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SIAM J. on Applied Mathematics

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1989

Volume 49, Issue 6, pp. 1567-1850


Singularly Perturbed Integral Equations with Endpoint Boundary Layers

W. E. Olmstead and J. S. Angell

SIAM J. Appl. Math. 49, pp. 1567-1584 (18 pages) | Cited 3 times

Online Publication Date: July 10, 2006

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A methodology is developed for certain singularly perturbed integral equations whose solution exhibits a boundary layer at either one or both endpoints of its domain. The scheme indicates how the width and magnitude of the boundary layer(s) are determined, and gives a construction of an asymptotic solution as well. Application to both linear and nonlinear equations is discussed and illustrated with examples.

The Barotropic Vorticity Equation Under Forcing and Dissipation: Bifurcations of Nonsymmetric Responses and Multiplicity of Solutions

G. Wolansky

SIAM J. Appl. Math. 49, pp. 1585-1607 (23 pages) | Cited 6 times

Online Publication Date: July 10, 2006

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A simplified model of barotropic flow in a bounded domain has been used, and stationary responses due to the balance between dissipation and external forcing have been studied. The forcing, as well as the domain, is assumed to admit some symmetry group. Sufficient conditions are deduced by which a branch of nonsymmetric stationary solutions bifurcate from a symmetric one and those results are applied to a specific problem of zonally symmetric forcing in an open channel.

The Stretching of a Slender, Axisymmetric, Viscous Inclusion–Part I: Asymptotic Analysis

P. Wilmott

SIAM J. Appl. Math. 49, pp. 1608-1616 (9 pages) | Cited 1 time

Online Publication Date: July 10, 2006

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A slender, axisymmetric inclusion of a highly viscous fluid is deformed by an external Stokes flow. The method of matched asymptotic expansions is used, and boundary conditions for the outer flow are found in terms of the cross-sectional area of the inclusion and its local extensional velocity along the axis. This problem is then formulated as an integral equation for which an exact solution is found.

The Stretching of a Slender, Axisymmetric, Viscous Inclusion–Part II: Numerical Solution and Results

A. D. Fitt and P. Wilmott

SIAM J. Appl. Math. 49, pp. 1617-1634 (18 pages) | Cited 2 times

Online Publication Date: July 10, 2006

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In Part I [SIAM J. Appl. Math., 49 (1989), pp. 1608–1616] of this paper, the stretching of a thin, axisymmetric, viscous inclusion was examined via the method of matched asymptotic expansions, resulting in a coupled system consisting of a nonlinear integrodifferential equation and a first-order conservation law. Now the numerical solution of this problem is considered, that relies on the approximation of the unknown functions in the integrodifferential equation in terms of the products of cubic splines coupled to a timestepping solution of the conservation law. Some results are given and comparisons made with the known exact solutions. Consideration is also given to cases where no exact solution is available.Key words. integrodifferential equations, slow viscous flow, numerical solution of equations

Convergence of a Multiple Reflection Method for Calculating Stokes Flow in a Suspension

Jonathan H. C. Luke

SIAM J. Appl. Math. 49, pp. 1635-1651 (17 pages) | Cited 1 time

Online Publication Date: July 10, 2006

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The calculation of solutions of Stokes equations in the complex geometry of a suspension with many particles of arbitrary shape and arrangement is reduced to a sequence of calculations of flows with single particle boundary conditions. The sequence of approximate solutions converge to the solution of the full problem in the energy dissipation norm at an exponential rate. The convergence proof is based on the observation that the flow in a suspension minimizes the rate of energy dissipation over a certain class of flows. Each flow in the approximating sequence minimizes the rate of dissipation over a subclass of these flows containing the previous flow, so the energy dissipation norm is decreasing monotonically. From the latter minimum principle, it follows that each approximate flow is obtained from the previous one through application of one of a finite number of orthogonal projection operators. The properties of the subspaces of flows associated with these projections assure that the sequence converges.

On the Development of Caustics in Shear Flows Over Rigid Walls

I. David Abrahams, Gregory A. Kriegsmann, and Edward L. Reiss

SIAM J. Appl. Math. 49, pp. 1652-1664 (13 pages)

Online Publication Date: July 10, 2006

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In this paper the scattered field produced by a source inside a shear layer of arbitrary profile, flowing above an infinite, rigid wall, is examined. The shear layer is topped by a uniform flow. A representation of the solution is obtained in terms of a pair of functions that satisfy a homogeneous second-order ordinary differential equation, with variable coefficients. The asymptotic form of these functions is obtained in the high-frequency limit. The representation yields, in this limit, a pair of parametric equations whose solutions describe the formation of infinite families of caustics inside the shear layer and downstream of the source. This is in agreement with previous results found by employing ray-theory techniques. Applications are given to the linear profile. The advantages of the present method and its application to other physical problems are discussed.

Analysis, Designs, and Behavior of Dissipative Joints for Coupled Beams

G. Chen, S G. Krantz, D. L. Russell, C. E. Wayne, H. H. West, and M. P. Coleman

SIAM J. Appl. Math. 49, pp. 1665-1693 (29 pages) | Cited 21 times

Online Publication Date: July 10, 2006

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In the construction of modern large flexible space structures, active and passive damping devices are commonly installed at joints of coupled beams to achieve the suppression of vibration. In order to successfully control such dynamic structures, the function and behavior of dissipative joints must be carefully studied.
These dissipative joints are analyzed by first classifying them into types according to the discontinuities of physical variables across a joint. The four important physical variables for beams are displacement $( y )$, rotation $( \theta )$, bending moment $( M )$, and shear $( V )$. Dissipative joints can be classified into the following four types: (1) $M$ and $V$ are continuous, $y$ and $\theta $ are discontinuous; (2) $y$ and $M$ are continuous, $\theta $ and $V$ are discontinuous; (3) $y$ and $\theta $ are continuous, $M$ and $V$ are discontinuous; (4) $\theta $ and $V$ are continuous, $y$ and $M$ are discontinuous,according to the conjugacy of these variables. Mechanical designs have been achieved for all these dissipative joints of the linear passive type.
The spectrum of two identical coupled beams with a linear dissipative joint shows an interesting pattern. It is proven that there are two families of eigenvalues, asymptotically appearing alternately and parallel to the imaginary axis with eigenfrequencies spaced vertically with gap $O( {n^2 } )$.
This interesting spectral behavior has also been observed and studied in experiments conducted at the Modelling, Information Processing and Control Facility of the University of Wisconsin. Numerical simulations using the Legendre spectral method have also confirmed these spectral properties.
All of the aforementioned mechanical designs, experimental, and numerical results are presented in this paper.

Exponential Stability Analysis of a Long Chain of Coupled Vibrating Strings with Dissipative Linkage

Kang-Sheng Liu, Fa-Lun Huang, and Goong Chen

SIAM J. Appl. Math. 49, pp. 1694-1707 (14 pages) | Cited 18 times

Online Publication Date: July 10, 2006

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Consider a long chain of coupled vibrating strings, where a stabilizer is installed at each internal node and perhaps also at a boundary point. The exponential stability of the stabilizers’ arrangement for this large dynamic structure will be determined.
Through a careful transformation of the coupled wave equations for this structure into an equivalent hyperbolic system and analysis of the eigendeterminant, it will be proven that the energy of the system decays uniformly exponentially if there is a stabilizer installed at a boundary point. If the stabilizers are installed only at internal nodes, it will be proven that the energy may decay either uniformly exponentially or nonuniformly, or may not decay at all, depending on the different wave speeds and the stabilizers– arrangement. All possible outcomes have been classified.

Comparison of One- and Three-Dimensional Models for Tracer Evolution in Tissue Cylinders

Leonard Sarason

SIAM J. Appl. Math. 49, pp. 1708-1721 (14 pages)

Online Publication Date: July 10, 2006

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Eigenvalues of the time-reduced operator for a one-dimensional model for diffusion and transport of tracers in tissue cylinders are shown to be limits of some of the eigenvalues of the corresponding three-dimensional operator under an appropriate limiting process. The proof depends on the constructionof approximate eigenfunctions for the three-dimensional operators from eigenfunctions of the one-dimensional operator. Solutions of time-dependent problems with inhomogeneous boundary data are shownto converge to solutions of a one-dimensional time-dependent problem.

Reaction-Diffusion Processes and Evolution to Harmonic Maps

Jacob Rubinstein, Peter Sternberg, and Joseph B. Keller

SIAM J. Appl. Math. 49, pp. 1722-1733 (12 pages) | Cited 14 times

Online Publication Date: July 10, 2006

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Initial-boundary value problems are considered for the reaction-diffusion equation $u_1 = \varepsilon \Delta u - \varepsilon ^{ - 1} f( u )$ with $x$ in a domain $\Omega $ in $R''$ and $u$ in $R'''$. First the asymptotic behavior of $u$ is determined for $\varepsilon $ small when $f( u ) = 0$ on a connected manifold $M$ of stable equilibrium points. It is found that a tends rapidly to $M$, being driven by reaction. Then $u$ evolves slowly by diffusion restricted to$M$. It tends ultimately to a limit that is a harmonic map of $\Omega $ into $M$ Next, the case where $f( u )$ has stable equilibrium points on two manifolds $M_1 $ and $M_2 $ is treated. In this case a front develops in $\Omega $, It separates the regions where $u$ is close to $M_1 $, from the regions where$u$ is close to $M_2 $. For $f( u ) = V_n ( u )$ a boundary layer solution is constructed for $u$ near the front, and the velocity of the front is found to be proportional to the jump in $V$ across it, to leading order in $\varepsilon $. When $V( u )$ has the same value on $M_1 $ and $M_2 $, this term is zero and the front velocity is$\varepsilon $ times its mean curvature. The case of a spherically symmetric potential $V( {| u |} )$ and the case $M = S^1 $ are presented to illustrate the results.

On Strongly Reverse Biased Semiconductor Diodes

Christian Schmeiser

SIAM J. Appl. Math. 49, pp. 1734-1748 (15 pages) | Cited 3 times

Online Publication Date: July 10, 2006

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A singular perturbation analysis of the drift-diffusion model for stationary flow of charge carriers in a semiconductor device is given. An extension of earlier results provides deeper insight into the structure of the solution and the possibility of computing approximations for leakage currents. The solvability of the approximating problem is proved.

Chaotic Behavior in the Josephson Equations with Periodic Force

Zhu-Jun Jing

SIAM J. Appl. Math. 49, pp. 1749-1758 (10 pages) | Cited 4 times

Online Publication Date: July 10, 2006

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The goal of this paper is to prove analytically the existence of chaos, for certain parameter values, in the dynamics of the Josephson equations \[\text{(E)(F)}\qquad\ddot \phi + \sin \phi + k\sin 2\phi + \begin{pmatrix} A( \cos \phi + 2k\cos 2\phi ) \\ \alpha \end{pmatrix} \dot \phi = \beta + B\sin \omega t\]. Combining previous results with new results, a more complete description of dynamics of the Josephson equations is obtained.

Limit Cycles in a Chemostat-Related Model

Yang Kuang

SIAM J. Appl. Math. 49, pp. 1759-1767 (9 pages) | Cited 14 times

Online Publication Date: July 10, 2006

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A Kolmogorov predator-prey system arising from the asymptotic analysis of a food chain in a chemostat is considered. This chemostat model has been investigated extensively by Butler, Hsu, and Waltman [J. Math. Biol., 17 (1983), pp.133–151]. Throughout this paper, it is assumed that there is a unique unstable interior equilibrium in a triangular region of attraction. In the first part of this paper, the goal is to locate the limit cycles. In the second part, it is shown that there is a range in the parameter space that guarantees the uniqueness of any limit cycle of this system.

Asymptotic Behavior for a Competitive Model with Genetic Variation

Eduardo M. Munoz and James F. Selgrade

SIAM J. Appl. Math. 49, pp. 1768-1778 (11 pages) | Cited 1 time

Online Publication Date: July 10, 2006

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This paper is a study of a three-dimensional system of autonomous ordinary differential equations that models two competing populations with genetic variation in one population. The genetically varying population is diploid and diallelic. The per capita growth rate of each genotype is assumed to be a linear function of the population densities (Lotka–Volterra competition). On one allele frequency fixation plane the competition is assumed to exhibit stable coexistence and on the other, mutual exclusion. It is shown that the three-dimensional dynamics depends on the sign of a quadratic combination $K$ of the competition parameters. Every solution orbit converges to an equilibrium solution regardless of the value of $K$. The asymptotic behavior for $K$ positive and negative is analyzed, and the existence of heteroclinic orbits is established. The degenerate transition behavior when $K$ equals zero is also discussed.

A Neural Network Modeled by an Adaptive Lotka-Volterra System

V. W. Noonburg

SIAM J. Appl. Math. 49, pp. 1779-1792 (14 pages) | Cited 12 times

Online Publication Date: July 10, 2006

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In this paper the asymptotic behavior of an $n$-dimensional Lotka–Volterra system with nonconstant coefficients that models the dynamics of an adaptive cellular network is analyzed. Each cell in the network is assumed to be connected to other cells in such a way that its output (level of activity) eitherinhibits or excites the growth of activity in the other cells. The “weight” of the connection between any two cells varies over time in proportion to the average level of interaction between the two cells over the recent past. If all of the connections are inhibitory, it is shown that the network tends to produce a spontaneous and interesting classification of its inputs.

An Asymptotic Solution to a Two-Dimensional Exit Problem Arising in Population Dynamics

H. Roozen

SIAM J. Appl. Math. 49, pp. 1793-1810 (18 pages) | Cited 5 times

Online Publication Date: July 10, 2006

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A study is made of a two-dimensional stochastic system with small stochastic fluctuations arising in population biology. At the boundary of the state space the diffusion matrix becomes singular. By an asymptotic analysis, expressions are derived that determine the probability of exit at each of the two boundaries and the expectation and variance of the exit time. These expressions contain constants that can be computed numerically.

First-Order Dynamics Driven by Rapid Markovian Jumps

M. M. Kłosek, B. J. Matkowsky, and Z. Schuss

SIAM J. Appl. Math. 49, pp. 1811-1833 (23 pages) | Cited 4 times

Online Publication Date: July 10, 2006

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The effect of rapid random jumps on the behavior of deterministically stable dynamics is considered. The random driving force (noise) is taken to be a state-dependent Markov jump process with either finite or infinite state space. While the jump rate is high $(O( 1/\varepsilon ) )$, the jump size is assumed to be$O( 1 )$. For a particle that is deterministically confined to a potential well, the stationary joint probability density function of the state and noise variables, the probability distribution of the exit points, and the mean first passage time from the well are computed. In contrast to the case of white noise, where these quantities depend on a barrier height determined solely by the potential, here they depend on an effective barrier height determined by both the potential and the noise process. The method used here is to introduce the small parameter $\varepsilon $, where $1/\varepsilon $ is a measure of the rapid jump rate, and to employ singular perturbation methods to solve the forward and backward master equations, for the above-mentioned quantities.

The Structure and Analysis of Spherical Time-Dependent Processes

D. R. Jensen and R V. Foutz

SIAM J. Appl. Math. 49, pp. 1834-1844 (11 pages) | Cited 3 times

Online Publication Date: July 10, 2006

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The class of all spherical processes is investigated irrespective of moments. Properties of these processes are studied including their structure, representations, and linear and nonlinear spaces. A basic representation is $\{ X( t ) = A^{1/2} Y( t );t \in T \}$, where $A$is nonnegative and $Y( t )$ is Gaussian. Time series analysesnot requiring moments are set forth in both the time and frequency domains, including spectral analysis. Solutions of forecasting problems are found to be linear when the process is spherical, with the property that errors of the optimal forecasts are stochasticalay most concentrated about zero. These developments apply to the $\alpha $-sub-Gaussian processes, and are advanced for use with certain heavy-tailed processes arising in economics, engineering, and the physical sciences.

A simple Duality Proof for Quadratically Constrained Entropy Functionals and Extension to Convex Constraints

Marc Teboulle

SIAM J. Appl. Math. 49, pp. 1845-1850 (6 pages) | Cited 2 times

Online Publication Date: July 10, 2006

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In this paper, a simple proof of duality results is presented that was recently derived by Zhang and Brockett [SIAM J. Appl. Math., 47 (1987), pp. 871–885] for the problem of minimizing the Kullback–Liebler discrimination information measure subject to quadratic and linear inequality constraints. The proof is a direct and simple application of Lagrangian Duality Theory and is extended to a more general class of entropy optimization problems with composite convex constraints.
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