Volume 13, Issue 3

In this paper some problem formulations of dynamic object parameters recovery described by non-autonomous system of ordinary differential equations with multipoint unshared edge conditions are investigated. Depending on the number of additional conditions the problem is reduced to an algebraic equations system or to a problem of quadratic programming. With this purpose the paper offers a new scheme of the edge conditions transfer method called by conditions shift. The method permits to get rid from differential links and multipoint unshared initially-edge conditions. The advantage of the proposed approach is concluded by capabilities of reduction of a parametric identification problem to essential simple problems of the solution of an algebraic system or quadratic programming.

The problem of reconstruction of vector and 2-tensor fields, given in refracting medium, is considered. The approach of direct reconstruction of a solenoidal part is presented. Least squares method is exploited for a reconstruction of solenoidal parts of original vector and 2-tensor fields by usage of certain solenoidal bases of polynomial type constructed for this purpose. The approach is justified theoretically by consideration of polynomial decompositions of polynomial vector and 2-tensor fields. The problems of existence, uniqueness and convergence of obtained approximation to the solenoidal part of original field are solved. The results of numerical simulation are presented and discussed.

Direct and inverse acoustic scattering problems involving smart obstacles are proposed and some ideas to study them are suggested. A smart obstacle is an obstacle that when hit by an incoming acoustic wave reacts circulating on its boundary a pressure current, that is a quantity dimensionally given by a pressure divided by a time, in order to generate a scattered wave that pursues a preassigned goal. In our models (see [6–10,12]) the smart obstacle pursues one of the following goals: i) to be undetectable, ii) to appear with a shape and/or acoustic boundary impedance different from its actual ones, iii) to appear with a shape and/or acoustic boundary impedance and in a location in space different from its actual ones. That is, in the first case the smart obstacle tries to be furtive, in the second case it tries to be masked that is it tries to appear as another obstacle that we call the mask and finally in the third case it tries to appear as another obstacle in a location in space different from its actual one. We refer to this last apparent obstacle as the ghost. The direct scattering problem considered is the following: given the incoming acoustic field, the obstacle, its acoustic impedance and its goal formulate an adequate mathematical model for the problems previously considered and find the optimal strategy to pursue the assigned goal within the proposed model. The inverse scattering problem considered is the following: given the knowledge of several far fields generated by the smart obstacle when hit by known incident acoustic fields it reacts with the optimal strategy and the knowledge of the goal pursued by the obstacle find the obstacle (i.e. find the shape, acoustic impedance and spatial location of the obstacle). For simplicity in this paper we limit our attention to the case of the obstacle that tries to be masked when the incoming acoustic field is time harmonic. Moreover in the inverse problem we assume that the acoustic boundary impedance of the obstacle and of the mask are known. In this case the direct scattering problem is translated in a constrained optimization problem and its solution is characterized as the solution of a set of auxiliary equations. The inverse scattering problem is translated in a two steps optimization procedure. Finally in a test case the inverse problem is solved numerically.

The aim of this work is to prove the existence and uniqueness theorems of the solution of an inverse problem and associated with them an initial boundary-value problem for a nonlinear model describing the process of coupling of electromagnetic and elastic waves. We start by introducing a simple model consisting from two differential equations, one of them is a hyperbolic equation (an analog of the Lamé system) and another one is a parabolic equation (an analog of the diffusion approximation of Maxwell's system) coupled by nonlinear terms in both equations. An application of this result can be found in explanation of nonlinear processes in the theory of electro-magnetoelastic interactions in continuum physics and geophysics.