Volume 58, Nº 4 (2018)

For a two-dimensional model of thermal scattering, inverse problems arising in the development of tools for cloaking material bodies on the basis of a mixed thermal cloaking strategy are considered. By applying the optimization approach, these problems are reduced to optimization ones in which the role of controls is played by variable parameters of the medium occupying the cloaking shell and by the heat flux through a boundary segment of the basic domain. The solvability of the direct and optimization problems is proved, and an optimality system is derived. Based on its analysis, sufficient conditions on the input data are established that ensure the uniqueness and stability of optimal solutions.

A system of loaded ordinary differential equations with multipoint conditions is considered. The problem under study is reduced to an equivalent boundary value problem for a system of ordinary differential equations with parameters. A system of linear algebraic equations for the parameters is constructed using the matrices of the loaded terms and the multipoint condition. The conditions for the unique solvability and well-posedness of the original problem are established in terms of the matrix made up of the coefficients of the system of linear algebraic equations. The coefficients and the righthand side of the constructed system are determined by solving Cauchy problems for linear ordinary differential equations. The solutions of the system are found in terms of the values of the desired function at the initial points of subintervals. The parametrization method is numerically implemented using the fourth-order accurate Runge–Kutta method as applied to the Cauchy problems for ordinary differential equations. The performance of the constructed numerical algorithms is illustrated by examples.

This paper examines the solution of convolution-type integral equations of the first kind by applying the Tikhonov regularization method with two-parameter stabilizing functions. The class of stabilizing functions is expanded in order to improve the accuracy of the resulting solution. The features of the problem formulation for identification and adaptive signal correction are described. A method for choosing regularization parameters in problems of identification and adaptive signal correction is suggested.

For the acoustic-sensing problem of determining the characteristics of a local inhomogeneity scattering a wave field in three-dimensional space, a numerical algorithm is proposed and justified that is efficient in terms of computational resources and CPU time. The algorithm is based on the fast Fourier transform, which is used under certain a priori assumptions on the character of the inhomogeneity and the observation domain of the scattered field. Typical numerical results obtained by solving this inverse problem with simulated data on a personal computer are presented, which demonstrate the capabilities of the algorithm.

The discrete source method is used to develop and implement a mathematical model for solving the problem of scattering electromagnetic waves by a three-dimensional plasmonic scatterer with nonlocal effects taken into account. Numerical results are presented whereby the features of the scattering properties of plasmonic particles with allowance for nonlocal effects are demonstrated depending on the direction and polarization of the incident wave.

The paper presents a mathematical model and results of numerical simulation of axisymmetric plasma flows in nozzle-type channels formed by two coaxial electrodes. Transonic accelerated flows, which are of interest for the development of plasma accelerators, are considered. The mathematical apparatus of the models are two-dimensional MHD problems solved numerically, the steady-state solutions of which are obtained in the process of relaxation. Some characteristics of the flow in narrow tubes between close trajectories are considered in the quasi-one-dimensional approximation. The main attention is paid to the influence of the longitudinal magnetic field and the curvilinear geometry of the electrodes on the properties of accelerated flows.

A compact metric space with a bounded Borel measure is considered. Any measurable set of diameter not exceeding r is called r-cluster. The existence of a collection consisting of a fixed number of 2r-clusters possessing the following properties is investigated: the clusters are located at the distance r from each other and the collection measure (the total measure of the clusters in the collection) is close to the measure of the entire space. It is proved that there exists a collection with a maximum measure among such collections. The concept of r-parametric discretization of the distribution of distances into short, medium, and long distances is defined. In terms of this discretization, a lower bound on the measure of the maximum-measure collection is obtained.