Analysis of Wave Propagation in 1D Inhomogeneous Media

In this paper, we consider the one-dimensional inhomogeneous wave equation with particular focus on its spectral asymptotic properties and its numerical resolution. In the first part of the paper, we analyze the asymptotic nodal point distribution of high-frequency eigenfunctions, which, in turn, gives further information about the asymptotic behavior of eigenvalues and eigenfunctions. We then turn to the behavior of eigenfunctions in the high- and low-frequency limit. In the latter case, we derive a homogenization limit, whereas in the first we show that a sort of self-homogenization occurs at high frequencies. We also remark on the structure of the solution operator and its relation to desired properties of any numerical approximation. We subsequently shift our focus to the latter and present a Galerkin scheme based on a spectral integral representation of the propagator in combination with Gaussian quadrature in the spectral variable with a frequency-dependent measure. The proposed scheme yields accurate resolution of both high- and low-frequency components of the solution and as a result proves to be more accurate than available schemes at large time steps for both smooth and nonsmooth speeds of propagation.     

Propagation of Reactions in Inhomogeneous Media

Consider reaction‐diffusion equation u _t =Δ u + f (x,u ) with and general inhomogeneous ignition reaction f ≥ 0 vanishing at u = 0,1. Typical solutions 0 ≤ u ≤ 1 transition from 0 to 1 as time progresses, and we study them in the region where this transition occurs. Under fairly general qualitative hypotheses on f we show that in dimensions d ≤ 3, the Hausdorff distance of the superlevel sets {u ≥ ε } and {u ≥ 1‐ε} remains uniformly bounded in time for each ε ? (0,1). Thus, u remains uniformly in time close to the characteristic function of in the sense of Hausdorff distance of superlevel sets. We also show that each {u ≥ ε} expands with average speed (over any long enough time interval) between the two spreading speeds corresponding to any x ‐independent lower and upper bounds on f . On the other hand, these results turn out to be false in dimensions d ≥ 4, at least without further quantitative hypotheses on f . The proof for d ≤ 3 is based on showing that as the solution propagates, small values of u cannot escape far ahead of values close to 1. The proof for d ≥ 4 is via construction of a counterexample for which this fails. Such results were before known for d =1 but are new for general non‐periodic media in dimensions d ≥ 2 (some are also new for homogeneous and periodic media). They extend in a somewhat weaker sense to monostable, bistable, and mixed reaction types, as well as to transitions between general equilibria of the PDE and to solutions not necessarily satisfying . © 2016 Wiley Periodicals, Inc.     

Admissibility of Control Operators in UMD Spaces and the Inverse Laplace Transform

This paper investigates the admissibility of control and observation operators in UMD spaces. Necessary and/or sufficient conditions for unbounded control operators to be admissible and weakly admissible in the Salamon–Weiss sense are presented. This is illustrated by an example which shows that the UMD-property is essential. In particular, we get a direct proof of the known result of Driouich and and El-Mennaoui (Arch Math 72:56–63, 1999) on the validity of the inverse formula of the Laplace transform for C ₀-semigroups on UMD-spaces and in Hilbert spaces, as proved earlier by Yao (SIAM J Math Anal 26(5):1331–1341, 1995). We outline how these results can be used to prove a partial validity of the inverse Laplace transform for semigroups in general Banach spaces. In particular, we obtain the result on the inverse Laplace transform due to Hille and Philllips (Am Math Soc Transl Ser 2, 1957).     

Goodness-of-Fit Tests for the Inverse Gaussian Distribution Based on the Empirical Laplace Transform

This paper considers two flexible classes of omnibus goodness-of-fit tests for the inverse Gaussian distribution. The test statistics are weighted integrals over the squared modulus of some measure of deviation of the empirical distribution of given data from the family of inverse Gaussian laws, expressed by means of the empirical Laplace transform. Both classes of statistics are connected to the first nonzero component of Neyman's smooth test for the inverse Gaussian distribution. The tests, when implemented via the parametric bootstrap, maintain a nominal level of significance very closely. A large-scale simulation study shows that the new tests compare favorably with classical goodness-of-fit tests for the inverse Gaussian distribution, based on the empirical distribution function.     

A New Numerical Method for the Inversion of the Laplace Transform

This paper presents a new numerical method for the inversionof the Laplace transform, where only real values of the transformare known. This method, based on the approximation of the transformby truncated series of orthogonal functions, related to theJacobi or Laguerre polynomials, avoids the loss of significantfigures inherent in earlier methods.     

Inverse Scattering Transform for the Coulomb Plasma with Negative Temperature

The boundary problem for the two-dimensional elliptic sinh-Gordon equation is studied. The exact solutions of these equations are found and trace identities are suggested. An application of this problem to the model of the Coulomb plasma with negative temperature is considered. Bibliography: 10 titles.     

NUMERICAL METHODS FOR MAXWELL＇S EQUATIONS IN INHOMOGENEOUS MEDIA WITH MATERIAL INTERFACES

In this paper, we will present some recent results on developing numerical methods for solving Maxwell‘s equations in inhomogeneous media with material interfaces. First,we will present a second order upwinding embedded boundary method - a Cartesian grid based finite difference method with special upwinding treatment near the material interfaces. Second, we will present a high order discontinuous spectral element with Dubinar orthogonal polynomials on triangles. Numerical results on electromagnetic scattering and photonic waveguide will be included.     

Zagier Duality for the Exponents of Borcherds Products for Hilbert Modular Forms

A certain sequence of weight modular forms arises in the theoryof Borcherds products for modular forms for SL₂(Z). Zagier proveda family of identities between the coefficients of these weight forms and a similar sequence of weight 3/2 modular forms, whichinterpolate traces of singular moduli. We obtain the analogousresults for modular forms arising from Borcherds products forHilbert modular forms.     

Dispersion and Strichartz estimates for some finite rank perturbations of the Laplace operator

We consider the dispersion properties in L^p spaces of Schrödinger hamiltonians with a large number of obstacles modelled by rank one perturbations. We obtain both for the dispersion an Strichartz estimates nonperturbative results with respect to the coupling constants.     

Nonlocality and the Inverse Scattering Transform for the Pavlov Equation

As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is nonlocal, and the proper choice of integration constants should be the one dictated by the associated inverse scattering transform (IST). Using the recently made rigorous IST for vector fields associated with the so‐called Pavlov equation , in this paper we establish the following. 1. The nonlocal term arising from its evolutionary form corresponds to the asymmetric integral . 2. Smooth and well‐localized initial data evolve in time developing, for , the constraint , where . 3. Because no smooth and well‐localized initial data can satisfy such constraint at , the initial () dynamics of the Pavlov equation cannot be smooth, although, because it was already established, small norm solutions remain regular for all positive times. We expect that the techniques developed in this paper to prove the above results should be successfully used in the study of the nonlocality of other basic examples of integrable dispersionless PDEs in multidimensions.     

A Quasi-Projection Analysis for Elastic Wave Propagation in Fluid-Saturated Porous Media

This paper deals with the superconvergence phenomena for Galerkin approximations of solutions of Biot's dynamic equations describing elastic wave propagation in fluid-saturated porous media. An asymptotic expansion to high order of Galerkin solutions is used to derive these results.     

Laplace Transform Method for the Ulam Stability of Linear Fractional Differential Equations with Constant Coefficients

Using the Laplace transform method, this paper deals with the Ulam stability of linear fractional differential equations with constant coefficients.     

Group Classification of Eikonal Equations for the Wave Equation in Inhomogeneous Media

We present a group classification of the family of equations of the form (∇ψ)² = 1/v ²(x, y, z), called the eikonal equations, which are equations of characteristics for equations describing acoustic and electromagnetic waves in inhomogeneous media. For some cases of equations with linear and quadratic functions v(x, y, z), having the sense of the speed of waves in a medium, we present explicit solutions (eikonals of point sources) and completely describe the geometry of rays.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

Mixed Method for Compressible Miscible Displacement with Dispersion in Porous Media

Compressible miscible displacement of one fluid by another in porous media is modelled by a nonlinear parabolic system. A finite element procedure is introduced to approximate the concentration of one fluid and the pressure of the mixture. The concentration is treated by a Galerkin method while the pressure is treated by a parabolic mixed finite element method. The effect of dispersion, which is neglected in [1], is considered. Optimal order estimates in L2 are derived for the errors in the approximate solutions.     

A Four for the Price of One Duality Principle for Distributive Spaces

In this paper we consider topological spaces as generalised orders and characterise those spaces which satisfy a (suitably defined) topological distributive law. Furthermore, we show that the category of these spaces is dually equivalent to a certain category of frames by simply observing that both sides represent the idempotents split completion of the same category.     

横观各向同性含液饱和多孔介质中应力波传播的特征分析

根据广义特征理论,对横观各向同性含液饱和多孔介质中应力波传播特性进行了特征分析．给出了特征曲面的微分方程以及沿次特征线的相容条件,得到了波阵面的解析表达式．详细地讨论了应力波在横观各向同性含液饱和多孔介质中传播时,其速度曲面和波阵面的形状及性质．分析结果亦表明,纯固体中应力波传播的特征方程,是含液饱和多孔介质中应力波特征方程的特例．