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1980

Volume 39, Issue 3, pp. 391-548


Reactive-Diffusive System with Arrhenius Kinetics: The Robin Problem

A. K. Kapila and B. J. Matkowsky

SIAM J. Appl. Math. 39, pp. 391-401 (11 pages) | Cited 7 times

Online Publication Date: July 12, 2006

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For a permeable catalyst pellet, the reactive-diffusive problem with first-order Arrhenius kinetics is studied. Steady solutions under Robin boundary conditions are derived for the cylindrical geometry in the limit of large activation energy. The response of the system exhibits three-fold and five-fold multiplicities as well as closed loops

A Sobolev Space Analysis of Picture Reconstruction

Frank Natterer

SIAM J. Appl. Math. 39, pp. 402-411 (10 pages) | Cited 11 times

Online Publication Date: July 12, 2006

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The problem of recovering the density function of a plane picture from its line integrals is considered. If a picture of finite extent whose density function belongs to the Sobolev space$H^\alpha $ of order $\alpha > \frac{1} {2}$ is to be reconstructed from $n$ line integrals with a root mean square error $\varepsilon $, then the root mean square error of the reconstruction is of the order $(\varepsilon ^{\alpha /(\alpha + 1/2)} + n^{ - \alpha / 2} )\| y \|_{H^\alpha }$ at least. We give a reconstruction method which achieves this optimal error bound.

The Thermal Explosion Confined by a Constant Temperature Boundary:I—The Induction Period Solution

D. R. Kassoy and Justin Poland

SIAM J. Appl. Math. 39, pp. 412-430 (19 pages) | Cited 17 times

Online Publication Date: July 12, 2006

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The time history of a spatially varying thermal explosion in a vessel with constant wall temperature is considered. A one-step irreversible, high activation energy reaction of the Arrhenius type is assumed to occur in a rigid, nondiffusing, combustible material. The induction period equations are solved numerically for a system confined to a slot-like region. A stiff-equation integrator is employed to delineate the nature of the thermal runaway process. A well defined hot spot is observed to form in the vicinity of the symmetry line. A precise description of the hot spot development is given in terms of an asymptotic theory valid close to the explosion time. The solution is constructed in terms of a slowly varying conduction-controlled outer region surrounding a much smaller zone in which the relatively rapid chemical kinetics determine how the hot spot therein develops. This analytical solution describes the final phase of the spatially varying induction period thermal runaway process, which cannot be obtained by numerical means alone. It is found that the dimension of the hot spot depends upon the square root of the product of the material thermal diffusivity and the time increment from the explosion time value. The hot spot development, which is kinetically controlled, is entirely independent of the vessel size.

Asymptotic Normality of Nonparametric Estimators of Derivatives of a Average of $\mu $-Densities

R. S. Singh

SIAM J. Appl. Math. 39, pp. 431-439 (9 pages)

Online Publication Date: July 12, 2006

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This paper investigates the variance-covariance and the distributional properties of nonparametric estimators of derivatives of average of $\mu $-densities proposed and discussed in Singh (1978). Let $X_j ,1\leqq j\leqq n$, be independent random variables with densities $f_i$ with respect to a $\sigma $-finite measure $\mu $ on the real line. For an integer $\nu \geqq 0$, Singh (SIAM J. Appl. Math., 35 (1978), pp. 637–649), gave a class of estimators $\hat \bar f^{(\nu )} $ of $\bar f^{(\nu )} = n^{ - 1} \sum _{i = 1}^n f_i^{(\nu )} $ based on $X_1 , \cdots X,X_n $, and pointed out various desirable properties and applications of these estimators. In this paper it is shown that the variance-covariance matrix of the vector ${\bf {\nu}} _n = (\hat \bar f^{(\nu )} (x_1 ), \cdots ,\hat \bar f^{(\nu )} (x_m ))$, where the $x_i $’s are distinct, is asymptotically equivalent to a diagonal matrix ${\bf{D}}_n $. The exact value of ${\bf{D}}_n $ is obtained. It is further shown that the distributions of${\bf {\nu}} _n $, with various standardizations, are asymptotically multivariate normal. A rate of closeness of the distribution of $\hat \bar f^{(\nu )} $ with the normal distribution is also obtained.

A new Asymptotic Method for Jump Phenomena

Edward L. Reiss

SIAM J. Appl. Math. 39, pp. 440-455 (16 pages) | Cited 8 times

Online Publication Date: July 12, 2006

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A variety of physical phenomena are concerned with rapid and sudden transitions. Typical examples of this are: the snap buckling of elastic shells; combustion and explosions; population outbreaks; transitions to and from the ice ages; and earthquakes. All these problems are characterized mathematically, as a small disturbance causing a large amplitude response. Thus, standard asymptotic and perturbation methods, such as the Poincaré–Linstedt perturbation method and the multi-time method, are not applicable to these problems. In these methods, small amplitude responses to small disturbances are studied. In this paper, a newmethod is presented for analyzing jump phenomena. It consists of a rational function representation of the response. Three applications of the method are given, to the snap buckling of an elastic arch, to a problem ofpopulation dynamics and to a simple combustion problem.

A second Order Procedure for One-Dimensional Velocity Inversion

Samuel H. Gray

SIAM J. Appl. Math. 39, pp. 456-462 (7 pages) | Cited 9 times

Online Publication Date: July 12, 2006

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A technique for one-dimensional velocity inversion is presented which represents a significant improvement over a currently used method due to Cohen and Bleistein, (SIAM J. App!. Math., 32 (1977), pp. 784–799). This technique yields corrections in both the size and the location of a velocity variation. It is shown that, when the velocity variation is of order $\varepsilon \ll 1$, the technique is correct through $O( \varepsilon ^2 )$. Also, numerical results which illustrate the improvements are discussed.

Flickering and ThermalFlicker Waves on Catalytic Wires and Gauzes and in Chemical Reactors

Donald S. Cohen and S. Rosenblat

SIAM J. Appl. Math. 39, pp. 463-474 (12 pages)

Online Publication Date: July 12, 2006

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We present a theoretical investigation of flickering (local temperature fluctuations) and thermal flicker waves in chemical reactors and on catalytic wires and gauzes. It is shown that these phenomena can be caused by harmonic coupling between the chemical kinetics and oscillations in the ambient field. The oscillations can be modeled either by fluctuations in the heat and mass transfer coefficients, or by fluctuations in the gas temperature and concentration.

Stability of Multi-Machine Power Systems with Nontrivial Transfer Conductances

Sherwin J. Skar

SIAM J. Appl. Math. 39, pp. 475-491 (17 pages) | Cited 2 times

Online Publication Date: July 12, 2006

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The local stability of second order vector differential equations with linear damping is examined by linearization. It is shown that without damping such systems are stable only if the eigenvalues of a certain matrix are real and nonpositive. Sufficient conditions for the asymptotic stability of the damped system are developed. The results are applied to power systems with nontrivial transfer conductances. An important consequence is that unstable equilibrium solutions for the power system swing equations may exist even though the rotor angles are less than $90^ \circ $ out of phase, that is, even though $|\delta _i - \delta _j - \alpha _{ij} | <' \pi/ 2$ for all rotor angle pairs$\delta _i ,\delta _j $ and all phases $( \alpha _{ij} + \pi/2$ in the transfer admittance matrix. It is also shown that there can be at most one equilibrium solution (up to a constant phase added to all rotor angles) of the swing equations with $|\delta _i - \delta _j - \alpha _{ij} | < \pi/ 2$ for all $\delta _i $, $\delta _j $, $\alpha _{ij} $.

Motion-Induced Singularities in Power Spectra Associated with Ocean Gravity-Wave Fluctuations

E. Y. Harper

SIAM J. Appl. Math. 39, pp. 492-511 (20 pages)

Online Publication Date: July 12, 2006

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We consider an observer with a measuring device having a scalar output that is linearly related to the random motion of ocean gravity waves. The device could measure waveheight, a component of flow velocity, etc. We study the power-spectral density of the time record obtained when the observer moves at constant speed in a horizontal plane. When the observer is at rest the spectrum is wideband, or incoherent. However when the observer moves, he may move with the envelope of a certain packet of waves, thereby introducing a strong coherence. This coherence manifests itself as a singularity in the power-spectral density of the time record. The location of the singularity in the frequency domain is predicted by the method of stationary phase, but the nature of the singularity is not. It is shown that for the case of swell the spectrum has a square-root singularity on the left, and a finite limit from the right, at the singular point. This peculiar behaviour is demonstrated experimentally. For the case of a wind-driven sea the singularity is logarithmic and unsymmetric about the singular point. For this case the location of the singularity in the frequency domain depends only on the observer’s speed, $U$, and is given by $g/8\pi U$ (cycles/unit time) where $g$ is the acceleration of gravity.

Similarity Solutions for Reactive Shock Hydrodynamics

J. David Logan and José de Jesús Pérez

SIAM J. Appl. Math. 39, pp. 512-527 (16 pages) | Cited 10 times

Online Publication Date: July 12, 2006

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It is shown how group-theoretic methods, or similarity methods, can be applied to determine the class of self-similar solutions for a one-dimensional, time-dependent problem in shock hydrodynamics,with a chemical reaction taking place behind the shock. The functional form of all reaction rates depending on the state variables is characterized under which the general system of differential equations and boundaryconditions admit self-similar solutions, It is shown that a subclass of these solutions can model certain processes in detonation physics.

Waves in Excitable Media

J. P. Keener

SIAM J. Appl. Math. 39, pp. 528-548 (21 pages) | Cited 42 times

Online Publication Date: July 12, 2006

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A general class of two-component reaction-diffusion systems with excitable dynamics is studied by means of singular perturbation theory. It is shown how stable traveling pulses and periodic wavetrains in one spatial dimension evolve from initial data. This information is applied to two-dimensional regions for which it is shown that steady rotating structures (spirals) exist.
The perturbation results are also used to show that a one-dimensional semi-infinite medium exhibits hysteresis when used as a periodic signaling device. Finally, other nonexcitable dynamics are analyzed, and their stable one-dimensional structures listed.
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