Alexander B. Samokhin

In the article, interpolation-based and mosaic-skeleton approaches for solving the problem of the potential flow of a plate are compared. They compress a dense matrix of a linear system, arising in the collocation method on the non-uniform grid. The first method relies on Fast Fourier Transform and linear interpolation from an additional uniform grid. The second uses block low-rank approximation of a matrix. Both techniques demonstrate effectiveness in time and memory but distinguish different structures in a matrix, which affects the solution of a linear system. In our implementation, the mosaic-skeleton method solves the problem quicker than its competitor. Still, it consumes more memory, and its duration grows more significantly with an increase in the size of a system.

This paper deals with volume singular integral equations describing the problems of low-frequency scattering of electromagnetic waves in bounded three-dimensional anisotropic structures. The spectrum of integral operators is studied. The domain of the operator spectrum in the complex plane for the low-frequency case is presented explicitly. The generalized simple iteration method is described for the application of which it is necessary to know the area of the operator spectrum in the complex plane. The collocation method on a uniform grid is used to discretize the integral equations. Using a fast discrete Fourier transform, this allows constructing an algorithm for fast multiplication of the matrix of a system of linear equations by a vector. The results of numerical solution of the considered problems are given.

A new method of digital signal processing of angle measuring systems based on digital antenna arrays is justified. It allows you to improve the image quality of one- and two-dimensional objects. The method can be applied in the optical, infrared, and terahertz ranges of electromagnetic wavelengths. In the process of numerical experiments, the effectiveness of the proposed method is compared with the known ones. The results of experiments on mathematical models are presented and discussed.

We consider the differential and integral statements of problems of acoustic scattering by three-dimensional bounded transparent structures described by an integral equation. The results of the numerical solution of the integral equation describing the class of problems under consideration are presented. A theorem on existence and uniqueness of the solution is proved.

The problems of scattering of electromagnetic waves by anisotropic dielectric structures are studied on the basis of volumetric singular integral equations. Existence and uniqueness theorems for solutions are given for a wide class of media, including lossless media. An efficient method for solving integral equations based on the collocation method and fast discrete Fourier transform algorithms is described.

Objectives. Integral equations have long been used in mathematical physics to demonstrate existence and uniqueness theorems for solving boundary value problems for differential equations. However, despite integral equations have a number of advantages in comparison with corresponding boundary value problems where boundary conditions are present in the kernels of equations, they are rarely used for obtaining numerical solutions of problems due to the presence of equations with dense matrices that arise that when discretizing integral equations, as opposed to sparse matrices in the case of differential equations. Recently, due to the development of computer technology and methods of computational mathematics, integral equations have been used for the numerical solution of specific problems. In the present work, two methods for numerical solution of two-dimensional and three-dimensional integral equations are proposed for describing several significant classes of problems in mathematical physics.Methods. The method of collocation on non-uniform and uniform grids is used to discretize integral equations. To obtain a numerical solution of the resulting systems of linear algebraic equations (SLAEs), iterative methods are used. In the case of a uniform grid, an efficient method for multiplying the SLAE matrix by vector is created.Results. Corresponding SLAEs describing the considered classes of problems are set up. Efficient solution algorithms using fast Fourier transforms are proposed for solving systems of equations obtained using a uniform grid.Conclusions. While SLAEs using a non-uniform grid can be used to describe complex domain configurations, there are significant constraints on the dimensionality of described systems. When using a uniform grid, the dimensionality of SLAEs can be several orders of magnitude higher; however, in this case, it may be difficult to describe the complex configuration of the domain. Selection of the particular method depends on the specific problem and available computational resources. Thus, SLAEs on a non-uniform grid may be preferable for many two-dimensional problems, while systems on a uniform grid may be preferable for three-dimensional problems.

We consider the application of the method of boundary integral equations to the problem of scattering of an electromagnetic field obeying the nonstationary Maxwell equations by a system of perfectly conducting objects. An integral representation of the electromagnetic field in terms of surface currents is used. The main result of the paper is the proof of the existence of boundary values for the electric field defined by such an integral representation that are expressed via a hypersingular integral understood in the sense of the Hadamard finite part. This permits reducing the problem to an integro-differential equation of evolution type with delay containing a hypersingular surface integral. It is also proved that if this equation has a solution in a certain function class, then the electric and magnetic fields determined by the corresponding integral representations are a solution of the original scattering problem for Maxwell’s equations.

We derive volume integral equations with time delay describing nonstationary problems of acoustic field scattering by transparent three-dimensional structures. An efficient method for the numerical solution of the resulting equations is proposed.

It is proven that under certain conditions, in a limited dielectric medium without losses, there exist nonzero solutions to the homogeneous problem. If the medium is anisotropic, then in the region of inhomogeneity there can be electromagnetic energy and, at the same time, there is no radiation into the surrounding space.