In this paper we consider the inverse problem of the permittivity determination of thin multisectional diaphragm in a rectangular waveguide. We perform a detailed analysis for one-, two- and three-sectional thin diaphragms. Numerical results are presented.
The eigenvalue problems for generalized natural modes of an inhomogeneous dielectric waveguide without a sharp boundary and a step-index dielectric waveguide with a smooth boundary of cross-section are formulated as problems for the set of time-harmonic Maxwell equations with partial radiation conditions at infinity in the cross-sectional plane. The original problems are reduced by the integral equation method to nonlinear spectral problems with Fredholm integral operators. Properties of the spectrum are investigated. The Galerkin and collocation methods for the calculations of generalized natural modes are proposed and convergence of the methods is proved. Some results of numerical experiments are discussed.
We consider the construction of guaranteed estimates of linear continuous function als of the unknown solutions and right-hand sides of the Helmholtz equation; the boundary value problems under study are associated with the wave diffraction by a bounded body D situated in a domain Omega is an element of R-n, n = 2, 3, whose boundary partial derivative Omega stretches to infinity (e.g., a wedge or a layer) and Green's function Phi(k) (x, y), (x, y is an element of Omega, x not equal y) corresponding to wave number k with k > 0 and boundary condition (I)k (x, y)vertical bar y is an element of Omega = 0 is known [4]. Here, for a function u(y) defined in (Omega) over bar Bu(y)vertical bar(y is an element of partial derivative Omega) + beta partial derivative u(y)/partial derivative y vertical bar(y is an element of partial derivative Q), alpha, beta = 0, 1, alpha + beta = 1, v is outward normal to aft We assume that right-hand sides of the equations entering the problem statement are not known; the only available information is that they belong to a bounded set of the space of square-integrable functions. In order to solve these estimation problems we need additional data: observations in the form of certain linear transformations of the solution distorted by noise. The latter are realizations of the random fields with the unknown second moment functions belonging to a given bounded set in the appropriate functional space. The approach set forth in and developed in this study allows us to obtain optimal estimates of the unknown solution or righthand sides of the equations and linear functionals, i.e., estimates sought in the class of functionals linear with respect to observations for which the maximal mean-square estimation error taken over all elements belonging to the aforementioned sets takes minimal value. Such estimates are called minimax or guaranteed estimates. We obtain representations for these estimates and estimation errors in terms of solutions to certain integro-differential or integral equations in bounded subdomains of domain Omega \ D.
We are looking for linear with respect to observations optimal estimates of solutions and right-hand sides of Maxwell equations called minimax or guaranteed estimates. We develop constructive methods for finding these estimates and estimation errors which are expressed in terms of solutions to special variational equations and prove that Galerkin approximations of the obtained variational equations converge to their exact solutions. © Springer International Publishing Switzerland 2013.
Generalized Chebyshev iteration (GCI) applied for solving linear equations with nonselfadjoint operators is considered. Sufficient conditions providing the convergence of iterations imposed on the domain of localization of the spectrum on the complex plane are obtained. A minimax problem for the determination of optimal complex iteration parameters is formulated. An algorithm of finding an optimal iteration parameter in the case of arbitrary location of the operator spectrum on the complex plane is constructed for the generalized simple iteration method. The results are applied to numerical solution of volume singular integral equations (VSIEs) associated with the problems of the mathematical theory of wave diffraction by 3D dielectric bodies. In particular, the domain of the spectrum location is described explicitly for low-frequency scattering problems and in the general case. The obtained results are discussed and recommendations concerning their applications are given. (C) 2013 Elsevier Inc. All rights reserved.
The problem of diffraction of an electromagnetic field by a locally nonhomogeneous body in a perfectly conducting waveguide of rectangular cross section isconsidered. This problem is reduced to solving a volume singular integral equation (VSIE). The examination of this equation is based on the analysis of the corresponding boundary value problem(BVP) for the system of Maxwell's equations and the equivalence of this BVP and VSIE. The existence and uniqueness for VSIE in the space of square-integrable functions are proved. A numerical Galerkin method for the solution of VSIE is proposed, and its convergence is proved