Integral equation methods in electromagnetic scattering from anisotropic media

We investigate scattering of time‐harmonic electromagnetic waves by an anisotropic inhomogeneous medium. The problem is equivalently transformed into a system of strongly singular integral equations. The uniqueness and existence of a solution is shown and we examine the regularity of the solution by means of integral equations. We also prove the analyticity of the scattered field with respect to the refractive matrix and give a characterization of the derivatives in terms of solutions to anisotropic scattering problems. Copyright © 2000 John Wiley & Sons, Ltd.     

Covolume techniques for anisotropic media

Summary The covolume method, a new approach applicable on general meshes, is extended to discretize and numerically solve the div-curl system in anisotropic media. The covolume method gives simple schemes and good approximations to the solution of the div-curl system. It works directly with the system and utilizes dual pairs of meshes that are orthogonally related. Central to the approach is the introduction of field components tangent and normal to the edges of one of the meshes, and the employment of dual discretization on the dual mesh pairs. The discretization procedures, schemes and error analysis are presented. The convergence of the method is proved.The work was partially done while this author was at Carnegie Mellon University     

Electromagnetism on anisotropic fractal media

Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows a generalization of the Green–Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Ampère laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell’s electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.     

Effective media for periodic anisotropic systems

One considers the wave propagation in periodic anisotropic systems. For these systems, with the aid of a matrix method, one determines the effective media.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 130–138, 1983.     

Inverse scattering in one-dimensional nonconservative media

The inverse scattering problem arising in wave propagation in one-dimensional non-conservative media is analyzed. This is done in the frequency domain by considering the Schrödinger equation with the potentialikP(x)+Q(x), wherek ² is the energy andP(x) andQ(x) are real integrable functions. Using a pair of uncoupled Marchenko integral equations,P(x) andQ(x) are recovered from an appropriate set of scattering data including bound-state information. Some illustrative examples are provided.Dedicated to M.G. Kreîn, one of the founding fathers of inverse scattering theory.     

Time-dependent potential scattering in chiral media

Electromagnetic waves propagating in a homogeneous three-dimensional unbounded chiral medium are considered. We define a chiral operator and study potential scattering relative to this operator. A spectral analysis of associated operators is obtained, based on the Plancherel theory of the Fourier transform. Using the generalised eigenfunction expansion theory, we give an integral representation of the solution. A discussion of asymptotic equality of solutions is provided and the associated wave operator introduced.     

The linear sampling method for anisotropic media

We consider the inverse scattering problem of determining the support of an anisotropic inhomogeneous medium from a knowledge of the incident and scattered time harmonic acoustic wave at fixed frequency. To this end, we extend the linear sampling method from the isotropic case to the case of anisotropic medium. In the case when the coefficients are real we also show that the set of transmission eigenvalues forms a discrete set.     

Entire solutions of sublinear elliptic equations in anisotropic media

We study the nonlinear elliptic problem −Δu=ρ(x)f(u) in RN (N?3), lim|x|→∞u(x)=?, where ??0 is a real number, ρ(x) is a nonnegative potential belonging to a certain Kato class, and f(u) has a sublinear growth. We distinguish the cases ?>0 and ?=0 and prove existence and uniqueness results if the potential ρ(x) decays fast enough at infinity. Our arguments rely on comparison techniques and on a theorem of Brezis and Oswald for sublinear elliptic equations.     

A mixed bvp for flow in strongly anisotropic media

Initial-boundary value problems for a class of linear parabolic equations are considered. The anisotropy of the medium is characterised by a small parameter. The solution structure is analysed by singular perturbation methods which include the construction of outer solutions and boundary and initial layer terms. The analysis is justified by convergence results     

Heat equation with memory in anisotropic and non-homogeneous media

A modified Fourier’s law in an anisotropic and non-homogeneous media results in a heat equation with memory, for which the memory kernel is matrix-valued and spatially dependent. Different conditions on the memory kernel lead to the equation being either a parabolic type or a hyperbolic type. Well-posedness of such a heat equation is established under some general and reasonable conditions. It is shown that the propagation speed for heat pulses could be either infinite or finite, depending on the different types of the memory kernels. Our analysis indicates that, in the framework of linear theory, heat equation with hyperbolic kernel is a more realistic model for the heat conduction, which might be of some interest in physics.     

Linear elliptic equations in composite media with anisotropic fibres

Linear elliptic equations in composite media with anisotropic fibres are concerned. The media consist of a periodic set of anisotropic fibres with low conductivity, included in a connected matrix with high conductivity. Inside the anisotropic fibres, the conductivity in the longitudinal direction is relatively high compared with that in the transverse directions. The coefficients of the elliptic equations depend on the conductivity. This work is to derive the Hölder and the gradient $L^p$ estimates (uniformly in the period size of the set of anisotropic fibres as well as in the conductivity ratio of the fibres in the transverse directions to the connected matrix) for the solutions of the elliptic equations. Furthermore, it is shown that, inside the fibres, the solutions have higher regularity along the fibres than in the transverse directions.     

On an integral related to biaxially anisotropic media

An integral arising in certain studies of theoretical electromagnetics is evaluated and its properties are discussed in some detail. The integral has three integer, two real, and one complex parameter. The integrand involves a product of Bessel functions of different argument and order. Several generalizations are discussed.     

Localization for random perturbations of anisotropic periodic media

We prove localization for random perturbations of periodic divergence form operators of the form ∇ · a_ω · ∇ near the band edges. Here a_ω is a matrix function which results from an Anderson type perturbation of a periodic matrix function.     

The stability of quasi-transverse shock waves in anisotropic elastic media

The stability of weak quasi-transverrse shock waves in a weakly anistropic elastic medium with respect to arbitrarily oriented perturbations is investigated in the linear approximation. It is shown that fast quasi-transverse shock waves are stable.     

Numerical modeling of fluid flow in anisotropic fractured porous media

A model of double porosity in the case of an anisotropic fractured porous medium is considered (Dmitriev, Maksimov; 2007). A function of fluid exchange between the fractures and porous blocks depending on flow direction is given. The flow function is based on the difference between the pressure gradients. This feature enables one to take into account anisotropic properties of filtration in a more general form. The results of numerical solving a model two-dimensional problem are presented. The computational algorithm is based on a finite-element space approximation and explicit-implicit time approximations.