Showing papers in "Quarterly of Applied Mathematics in 2014"
TL;DR: In this paper, the authors established local derivative estimates for solutions to a class of elliptic systems arising from studies of fiber-reinforced composite materials, where the maximal strains can be strongly influenced by the distances between the fibers.
Abstract: In this paper we establish local derivative estimates for solutions to a class of elliptic systems arising from studies of fiber-reinforced composite materials. From the structure of the composite, there are a relatively large number of fibers which are touching or nearly touching. The maximal strains can be strongly influenced by the distances between the fibers. Stimulated by some works on damage analysis of fiber composites ([6]), there have been a number of papers, starting from [9], [15] and [16], on gradient estimates for solutions of elliptic equations and systems with piecewise smooth coefficients which are relevant in such studies. See, e.g. [1, 2, 3, 4, 5], [7, 8], [10], [12], [17], [18, 19]. Earlier studies on such and closely related issues can be found in [11], [13, 14]. In a recent paper [2], some gradient estimates were obtained concerning the conductivity problem where the conductivity is allowed to be ∞ (perfect conductor).
57 citations
TL;DR: In this paper, the option pricing under the FMLS (finite moment log stable) model, which can effectively capture the leptokurtic feature observed in many financial markets, is investigated.
Abstract: This paper investigates the option pricing under the FMLS (finite moment log stable) model, which can effectively capture the leptokurtic feature observed in many financial markets. However, under the FMLS model, the option price is governed by a modified Black-Scholes equation with a spatial-fractional derivative. In comparison with standard derivatives of integer order, the fractional-order derivatives are characterized by their “globalness”, i.e., the rate of change of a function near a point is affected by the property of the function defined in the entire domain of definition rather than just near the point itself. This has added an additional degree of difficulty not only when a purely numerical solution is sought but also when an analytical method is attempted. Despite this difficulty, we have managed to find an explicit closed-form analytical solution for European-style options after successfully solving the FPDE (fractional partial differential equation) derived from the FMLS model. After the validity of the put-call parity under the FMLS model is verified both financially and mathematically, we have also proposed an efficient numerical evaluation technique to facilitate the implementation of our formula so that it can be easily used in trading practice. Received August 2, 2012 and, in revised form, June 24, 2013. 2010 Mathematics Subject Classification. Primary 97M30, 35R11.
55 citations
TL;DR: In this paper, the free boundary problem for the plasma vacuum interface model in ideal incompressible magneto-hydrodynamics is considered and a suitable a priori estimate, uniform with respect to the small parameter of the regularization, is derived in the anisotropic Sobolev space.
Abstract: We consider the free boundary problem for the plasma vacuum interface model in ideal incompressible magneto-hydrodynamics. Under a suitable stability condition on the initial discontinuity, the well-posedness of the linearized problem, around a non constant basic state sufficiently smooth, is investigated. Since the latter amounts to be a non standard initial-boundary value problem of mixed hyperbolic-elliptic type, for its resolution we introduce a fully ”hyperbolic” regularized problem. For the regularized problem, a suitable a priori estimate, uniform with respect to the small parameter of the regularization, is derived in the anisotropic Sobolev space H 1 .
54 citations
TL;DR: In this paper, the authors considered the Bresse system with past history acting in the shear angle displacement and showed the exponential decay of the solution if and only if the wave speeds are the same.
Abstract: In this paper we consider the Bresse system with past history acting in the shear angle displacement. We show the exponential decay of the solution if and only if the wave speeds are the same. On the contrary, we show that the Bresse system is polynomial stable with optimal decay rate. The systems of equations considered here introduce new mathematical difficulties in order to determine the asymptotic behavior. As far as the authors know, there have been no contributions made in this sense.
41 citations
TL;DR: In this paper, the exact Riemann solutions for the shallow water equations with a bottom step including the dry bed problem were determined completely with a single bottom step, and the L-M and R-M curves were constructed in the state space.
Abstract: We determine completely the exact Riemann solutions for the shallow water equations with a bottom step including the dry bed problem The nonstrict hyperbolicity of this first order system of partial differential equations leads to resonant waves and non unique solutions To address these difficulties we construct the L–M and R–M curves in the state space For the bottom step elevated from left to right, we classify the L–M curve into five different cases and the R–M curve into two different cases based on the subcritical and supercritical Froude number of the Riemann initial data as well as the jump of the bottom step The behaviors of all basic cases of the L–M and R–M curves are fully analyzed We observe that the non–uniqueness of the Riemann solutions is due to bifurcations on the L–M or R–M curves The possible Riemann solutions include classical waves and resonant waves as well as dry bed solutions that are solved in a uniform framework for any given initial data
32 citations
TL;DR: In this paper, the authors considered a mathematical model which describes the frictionless contact between a viscoplastic body and an obstacle, the so-called foundation, and provided a mixed variational formulation of the model which involves a dual Lagrange multiplier.
Abstract: We consider a mathematical model which describes the frictionless contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic and the contact is modeled with normal compliance and unilateral constraint. We provide a mixed variational formulation of the model which involves a dual Lagrange multiplier, and then we prove its unique weak solvability. We also prove an estimate which allows us to deduce the continuous dependence of the weak solution with respect to both the normal compliance function and the penetration bound. Finally, we provide a numerical validation of this convergence result.
18 citations
TL;DR: In this paper, the authors derived the finite Larmor radius approximation, when collisions are taken into account, corresponding to different collision kernels: the relaxation and the Fokker-Planck operators.
Abstract: The subject matter of this paper concerns the derivation of the finite Lar-mor radius approximation, when collisions are taken into account. Several studies are performed, corresponding to different collision kernels : the relaxation and the Fokker-Planck operators. Gyroaveraging the relaxation operator leads to a position-velocity integral operator, whereas gyroaveraging the linear Fokker-Planck operator leads to diffusion in velocity but also with respect to the perpendicular position coordinates.
14 citations
TL;DR: In this paper, the authors investigate the existence of traveling wave solutions to the onedimensional reaction-diffusion system and show that this system has a unique, up to translation, traveling wave solution with speed c iff c ≥ 2/ √ β + 1.
Abstract: We investigate the existence of traveling wave solutions to the onedimensional reaction-diffusion system ut = δuxx − 2uv/(β + u), vt = vxx + uv/(β + u), which describes the acidic nitrate-ferroin reaction. Here β is a positive constant, u and v represent the concentrations of the ferroin and acidic nitrate respectively, and δ denotes the ratio of the diffusion rates. We show that this system has a unique, up to translation, traveling wave solution with speed c iff c ≥ 2/ √ β + 1.
14 citations
TL;DR: In this article, the formation of singularities in thermoelasticity with second sound was investigated and it was shown that there are in general no global smooth solutions for large initial data.
Abstract: We investigate the formation of singularities in thermoelasticity with second sound. Transforming into Euler coordinates and combining ideas from Sideris [14], used for compressible fluids, and Tarabek [15], used for small data large time existence in second sound models, we are able to show that there are in general no global smooth solutions for large initial data. In contrast to the situation for classical thermoelasticity, we require largeness of the data itself, not of its derivatives.
14 citations
TL;DR: In this paper, the authors obtained a completely explicit form for the gyroaverage of the Fokker-Planck-Landau kernel, accounting for diffusion and convolution with respect to both velocity and position coordinates.
Abstract: This paper is devoted to the finite Larmor radius approximation of the Fokker-Planck-Landau equation, which plays a major role in plasma physics. We obtain a completely explicit form for the gyroaverage of the Fokker-Planck-Landau kernel, accounting for diffusion and convolution with respect to both velocity and (perpendicular) position coordinates. We show that the new collision operator enjoys the usual physical properties ; the averaged kernel balances the mass, momentum, kinetic energy and dissipates the entropy, globally in velocity and perpendicular position coordinates.
13 citations
TL;DR: In this article, the authors studied the quenching behavior of the solution of a semilinear heat equation with a singular boundary outflux and proved a finite-time quench for the solution.
Abstract: In this paper, we study the quenching behavior of the solution of a semilinear heat equation with a singular boundary outflux. We prove a finite-time quenching for the solution. Further, we show that quenching occurs on the boundary under certain conditions and we show that the time derivative blows up at a quenching point. Finally, we get a quenching rate and a lower bound for the quenching time.
TL;DR: In this article, an asymptotic flocking estimate for the Cucker-Smale flocking model under the rooted leadership in a large coupling limit is presented, which is based on Tikhonov singular perturbation theory.
Abstract: We present an asymptotic flocking estimate for the Cucker-Smale flocking model under the rooted leadership in a large coupling limit. For this, we reformulate the Cucker-Smale model into a fast-slow dynamical system involving a small parameter which corresponds to the inverse of a coupling strength. When the coupling strength tends to infinity, the spatial configuration will be frozen instantaneously, whereas the velocity configuration shrinks to the global leader’s velocity immediately. For the rigorous explanation of this phenomenon, we use Tikhonov’s singular perturbation theory. We also present several numerical simulations to confirm our analytical theory.
TL;DR: In this paper, a family of exact Gerstner type solutions for the geophysical equatorial deep water wave problem in the f-plane approximation is described and the particle paths in the presence and absence of the Coriolis force are analyzed in dependence of the current strength.
Abstract: We describe a family of exact Gerstner type solutions for the geophysical equatorial deep water wave problem in the f-plane approximation. These Gerstner type waves are two-dimensional and travel with constant speed over a uniform horizontal current. The particle paths in the presence and absence of the Coriolis force are also analyzed in dependence of the current strength.
TL;DR: This work presents a novel dynamical systems modeling approach to understand the processes governing an individual’s behavior in the context of problem drinking, and discusses the construction of an initial mathematical model for two starkly distinct individuals.
Abstract: We consider a conceptual and quantitative modeling approach for investigating dynamic behavior change. While the approach is applicable to behavior change in eating disorders, smoking, substance abuse and other behavioral disorders, here we present our novel dynamical systems modeling approach to understand the processes governing an individual’s behavior in the context of problem drinking. Recent advances in technology have resulted in large intensive longitudinal data sets which are particularly well suited for study within such frameworks. However, the lack of previous work in this area (specifically, on the interand intra-personal factors governing drinking behavior of individuals) renders this a daunting and unique challenge. As a result, issues which are typically routine in mathematical modeling require considerable effort such as the determination of key quantities of interest, and the timescale on which to represent them. We discuss the construction of an initial mathematical model for two starkly distinct individuals and make a case for the potential for such efforts to help in understanding the underlying mechanisms responsible for behavior change in problem drinkers. AMS Subject Classification: 91E10, 34K29, 62G10.
TL;DR: In this paper, the authors present the results of a study at the Departamento de Matematica Faculdade de Engenharia de Ilha Solteira UNESP - Univ Estadual Paulista, Rua Rio de Janeiro, 266
Abstract: Departamento de Matematica Faculdade de Engenharia de Ilha Solteira UNESP - Univ Estadual Paulista, Rua Rio de Janeiro, 266
TL;DR: In this paper, the first initial boundary value problems of a class of forward-backward convection-diffusion equations were studied and the existence theorem and the continuous dependence theorem of Young measure solutions were established.
Abstract: This paper is devoted to the first initial boundary value problems of a class of forward-backward convection-diffusion equations. The existence theorem and the continuous dependence theorem of Young measure solutions are established.
TL;DR: In this paper, general translational addition theorems for spherical scalar Laplacian functions are presented for boundary value problems, and their application to boundary value problem is illustrated.
Abstract: General translational addition theorems are presented for spherical scalar Laplacian functions, and their application to boundary value problems is illustrated. By these theorems, the eigenfunction solutions in a system of spherical coordinates are expressed in terms of the spherical coordinates in another system, translated with respect to the first one. This allows for a rigorous analytic solution to be obtained for Laplacian and Poissonian fields in the presence of arbitrary configurations of spheres by imposing the exact boundary conditions. Complete formulations and solutions are presented for systems of electrically charged spheres and for arrays of perfect conductor spheres in external electric and magnetic fields. Illustrative computation examples are given for three-sphere systems. Numerical results of specified accuracy are generated, which are useful for validating various approximate numerical methods.
TL;DR: In this paper, a multiple scales asymptotic analysis is used to describe the attenuation of a water hammer pressure wave in the Brunone model of unsteady friction.
Abstract: A multiple-scales asymptotic analysis is used to describe the attenuation of a water hammer pressure wave in the Brunone model of unsteady friction. The method is applied to water hammer caused by sudden valve closure in water reservoir pipelines. The analytical results explain the parametric dependence of the Brunone unsteady friction pressure-wave attenuation. It is also found that viscous head in an extended steady friction model may provide an alternative to the unsteady friction basis for increased attenuation in cases where the attenuation has a weak spatial dependence and is primarily time-dependent. All results are numerically verified using the method of characteristics.
TL;DR: In this paper, a kernel method is proposed to estimate the condensed density of the generalized eigenvalues of pencils of Hankel matrices whose elements have a joint noncentral Gaussian distribution with nonidentical covariance.
Abstract: A kernel method is proposed to estimate the condensed density of the generalized eigenvalues of pencils of Hankel matrices whose elements have a joint noncentral Gaussian distribution with nonidentical covariance. These pencils arise when the complex exponentials approximation problem is considered in Gaussian noise. Several moments problems can be formulated in this framework and the estimation of the condensed density above is the main critical step for their solution. It is shown that the condensed density satisfies approximately a diffusion equation, which allows to estimate an optimal bandwidth. It is proved by simulation that good results can be obtained even when the signal-to-noise ratio is so small that other methods fail.
TL;DR: In this article, a boundary value problem (BVP) derived from a model of boundary layer flow past a suddenly heated vertical surface in a saturated porous medium was investigated and it was shown that precisely one solution of this continuum approaches the boundary condition at infinity exponentially while all others approach algebraically.
Abstract: In this paper we investigate a boundary value problem (BVP) derived from a model of boundary layer flow past a suddenly heated vertical surface in a saturated porous medium. The surface is heated at a rate proportional to x where x measures distance along the wall and k > −1 is constant. Previous results have established the existence of a continuum of solutions for −1 < k < −1/2. Here we further analyze this continuum and determine that precisely one solution of this continuum approaches the boundary condition at infinity exponentially while all others approach algebraically. Previous results also showed that the solution to the BVP is unique for −1/2 ≤ k < 0. Here we extend the range of uniqueness to 0 ≤ k ≤ 1. Finally, the physical implications of the mathematical results are discussed and a comparison is made to the solutions for the related case of prescribed surface temperature on the surface.
TL;DR: In this paper, an analytical approach is proposed to study the contribution of edge effects in the multiple high-frequency diffraction, according to guidelines of classical Kirchhoff theory in (scalar) wave propagation.
Abstract: An analytical approach is proposed to study the contribution of edge effects in the multiple high-frequency diffraction, according to guidelines of classical Kirchhoff theory in (scalar) wave propagation. We start from a suitable asymptotic analysis of the Kirchhoff diffraction integral, here set up in a generalized (iterated) form to describe the multiple reflections from an arbitrary sequence of curved reflecting (smooth) surfaces. The explicit formula obtained for a concrete example of double reflection is compared with the results from direct numerical simulation.
TL;DR: In this paper, the existence and uniqueness of solution to the Cauchy problem for a system of multi-time conservation laws with two independent time variables in one space dimension were shown, based on a suitable generalization of the Lax-Oleinik formula.
Abstract: Motivated by the work of P.L. Lions and J-C. Rochet [12], concerning multi-time Hamilton-Jacobi equations, we introduce the theory of multitime systems of conservation laws. We show the existence and uniqueness of solution to the Cauchy problem for a system of multi-time conservation laws with two independent time variables in one space dimension. Our proof relies on a suitable generalization of the Lax-Oleinik formula.
TL;DR: In this article, the Vlasov equation was used to model the solar wind interacting with a magnetized obstacle with the domain considered is a disk in the plane, and inflowing boundary conditions were given for the particle density.
Abstract: The solar wind interacting with a magnetized obstacle is modeled with the Vlasov equation. The domain considered is a disk in the plane. Inflowing boundary conditions are given for the particle density. A magnetic field is prescribed, and the electric field is computed self consistently with potential zero on the boundary. Taking the boundary condition for the particle density to be sufficiently small, it is shown that there is a natural smooth steady solution. The speed of the inflowing plasma and the magnetic field are not size restricted.
TL;DR: In this article, the authors studied the stability of the solution of the Gelfand equation with respect to the singular solution as r goes to infinity and proved asymptotic stability by considering the initial value problem.
Abstract: We study the asymptotic behaviors of the solution to the Gelfand equation. The Gelfand equation appears in the kinetic theory of gravitational steady state and the theory of nonlinear diffusion. We present a convergence rate of the solutions of the Gelfand equation to the unique singular solution as r goes to infinity and prove asymptotic stability of the solution by considering the initial value problem for the Gelfand equation. To obtain the convergence rate and the point-wise stability estimate, we construct a uniform lower bound function and use the solution for the linearized Gelfand equation.