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2009

Volume 69, Issue 6, pp. 1501-1804


The Riemann Problem for a Nonisentropic Fluid in a Nozzle with Discontinuous Cross-Sectional Area

Mai Duc Thanh

SIAM J. Appl. Math. 69, pp. 1501-1519 (19 pages)

Online Publication Date: March 04, 2009

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We present a full investigation of the Riemann problem for a nonisentropic polytropic fluid in a nozzle with piecewise constant cross-section. First, we introduce the concept of elementary waves which turn out to make up Riemann solutions. Second, we study a procedure to select admissible stationary waves relying on the monotone criterion. By projecting all the wave curves in the $(p,u)$-plane, we construct Riemann solutions. Existence of Riemann solutions can be obtained for large initial data. Furthermore, we establish the uniqueness of Riemann solutions in strictly hyperbolic domains. Our argument can lead to estimate regions where the Riemann problem admits a unique solution.

Scattering of Surface Water Waves by a Floating Elastic Plate in Two Dimensions

Rupanwita Gayen and B. N. Mandal

SIAM J. Appl. Math. 69, pp. 1520-1541 (22 pages)

Online Publication Date: March 04, 2009

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A new method is developed to study the problem of water wave scattering by a thin elastic plate of arbitrary width floating in deep water assuming linear theory. Using Havelock's expansion of water wave potentials, the boundary value problem describing the potentials is reduced to solving singular integral equations of Carleman type. With the introduction of some integral operators the problem is further reduced to twelve Fredholm integral equations of second kind with regular kernels, and the numerical solutions of these integral equations are used to compute the reflection and transmission coefficients. The numerical estimates for the reflection coefficient are presented in a number of figures given varying different physical parameters. It is shown that the present analysis produces known results for the reflection coefficient.

Interactions of Elementary Waves for the Aw–Rascle Model

Meina Sun

SIAM J. Appl. Math. 69, pp. 1542-1558 (17 pages)

Online Publication Date: March 04, 2009

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In this paper, we study the interactions of elementary waves for the traffic flow model proposed by Aw and Rascle in [SIAM J. Appl. Math., 60 (2000), pp. 916–938]. The solutions are obtained constructively when the initial data are three piecewise constant states. In particular, a new wave $SJ$ in which a shock wave $S$ and a contact discontinuity $J$ coincide with each other is obtained during the process of interaction. Moreover, by studying the limits of the solutions as the perturbed parameter $\varepsilon$ tends to zero, it can be found that the Riemann solutions are stable for such perturbations with the initial data.

Drift-Diffusion Past a Circle: Shadow Region Asymptotics

Sean Lynch and Charles Knessl

SIAM J. Appl. Math. 69, pp. 1559-1579 (21 pages)

Online Publication Date: March 11, 2009

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We consider the steady state concentration of some diffusing substance, subject to a uniform drift field, past a circular obstacle. We obtain some exact representations of the concentration profile of the substance exterior to the obstacle. These representations are particularly useful for studying the solution in ranges of space where the concentration is very small (the “shadow” regions). We assume then that the drift dominates diffusion and obtain various asymptotic expansions in the shadow regions.

New Conditions on the Existence and Stability of Periodic Solution in Lotka–Volterra's Population System

Yonghui Xia and Maoan Han

SIAM J. Appl. Math. 69, pp. 1580-1597 (18 pages)

Online Publication Date: March 11, 2009

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In this paper, we revisit the famous periodic Lotka–Volterra competitive system. Some new and interesting sufficient conditions are obtained to guarantee the existence and global asymptotic stability of the periodic solution in the Lotka–Volterra competitive system. Our method is based on Mawhin's coincidence degree, matrix's spectral theory, and some new estimation techniques for the priori bounds of unknown solutions to the equation $Lx=\lambda Nx$. Due to this new method, our new results are much different from the known results in the previous literature. Finally, some examples and their simulations show the feasibility of our results.

Bounds for the Effective Stress of Classical and Strain Gradient Plastic Composites

Viet Ha Hoang

SIAM J. Appl. Math. 69, pp. 1598-1617 (20 pages)

Online Publication Date: March 18, 2009

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Given an average strain, rigorous bounds are established for the stress in a deformation of a plastic composite material, which follows a power law. The deformation theory of strain gradient plasticity, which introduces an internal material length scale, is used. It falls into the classical deformation theory of elasto-plasticity when this length scale equals zero. The method employs the idea by Milton and Serkov [J. Mech. Phys. Solids, 48 (2000), pp. 1259–1324] and other techniques for bounding effective energy. We derive two stress bounds which closely relate to the Reuss lower bound and the Hashin–Shtrikman upper bound for the energy. We then study numerically the dependence on the internal length scale of the magnitude of the stress and the region in the stress space determined by these two bounds in which the macro stress must lie. The results confirm the prediction made by Fleck and Willis [J. Mech. Phys. Solids, 52 (2004), pp. 1855–1888] for the macroscopic uniaxial response by differentiating their energy bounds.

Effective Equations for Localization and Shear Band Formation

Theodoros Katsaounis and Athanasios E. Tzavaras

SIAM J. Appl. Math. 69, pp. 1618-1643 (26 pages) | Cited 1 time

Online Publication Date: March 18, 2009

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We develop a quantitative criterion determining the onset of localization and shear band formation at high strain-rate deformations of metals. We introduce an asymptotic procedure motivated by the theory of relaxation and the Chapman–Enskog expansion and derive an effective equation for the evolution of the strain rate, consisting of a second order nonlinear diffusion regularized by fourth order effects and with parameters determined by the degree of thermal softening, strain hardening, and strain-rate sensitivity. The nonlinear diffusion equation changes type across a threshold in the parameter space from forward parabolic to backward parabolic, what highlights the stable and unstable parameter regimes. The fourth order effects play a regularizing role in the unstable region of the parameter range.

A Nonautonomous Juvenile-Adult Model: Well-Posedness and Long-Time Behavior via a Comparison Principle

Azmy S. Ackleh and Keng Deng

SIAM J. Appl. Math. 69, pp. 1644-1661 (18 pages)

Online Publication Date: March 27, 2009

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A nonautonomous nonlinear continuous juvenile-adult model where juveniles and adults depend on different resources is developed. It is assumed that juveniles are structured by age, while adults are structured by size. Existence-uniqueness results are proved using the monotone method based on a comparison principle established in this paper. Conditions on the model parameters that lead to extinction or persistence of the population are obtained via the upper-lower solution technique.

Detecting Inclusions in Electrical Impedance Tomography Without Reference Measurements

Bastian Harrach and Jin Keun Seo

SIAM J. Appl. Math. 69, pp. 1662-1681 (20 pages) | Cited 2 times

Online Publication Date: March 27, 2009

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We develop a new variant of the factorization method that can be used to detect inclusions in electrical impedance tomography from either absolute current-to-voltage measurements at a single, nonzero frequency or from frequency-difference measurements. This eliminates the need for numerically simulated reference measurements at an inclusion-free body and thus greatly improves the method's robustness against forward modeling errors, e.g., in the assumed body's shape.

Monte Carlo Malliavin Computation of the Sensitivities of Solutions of SPDEs

René Carmona and Lixin Wang

SIAM J. Appl. Math. 69, pp. 1682-1711 (30 pages)

Online Publication Date: April 01, 2009

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This paper deals with an application of the Malliavin calculus to a stochastic partial differential equation of the Schrödinger type. This equation appears as the major building block in the analysis of the focusing properties of time-reversed waves in a random medium in the asymptotic regime where the parabolic approximation is valid. We consider the sensitivities of the solutions with respect to several parameters, and we provide closed form formulae in terms of Skorohod integrals with respect to an infinite dimensional Wiener process. We construct finite dimensional approximation schemes for these integrals. These schemes are based on a sieve of Wiener chaos expansions mixed with Galerkin approximations in a natural Fourier basis. In two space dimensions, our computational algorithm seems to perform better than those we found in the literature. Moreover, because it avoids finite difference methods, it can be implemented in three space dimensions without much ado.

On the Phase Diagram for Microphase Separation of Diblock Copolymers: An Approach via a Nonlocal Cahn–Hilliard Functional

Rustum Choksi, Mark A. Peletier, and J. F. Williams

SIAM J. Appl. Math. 69, pp. 1712-1738 (27 pages) | Cited 5 times

Online Publication Date: April 01, 2009

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We consider analytical and numerical aspects of the phase diagram for microphase separation of diblock copolymers. Our approach is variational and is based upon a density functional theory which entails minimization of a nonlocal Cahn–Hilliard functional. Based upon two parameters which characterize the phase diagram, we give a preliminary analysis of the phase plane. That is, we divide the plane into regions wherein a combination of analysis and numerics is used to describe minimizers. In particular we identify a regime wherein the uniform (disordered state) is the unique global minimizer; a regime wherein the constant state is linearly unstable and where numerical simulations are currently the only tool for characterizing the phase geometry; and a regime of small volume fraction wherein we conjecture that small well-separated approximately spherical objects are the unique global minimizer. For this last regime, we present an asymptotic analysis from the point of view of the energetics which will be complemented by rigorous $\Gamma$-convergence results to appear in a subsequent article. For all regimes, we present numerical simulations to support and expand on our findings.

Nonlinear Stability for Diffusion Models in Biology

Giuseppe Mulone and Brian Straughan

SIAM J. Appl. Math. 69, pp. 1739-1758 (20 pages)

Online Publication Date: April 09, 2009

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The reduction method for studying optimal nonlinear stability of constant solutions to some ecological models with diffusion, which include the Cantrell–Cosner and the May–Leonard systems, is given. A new canonical energy (Lyapunov function) is introduced, and it is proved that the regions of linear and nonlinear stability coincide with a known radius of attraction for the initial data. Attention is focused on a May–Leonard system with circular symmetry, an asymmetric May–Leonard system with diffusion, and a system for aggregation of glia in the brain.

A Nondestructive Evaluation Method for Concrete Voids: Frequency Differential Electrical Impedance Scanning

Sungwhan Kim, Jin Keun Seo, and Taeyoung Ha

SIAM J. Appl. Math. 69, pp. 1759-1771 (13 pages)

Online Publication Date: April 09, 2009

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This paper proposes a new nondestructive evaluation method for detecting cracks, voids, and other hidden defects inside concrete structures, called “frequency differential electrical impedance scanning (fdEIS).” The primary benefit of fdEIS over the conventional nondestructive methods is that it is possible to determine the thickness of the voids. In fdEIS, we inject a sequence of electrical currents with various frequencies through the tested concrete wall by applying a sinusoidal voltage difference between a surface electrode and a scan probe, which are placed on opposite surfaces of the wall. Through the probe, we measure the derivative $\frac{d}{d\omega}g_{\omega}$ of exit currents (Neumann data) with respect to the angular frequency variable $\omega$. We find the fundamental concept in fdEIS relating the thickness of the voids to $\frac{d}{d\omega}g_{\omega}$ and derive an approximation formula for estimating the thickness of the voids. We demonstrate the performance of our method in numerical simulations.

Mutation-Selection Balance with Recombination: Convergence to Equilibrium for Polynomial Selection Costs

Aubrey Clayton and Steven N. Evans

SIAM J. Appl. Math. 69, pp. 1772-1792 (21 pages)

Online Publication Date: April 09, 2009

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We study a continuous-time dynamical system that models the evolving distribution of genotypes in an infinite population where genomes may have infinitely many or even a continuum of loci, mutations accumulate along lineages without back-mutation, added mutations reduce fitness, and recombination occurs on a faster time scale than mutation and selection. Some features of the model, such as existence and uniqueness of solutions and convergence to the dynamical system of an approximating sequence of discrete-time models, were presented in earlier work by Evans, Steinsaltz, and Wachter [A Mutation-Selection Model for General Genotypes with Recombination, 2006, preprint available online at http://arxiv.org/abs/q-bio/0609046] for quite general selective costs. Here we study a special case where the selective cost of a genotype with a given accumulation of ancestral mutations from a wild type ancestor is a sum of costs attributable to each individual mutation plus successive interaction contributions from each $k$-tuple of mutations for $k$ up to some finite “degree.” Using ideas from complex chemical reaction networks and a novel Lyapunov function, we establish that the phenomenon of mutation-selection balance occurs for such selection costs under mild conditions. That is, we show that the dynamical system has a unique equilibrium and that it converges to this equilibrium from all initial conditions.

A Free Boundary Problem of a Real Option Model

C. Atkinson

SIAM J. Appl. Math. 69, pp. 1793-1804 (12 pages)

Online Publication Date: April 09, 2009

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An analytical solution of a real option problem is presented. Explicit solutions are presented for each side of a critical asset boundary, and to find this unknown boundary, a nonlinear Volterra integral equation is derived. Asymptotic solutions of this equation are found, and details of a numerical method to solve this equation are presented.
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