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2010

Volume 70, Issue 8, pp. 2797-3362


Numerical Solution of the Small Dispersion Limit of the Camassa–Holm and Whitham Equations and Multiscale Expansions

S. Abenda, T. Grava, and C. Klein

SIAM J. Appl. Math. 70, pp. 2797-2821 (25 pages)

Online Publication Date: August 26, 2010

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The small dispersion limit of solutions to the Camassa–Holm (CH) equation is characterized by the appearance of a zone of rapid modulated oscillations. An asymptotic description of these oscillations is given, for short times, by the one-phase solution to the CH equation, where the branch points of the corresponding elliptic curve depend on the physical coordinates via the Whitham equations. We present a conjecture for the phase of the asymptotic solution. A numerical study of this limit for smooth hump-like initial data provides strong evidence for the validity of this conjecture. We present a quantitative numerical comparison between the solution to the CH equation and the asymptotic solution. The dependence on the small dispersion parameter $\epsilon$ is studied in the interior and at the boundaries of the Whitham zone. In the interior of the zone, the difference between the solution to the CH equation and the asymptotic solution is of the order $\epsilon$, at the trailing edge of the order $\sqrt{\epsilon}$, and at the leading edge of the order $\epsilon^{1/3}$. For the latter we present a multiscale expansion which describes the amplitude of the oscillations in terms of the Hastings–McLeod solution of the Painlevé II equation. We show numerically that this multiscale solution provides an enhanced asymptotic description near the leading edge.

Turing Patterns and Wavefronts for Reaction-Diffusion Systems in an Infinite Channel

Chao-Nien Chen, Shin-Ichiro Ei, and Ya-Ping Lin

SIAM J. Appl. Math. 70, pp. 2822-2843 (22 pages)

Online Publication Date: September 16, 2010

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This paper deals with reaction-diffusion systems on an infinitely long strip in $\boldsymbol{R}^2$. Through a pitchfork bifurcation, spatially heterogeneous patterns exist in a neighborhood of Turing instability. Motivated by the works of Kondo and Asai, we study wavefront solution heteroclinic to Turing patterns. It will be seen that the dynamics of a wavefront can be approximated by a fourth order equation of buckling type.

An Efficient $\mathcal{Q}$-Tensor-Based Algorithm for Liquid Crystal Alignment away from Defects

K. R. Daly, G. D'Alessandro, and M. Kaczmarek

SIAM J. Appl. Math. 70, pp. 2844-2860 (17 pages)

Online Publication Date: September 16, 2010

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We develop a fast and accurate approximation of the normally stiff equations which minimize the Landau–de Gennes free energy of a nematic liquid crystal. The resulting equations are suitable for all configurations in which defects are not present, making them ideal for device simulation. Specifically they offer an increase in computational efficiency by a factor of 100 while maintaining an error of order $(10^{-4})$ when compared to the full stiff equations. As this approximation is based on a $\mathcal{Q}$-tensor formalism, the sign reversal symmetry of the liquid crystal is respected. In this paper we derive these equations for a simple two-dimensional case, where the director is restricted to a plane, and also for the full three-dimensional case. An approximation of the error in the perturbation scheme is derived in terms of the first order correction, and a comparison to the full stiff equations is given.

Wave Propagation in Multicomponent Flow Models

Tore FlÅtten, Alexandre Morin, and Svend Tollak Munkejord

SIAM J. Appl. Math. 70, pp. 2861-2882 (22 pages) | Cited 1 time

Online Publication Date: September 23, 2010

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We consider systems of hyperbolic balance laws governing flows of an arbitrary number of components equipped with general equations of state. The components are assumed to be immiscible. We compare two such models: one in which thermal equilibrium is attained through a relaxation procedure, and a fully relaxed model in which equal temperatures are instantaneously imposed. We describe how the relaxation procedure may be made consistent with the second law of thermodynamics. Exact wave velocities for both models are obtained and compared. In particular, our formulation directly proves a general subcharacteristic condition: For an arbitrary number of components and thermodynamically stable equations of state, the mixture sonic velocity of the relaxed system can never exceed the sonic velocity of the relaxation system.

A Posteriori Error Estimate and Convergence Analysis for Conductivity Image Reconstruction in MREIT

Jijun Liu, Jinkeun Seo, and EungJe Woo

SIAM J. Appl. Math. 70, pp. 2883-2903 (21 pages)

Online Publication Date: September 23, 2010

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Magnetic resonance electrical impedance tomography (MREIT) takes advantage of internal information to solve its nonlinear inverse problem of recovering a conductivity distribution inside an imaging object. When we inject current into the imaging object, there occurs a distribution of internal magnetic flux density $\mathbf{B}=(B_x,B_y,B_z)$. In MREIT we utilize a magnetic resonance imaging scanner with its main magnetic field in the $z$ direction to acquire $B_z$ data. The harmonic $B_z$ algorithm was invented in 2001 to reconstruct cross-sectional conductivity images from $B_z$ data sets subject to multiple injection currents. Utilizing internal $B_z$ data, it overcomes the inherent ill posedness in electrical impedance tomography. We can set up the inverse problem in MREIT as a coefficient identification problem of finding $\sigma$ appearing in $\nabla\cdot(\sigma\nabla u)=0$ from acquired data of the $z$ component of $\nabla\times(\sigma\nabla u)$. The harmonic $B_z$ algorithm has shown an excellent performance in numerical simulations and phantom experiments. Experimental MREIT studies have now reached the stage of in vivo animal and human imaging experiments. However, there is not much work on rigorous mathematical theories of error estimate and convergence analysis yet. The purpose of this paper is to provide a posteriori error estimate in MREIT conductivity image reconstructions. This enables us to evaluate a difference between a reconstructed conductivity image and the unknown true conductivity image. We also describe a convergence analysis of the harmonic $B_z$ algorithm, which improves the previous result of [J. J. Liu, J. K. Seo, M. Sini, and E. J. Woo, SIAM J. Appl. Math., 67 (2007), pp. 1259–1282] in the sense that assumptions on the conductivity are much relaxed.

A Surface Phase Field Model for Two-Phase Biological Membranes

Charles M. Elliott and Björn Stinner

SIAM J. Appl. Math. 70, pp. 2904-2928 (25 pages) | Cited 1 time

Online Publication Date: September 23, 2010

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We study vesicles formed by lipid bilayers that are governed by an elastic bending energy and on which the lipids laterally separate forming two different phases. The energy laden phase interfaces may be modeled as curves on the hypersurface representing the membrane (sharp interface model). The phase field methodology is another powerful tool to model such phase separation phenomena where thin layers describe the interfaces (diffuse interface model). For both approaches we characterize equilibrium shapes in terms of the Euler–Lagrange equations of the total membrane energy subject to constraints on the area of the two phases and the volume. We further show by matching appropriate formal asymptotic expansions that the sharp interface model is obtained from the diffuse interface model as the thickness of the phase interface tends to zero. The essential challenge lies in the fact that also the geometry of the membrane is unknown and depends on a small parameter representing the interface thickness.

Derivation of the Wenzel and Cassie Equations from a Phase Field Model for Two Phase Flow on Rough Surface

Xianmin Xu and Xiaoping Wang

SIAM J. Appl. Math. 70, pp. 2929-2941 (13 pages) | Cited 1 time

Online Publication Date: October 05, 2010

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In this paper, the equilibrium behavior of an immiscible two phase fluid on a rough surface is studied from a phase field equation derived from minimizing the total free energy of the system. When the size of the roughness becomes small, we derive the effective boundary condition for the equation by the multiple scale expansion homogenization technique. The Wenzel and Cassie equations for the apparent contact angles on the rough surfaces are then derived from the effective boundary condition. The homogenization results are proved rigorously by the $\Gamma$-convergence theory.

Single Phytoplankton Species Growth with Light and Advection in a Water Column

Sze-Bi Hsu and Yuan Lou

SIAM J. Appl. Math. 70, pp. 2942-2974 (33 pages)

Online Publication Date: October 05, 2010

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We investigate a nonlocal reaction-diffusion-advection equation which models the growth of a single phytoplankton species in a water column where the species depends solely on light for its metabolism. We study the combined effect of death rate, sinking or buoyant coefficient, water column depth, and vertical turbulent diffusion rate on the persistence of a single phytoplankton species. Under a general reproductive rate which is an increasing function of light intensity, we establish the existence of a critical death rate; i.e., the phytoplankton survives if and only if its death rate is less than the critical death rate. The critical death rate is a strictly monotone decreasing function of the sinking or buoyant coefficient and water column depth, and it is also a strictly monotone decreasing function of the turbulent diffusion rate for buoyant species. In contrast to the critical death rate, a critical sinking or buoyant velocity, a critical water column depth, and a critical turbulent diffusion rate may or may not exist. For instance, if the death rate is suitably small with respect to the water column depth, the phytoplankton can persist for any sinking or buoyant velocity; i.e., there is no critical sinking or buoyant velocity under such a situation. We further show that a critical water column depth, a critical sinking or buoyant velocity, and a critical turbulent diffusion rate for both buoyant species and species with large sinking rates can exist for some intermediate range of phytoplankton death rates and, whenever they exist, are always unique. In strong contrast, there may exist two critical turbulent diffusion rates for species with small sinking rates. The phytoplankton forms a thin layer at the surface of the water column for large buoyant rates, and it forms a thin layer at the bottom of the water column for large sinking rates. Precise characterizations of these thin layers are also given.

Guided Surface Waves on One- and Two-Dimensional Arrays of Spheres

I. Thompson and C. M. Linton

SIAM J. Appl. Math. 70, pp. 2975-2995 (21 pages)

Online Publication Date: October 05, 2010

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Guided acoustic waves propagating along one- and two-dimensional arrays of rigid spheres are studied semianalytically. The quasi-periodic wavefield is constructed as a superposition of spherical wave functions, and then application of the boundary condition on the sphere surfaces leads to an infinite system of real linear algebraic equations. The vanishing of the determinant of the associated infinite matrix provides the condition for surface waves to exist, and these are determined numerically. In the case of a two-dimensional array, we consider arbitrary skew lattices and compute surface modes which are either symmetric or antisymmetric about the plane of the array. Our numerical calculations make extensive use of previous work by the authors on the accurate and efficient computation of lattice sums.

On Acoustic Cloaking Devices by Transformation Media and Their Simulation

Ulrich Hetmaniuk and Hongyu Liu

SIAM J. Appl. Math. 70, pp. 2996-3021 (26 pages)

Online Publication Date: October 14, 2010

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By transformation optics, we construct and investigate three-dimensional acoustic cloaking devices for which the device itself and the cloaked region can be mapped into two nested $l^p$-balls ($1\leq p<\infty$). Finite energy solutions for these cloaking devices are studied in weighted Sobolev spaces with singular weights. We show that the problems in the cloaking layer and the cloaked region can be decoupled. The problem in the cloaking layer has the same exterior Cauchy data as the one in the background medium, making the cloaking device together with the cloaked region invisible to exterior measurements. The cloaking works at any fixed frequency and can deal with sources/sinks as well. Finally, a conforming finite element method incorporating the Heaviside step function is proposed for simulating the devices. Numerical experiments illustrate the effectiveness of this discretization.

Comparison Study of Dynamics in One-Sided and Two-Sided Solid-Combustion Models

Y. Yang, L. K. Gross, and J. Yu

SIAM J. Appl. Math. 70, pp. 3022-3038 (17 pages)

Online Publication Date: October 21, 2010

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Comparing two-sided and one-sided solid-combustion models, this paper concerns nonlinear transition behavior of small disturbances of front propagation and temperature as they evolve in time. Features include linear instability of basic solutions and weakly nonlinear evolution of small perturbations, as well as the complex dynamics of period doubling, quadrupling, and eventual chaotic oscillations. Both asymptotic and numerical methods are used for different solution regimes. First, multiscale weakly nonlinear analysis takes into account the cumulative effect of small nonlinearities to obtain a correct description of the evolution over long times. For a range of parameters, the asymptotic method with some dominant modes captures the formation of coherent structures. In other cases, numerical solutions reveal period-folding behaviors. In general, the one- and two-sided models agree qualitatively for all solution regimes, which is consistent with prior numerical comparisons and extends our results from [L. K. Gross and J. Yu, SIAM J. Appl. Math., 65 (2005), pp. 1708–1725].

Stimulus-Driven Traveling Solutions in Continuum Neuronal Models with a General Smooth Firing Rate Function

G. Bard Ermentrout, Jozsi Z. Jalics, and Jonathan E. Rubin

SIAM J. Appl. Math. 70, pp. 3039-3064 (26 pages) | Cited 1 time

Online Publication Date: October 21, 2010

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We examine the existence of traveling wave solutions for a continuum neuronal network modeled by integro-differential equations. First, we consider a scalar field model with a general smooth firing rate function and a spatiotemporally varying stimulus. We prove that a traveling front solution that is locked to the stimulus exists for a certain interval of stimulus speeds. Next, we include a slow adaptation equation and obtain a formula, which involves a certain adjoint solution, for the stimulus speeds that induce locked traveling pulse solutions. Further, we use singular perturbation analysis to characterize an approximation to the adjoint solution that we compare to a numerically computed adjoint. Numerical simulations are used to illustrate the traveling fronts and pulses that we study and to make comparisons with our analytically computed bounds for stimulus-locked wave behavior.

Diffraction by a Two-Dimensional Traction-Free Elastic Wedge

A. K. Gautesen and L. Ju. Fradkin

SIAM J. Appl. Math. 70, pp. 3065-3085 (21 pages)

Online Publication Date: October 28, 2010

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This paper addresses the canonical two-dimensional problem of diffraction of the elastic plane wave by a traction-free straight-edged wedge made of an isotropic solid. Its solution can be used in applications to model diffraction from curved surface-breaking cracks with a curvature that is small compared to a wavelength. We use the Fourier transform to obtain a semianalytical solution, conduct internal checks, crossvalidate the code based on this scheme with the one based on the Sommerfeld integral, and describe experimental validation of the codes. Finally, we draw attention to high sensitivity of the backscatter diffraction coefficients to the Poisson ratio.

Frontal Reaction in a Layered Polymerizing Medium

Dmitry Golovaty, L. K. Gross, and James T. Joyner

SIAM J. Appl. Math. 70, pp. 3086-3104 (19 pages)

Online Publication Date: November 02, 2010

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We analyze the dynamics of a reaction propagating along a two-dimensional medium of nonuniform composition. We consider the context of a self-sustaining reaction front that converts a monomer-initiator mixture into an inhomogeneous polymeric material. We model the system with one-step effective kinetics, assuming large activation energy. Using asymptotic methods, we find the analytical expressions for the front profile as well as monomer and temperature distributions. Further, we demonstrate that the predictions of the asymptotic theory match well with the numerical simulations.

Direct and Inverse Obstacle Scattering Problems in a Piecewise Homogeneous Medium

Xiaodong Liu and Bo Zhang

SIAM J. Appl. Math. 70, pp. 3105-3120 (16 pages)

Online Publication Date: November 04, 2010

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This paper is concerned with the problem of scattering of time-harmonic acoustic waves from an impenetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is established, employing the integral equation method, and then used, in conjunction with the representation in a combination of layer potentials of the solution, to prove a priori estimates of solutions on some part of the interface between the layered media. The inverse problem is also considered in this paper. A uniqueness result is obtained for the first time in determining both the penetrable interface and the impenetrable obstacle with its physical property from a knowledge of the far field pattern for incident plane waves. In doing so, an important role is played by the a priori estimates of the solution for the direct problem.

An Age-Structured Model for the Transmission Dynamics of Hepatitis B

Lan Zou, Shigui Ruan, and Weinian Zhang

SIAM J. Appl. Math. 70, pp. 3121-3139 (19 pages) | Cited 1 time

Online Publication Date: November 04, 2010

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Hepatitis B virus (HBV) infection is endemic in many parts of the world. One of the characteristics of HBV transmission is the age structure of the host population. In this paper, we propose an age-structured model for the transmission dynamics of HBV. The host population is stratified by age and is divided into six subclasses: susceptible, latently infected, acutely infectious, carrier, recovered, and vaccinated individuals. By determining the basic reproduction number, we study the existence and stability of the disease-free and endemic steady state solutions of the model. Numerical simulations are performed to find optimal strategies for controlling the transmission of HBV.

Critical Transonic Shock and Supersonic Bubble in Oblique Rarefaction Wave Reflection along a Compressive Corner

Wancheng Sheng, Guodong Wang, and Tong Zhang

SIAM J. Appl. Math. 70, pp. 3140-3155 (16 pages)

Online Publication Date: November 11, 2010

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The oblique rarefaction wave reflection along a compressive corner is considered by using the numerical generalized characteristic analysis method (NGCAM). We have clarified that there are two patterns: regular reflection-like and Mach reflection-like. In both patterns, the reflection wave is always a compression simple wave, which ultimately forms a critical transonic shock. The critical transonic shock is a special kind of transonic shock, and the flow at the back bank of the shock is just sonic. A supersonic bubble near the compression corner grows in the second pattern and will break through as the rarefaction wave size increases.

Cucker–Smale Flocking under Rooted Leadership with Fixed and Switching Topologies

Zhuchun Li and Xiaoping Xue

SIAM J. Appl. Math. 70, pp. 3156-3174 (19 pages)

Online Publication Date: November 11, 2010

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In recent years, a number of articles have focused on mathematical models for emergent phenomena, for instance, the flocking of birds or the schooling of fish. In 2007, Cucker and Smale proposed an ingenious model which captures many of the observed features of moving animals. Subsequently, Shen extended the result to hierarchically structured flocks. Motivated by these works, in this paper we study the discrete Cucker–Smale flocking under rooted leadership, which means that there exists an overall leader such that any other agent is led, directly or indirectly, by the leader. The feature of our proposal, departing from the existing models, is that both the assumption of symmetry and the partial ordering of a hierarchy are dropped. The rooted leadership topology is a necessary condition for the group to converge towards a single leader's fixed constant velocity. The rates of convergence are established for flocks with fixed and switching topologies. The results may reveal the applicability and advantage of cooperation, or exchange of information, inside the group.

Analysis of Adaptive Response to Dosing Protocols for Biofilm Control

Barbara Szomolay, Isaac Klapper, and Martin Dindos

SIAM J. Appl. Math. 70, pp. 3175-3202 (28 pages)

Online Publication Date: November 11, 2010

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Biofilms are sessile populations of microbes that live within a self-secreted matrix of extracellular polymers. They exhibit high tolerance to antimicrobial agents, and experimental evidence indicates that in many instances repeated doses of antimicrobials further reduce disinfection efficiency due to an adaptive stress response. In this investigation, a mathematical model of bacterial adaptation is presented consisting of an adapted-unadapted population system embedded within a moving boundary problem coupled to a reaction-diffusion equation. The action of antimicrobials on biofilms under different dosing protocols is studied both analytically and numerically. We find the limiting behavior of solutions under periodic and on-off dosing as the period is made very large or very small. High dosages often carry undesirable side effects so we specially consider low dosing regimes. Our results indicate that on-off dosing for small doses of biocide is more effective than constant dosing. Moreover, in a specific case, on-off dosing for short periods is again more effective regardless of the biocide dose. We also provide sufficient conditions for the eradication of biofilms under a constant dosing regime.

A Robust Moment Method for Evaluation of the Disappearance Rate of Evaporating Sprays

Marc Massot, Frédérique Laurent, Damien Kah, and Stéphane de Chaisemartin

SIAM J. Appl. Math. 70, pp. 3203-3234 (32 pages)

Online Publication Date: November 17, 2010

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In this paper we tackle a critical issue in the numerical modeling, by Eulerian moment methods, of polydisperse multiphase systems, constituted of dispersed particles or droplets, a general class of systems which includes aerosols. Their modeling starts at a mesoscopic scale with an equation on the number density function (NDF) of particles/droplets which satisfies a population balance equation. In order to limit the computational cost, moment methods provide a system of conservation equation with an eventual closure problem, which can be solved using quadrature methods in order to retrieve the unclosed terms from the considered set of moments. However, a drift velocity, that is, the rate of change due to continuous phenomena of the internal coordinate, such as the size of the particles, has sometimes to be taken into account; it can be either positive like molecular growth, or negative such as for evaporation of droplets in aerosols or oxidation of soots. When negative, it leads to the disappearance of droplets/particles, thus creating a negative flux at zero size. Its closure requires an evaluation of the reconstructed NDF at zero size from the knowledge of a given finite set of moments. The nature of this information, pointwise in internal coordinates, and its influence on moment dynamics results is a difficulty from both a modeling and a numerical point of view. We obtain a comprehensive solution to this important issue. Since we introduce some new tools in order to resolve the flux evaluation, we also introduce a new Eulerian type of description, which will combine both the flexibility of Eulerian models for which the size phase space is discretized into “sections” (i.e., size intervals) and the efficiency of direct quadrature method of moments (DQMOM). It yields a precise and stable description of moment dynamics with a minimal number of variables, which will lead to a low computational cost in multidimensional configurations.

An Operator-Like Description of Love Affairs

Fabio Bagarello and Francesco Oliveri

SIAM J. Appl. Math. 70, pp. 3235-3251 (17 pages)

Online Publication Date: November 17, 2010

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We adopt the so-called occupation number representation, originally used in quantum mechanics and recently considered in the description of stock markets, in the analysis of the dynamics of love relations. We start with a simple model, involving two actors (Alice and Bob): in the linear case we obtain periodic dynamics, whereas in the nonlinear regime, either periodic or quasi-periodic solutions are found. Then we extend the model to a love triangle involving Alice, Bob, and a third actress, Carla. Interesting features appear, and in particular we find analytical conditions for the linear model of the love triangle to have periodic or quasi-periodic solutions. Numerical solutions are exhibited in the nonlinear case.

On Nonlinear Models of Markets with Finite Liquidity: Some Cautionary Notes

Kristoffer J. Glover, Peter W. Duck, and David P. Newton

SIAM J. Appl. Math. 70, pp. 3252-3271 (20 pages)

Online Publication Date: December 08, 2010

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The recent financial crisis and related liquidity issues have illuminated an urgent need for a better understanding of the effects of limited liquidity on all aspects of the financial system. This paper considers such effects on the Black–Scholes–Merton financial model, which for the most part result in highly nonlinear partial differential equations (PDEs). We investigate in detail a model studied by Schönbucher and Wilmott (2000) which incorporates the price impact of option hedging strategies. First, we consider a first-order feedback model, which leads to the exceptional case of a linear PDE. Numerical results, and more particularly an asymptotic approach close to option expiry, reveal subtle differences from the Black–Scholes–Merton model. Second, we go on to consider a full-feedback model in which price impact is fully incorporated into the model. Here, standard numerical techniques lead to spurious results in even the simplest cases. An asymptotic approach, valid close to expiry, is mounted, and a robust numerical procedure, valid for all times, is developed, revealing two distinct classes of behavior. The first may be attributed to the infinite second derivative associated with standard option payoff conditions, for which it is necessary to admit solutions with discontinuous first derivatives; perhaps even more disturbingly, negative option values are a frequent occurrence. The second failure (applicable to smoothed payoff functions) is caused by a singularity in the coefficient of the diffusion term in the option-pricing equation. Our conclusion is that several classes of model in the literature involving permanent price impact irretrievably break down (i.e., there is insufficient “financial modeling” in the pricing equation). Our analysis should provide the information necessary to avoid such pitfalls in the future.

New Bounds on Strong Field Magneto-Transport in Multiphase Columnar Composites

Marc Briane and Graeme W. Milton

SIAM J. Appl. Math. 70, pp. 3272-3286 (15 pages) | Cited 1 time

Online Publication Date: December 14, 2010

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This paper provides new bounds on the effective properties in dimension two of multiphase composites with isotropic phases having nonsymmetric conductivities. This is relevant to classical strong field magneto-transport in heterogeneous conductors with columnar microstructure, with a magnetic field in the direction of the columns. On the one hand, a polygon-like $G$-closure for multiphase composites without prescribed volume fractions is derived for isotropic effective resistivities. On the other hand, the $G$-closure of the possible average currents associated with a given applied electric field is found for anisotropic multiphase composites with prescribed volume fractions. This extends the $G$-closure result obtained by Raĭtum and Tartar for symmetric conductivities.

Premixed Flame Propagation in Channels of Varying Width

Hazem El-Rabii, Guy Joulin, and Kirill A. Kazakov

SIAM J. Appl. Math. 70, pp. 3287-3318 (32 pages) | Cited 1 time

Online Publication Date: December 14, 2010

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A theory of flame propagation in two-dimensional channels with adiabatic and impermeable curved walls is developed within the framework of the on-shell description of premixed flames. Employing the Green function appropriate to the given channel geometry, an implicit integral representation for the burnt gas velocity is constructed. It is next used to derive an explicit expression for the rotational component of the gas velocity near the flame front by successive separation of irrotational contributions. We prove that this separation can be performed in a way consistent with boundary conditions at the channel walls. As a result, the unknown irrotational component can be projected out by applying a dispersion relation stemming from its analyticity, thus leading to a closed system of equations for the on-shell fresh gas velocity and the flame front position. These equations show that, in addition to the usual nonlocality associated with potential flows, vorticity produced by a curved flame leads to specific nonlocal spatial and temporal influence of the channel geometry on the flame evolution. To elucidate this influence, three special cases are considered in more detail: a steady flame stabilized by incoming flow in a bottle-shaped channel, quasi-steady flames, and unsteady flames with small gas expansion propagating in channels of slowly varying width. In the last case, analytical solutions of the equations derived for the front shape are obtained in the first post-Sivashinsky approximation on using the method of pole decomposition.

Singular Solutions of the Biharmonic Nonlinear Schrödinger Equation

G. Baruch, G. Fibich, and E. Mandelbaum

SIAM J. Appl. Math. 70, pp. 3319-3341 (23 pages)

Online Publication Date: December 22, 2010

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We consider singular solutions of the $L^2$-critical biharmonic nonlinear Schrödinger equation. We prove that the blowup rate is bounded by a quartic-root, the solution approaches a quasi–self-similar profile, and a finite amount of $L^2$-norm, which is no less than the critical power, concentrates into the singularity. We also prove the existence of a ground-state solution. We use asymptotic analysis to show that the blowup rate of peak-type singular solutions is slightly faster than that of a quartic-root, and the self-similar profile is given by the ground-state standing wave. These findings are verified numerically (up to focusing levels of $10^8$) using an adaptive grid method. We also use the spectral renormalization method to compute the ground state of the standing-wave equation, and the critical power for collapse, in one, two, and three dimensions.

Reconstruction of Planar Conductivities in Subdomains from Incomplete Data

Adrian Nachman, Alexandru Tamasan, and Alexandre Timonov

SIAM J. Appl. Math. 70, pp. 3342-3362 (21 pages) | Cited 1 time

Online Publication Date: December 22, 2010

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We consider the problem of recovering a sufficiently smooth isotropic conductivity from interior knowledge of the magnitude of the current density field $|J|$ generated by an imposed voltage potential $f$ on the boundary. In any dimension $n\geq2$, we show that equipotential sets are global area minimizers in the conformal metric determined by $|J|$. In two dimensions, assuming the boundary voltage is almost two-to-one, we prove uniqueness of the minimization problem. This yields two results on reconstruction from incomplete data. In the first case, $|J|$ is known in all of $\Omega$, but the almost two-to-one $f$ is know only on subintervals of the boundary. The second case assumes that $|J|$ is known only in an appropriate subdomain $\tilde{\Omega}$: our method works provided that $\tilde{\Omega}$ contains entire equipotential curves joining boundary points. Based on solving two point boundary value problems for the geodesic system, we give a procedure to determine whether $\tilde{\Omega}$ satisfies this property, to construct the equipotential curves lying entirely in the interior of $\tilde{\Omega}$, and to obtain the conductivity in the region spanned by these curves. We also conduct a numerical study to illustrate the computational feasibility of the method.
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