, 2006

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A mathematical model proposed by Alt [J. Math. Biol., 9 (1980), pp. 147–177] to describe chemotactic bacterial migration is studied, and solutions to this model are compared to solutions of a simpler model proposed by Rivero et al. [Chemical Engineering Science, 44 (1989), pp. 2881–2897]. It is found that a discretized version of Alt’s model produces solutions similar to the continuous model, even at very coarse discretization. The relationship between the discretized Alt model and the model of Rivero et al. is elucidated, and it is found that a slightly modified version of the Rivero et al. model produces solutions similar to those of the Alt model for systems with one- and two-dimensional attractant gradients, suggesting that the simple and easy-to-use Rivero et al. model (after slight modification) is adequate for modeling bacterial behavior within the parameter ranges investigated. A preliminary investigation of the use of the lattice Boltzmann method for the study of bacterial migration is reported.

The Effect of a Thin Coating on the Scattering of a Time-Harmonic Wave for the Helmholtz Equation

A. Bendali and K. Lemrabet

SIAM J. Appl. Math. 56, pp. 1664-1693 (30 pages) | Cited 20 times

Online Publication Date: July 05, 2006

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A model problem in the scattering of a time-harmonic wave by an obstacle coated with a thin penetrable shell is examined. In previous studies, the contrast coefficients of the thin shell are assumed to tend to infinity in order to compensate for the thickness considered. In this paper, these coefficients are assumed to remain finite. Such a treatment leads to a singular perturbation term that creates a typical difficulty for the asymptotic analysis of the problem with respect to the thickness of the coating. As a result, the asymptotic analysis is essentially based on a suitable handling of the stability of the solution relative to the thickness. As a consequence, it is shown how effective boundary conditions which can be substituted to the thin shell can then be obtained and analyzed in a simple way.

A General Fractal Distribution Function for Rough Surface Profiles

Denis Blackmore and Jack G. Zhou

SIAM J. Appl. Math. 56, pp. 1694-1719 (26 pages) | Cited 6 times

Online Publication Date: July 05, 2006

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Starting with a very general functional description, involving fractal parameters, of the height along a given line on a rough surface, a distribution function for the corresponding surface profile is derived. This distribution is found to differ from Gaussian form by a convergent power series and to be directly dependent on two fractal parameters: the fractal dimension and topothesy. It is shown how the distribution function can be used to determine the effects of varying the fractal parameters on the height of the bearing-area curve (a standard measure of surface roughness). By truncating the series representation for the distribution function for the surface profiles, two approximate models for the height distribution are obtained. These models are shown to compare favorably with experimentally obtained distributions.

The Melnikov Theory for Subharmonics and Their Bifurcations in Forced Oscillations

Kazuyuki Yagasaki

SIAM J. Appl. Math. 56, pp. 1720-1765 (46 pages) | Cited 8 times

Online Publication Date: July 05, 2006

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The subharmonic Melnikov theory for periodic perturbations of planar Hamiltonian systems is improved. An approximation to the associated Poincaré map in action-angle coordinates is explicitly constructed, and existence, stability, and bifurcation theorems for subharmonics are obtained. In particular, simple formulas for determining the stability of subharmonics and invariant circles bifurcating from them at Hopf bifurcations are obtained, and a degenerate resonance case, which was not appropriately treated in previous references, is discussed. Furthermore, the weak nonlinearity case, in which the unperturbed system is linear, is studied. The results are also useful to describe dynamics near the unperturbed centers in strongly nonlinear systems. Several examples are given to illustrate our theory.

Asymptotic Series for Singularly Perturbed Kolmogorov–Fokker–Planck Equations

R. Z. Khasminskii and G. Yin

SIAM J. Appl. Math. 56, pp. 1766-1793 (28 pages) | Cited 9 times

Online Publication Date: July 05, 2006

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We derive limit theorems for the transition densities of diffusion processes and develop asymptotic expansions for solutions of a class of singularly perturbed Kolmogorov–Fokker–Planck equations. The model under consideration can be viewed as a Markov process having two time scales. One of them is a rapidly changing scale, and the other is a slowly varying one. The study is motivated by a wide range of applications involving singularly perturbed Markov processes in manufacturing systems, reliability analysis, queueing networks, statistical physics, population biology, financial economics, and many other related fields. In this work, the asymptotic expansion is constructed explicitly. It is shown that the initial layer terms in the expansion decay at an exponential rate. Error bounds on the remainder terms also are obtained. The validity of the expansion is rigorously justified.

On Transition Densities of Singularly Perturbed Diffusions with Fast and Slow Components

R. Z. Khasminskii and G. Yin

SIAM J. Appl. Math. 56, pp. 1794-1819 (26 pages) | Cited 5 times

Online Publication Date: July 05, 2006

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We derive asymptotic properties of transition densities for singularly perturbed diffusion processes with fast and slow components. Our study focuses on the Kolmogorov–Fokker–Planck equations. The model can be viewed as a diffusion process having two time scales and is motivated by a wide variety of applications involving singularly perturbed Markov processes in manufacturing systems, homogenization, reliability analysis, queueing networks, statistical physics, population biology, financial economics, and many other related fields. By virtue of the methods of matched singular perturbation, asymptotic expansion is constructed for the transition density. The expansion includes both regular part and boundary layer corrections. Detailed justification of the asymptotic expansion is given, and error bounds are also provided.
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