University proceedings. Volga region. Physical and mathematical sciences

Background. Hypersingular integral equations on a segment that arise in many issues of mathematical physics are considered. Materials and methods. Hypersingular equations are studied in special classes of functions, which are represented by Fourier series of Chebyshev polynomials of the 2nd kind. Results and conclusions. The criteria of compactness of operators in special classes of functions are proved. The main result is the proof of the Fredholm property of hypersingular operator in special classes of functions, which is important in the formulation and implementation of a numerical method for solving hypersingular equations.

Background. In the theory of linear differential equations, transformations generated by differential substitutions of dependent variables play an essential role. The study of these transformations led to create a general theory of differential symmetry algebras of homogeneous linear systems of differential equations and to the theory of differential homomorphisms. These theories turned out to be closely related to the concept of the theorem on the zeros of linear differential operators (LDO). To date, several theorems on the zeros of linear equations have been proved, but these theorems are not sufficient to study the algebras of differential symmetry and the relations between different types of linear homogeneous systems of differential equations. The formulation and proof of new theorems about the zeros of LDO is an urgent task. The main objective of the work is to formulate and prove a version of the formal theorem on the zeros of linear differential operators. Another important objective is to construct examples of the theorem’s application that confirm its usefulness and validity. Materials and methods. Section 1 (Introduction) provides general information on the works that present theorems on zeros of LDO. The meaning of formal theorems on zeros and the role that such particular theorems can play in the general theory are explained. Section 2 presents the basic notations and concepts, and provides a definition of the theorem on zeros of linear differential operators for a family of modules over the ring of scalar linear differential operators. Section 3 describes the elements of the theory of pseudoinverses of matrices and operators, which are used in proving the main theorem of the work. Results. Section 4 formulates and proves a version of the formal theorem on zeros (Theorem 1). Section 5 gives examples of families of linear differential operators for which the conditions of Theorem 1 are satisfied (Theorems 2, 3, 4). In addition, a method for constructing local sections in the general pseudoinverse problem is described; a pseudoinverse matrix is applied in a new situation; a special basis is used in which the coordinates of the LDO coincide with its coefficients; a useful concept of the matrix of the main symbols of the LDO by columns is introduced. Conclusions. The results of the work can serve as the basis for proving the validity of the formal theorem on zeros for a set of specific linear differential operators and families of operators.

Background. Boundary value problems for Maxwell's equations are widely used in various fields of electrodynamics due to their ability to model complex physical situations associated with the interaction of electromagnetic waves with boundaries and thin layers of materials. The objective of this work is to derive and analyze a system of integral equations for the problem of electromagnetic wave diffraction on a dielectric ball coated with graphene, and to prove the existence and uniqueness of a solution to the boundary value problem. Materials and methods. Using a combination of Stratton-Chu formulas, a system of vector integral equations over the surface of a sphere is obtained. Results. A system of scalar singular integral equations is obtained for searching for four unknown functions. The theorem on the existence and uniqueness of the solution of the system of equations, aswell as the existence and uniqueness of the solution of the boundary value problem of diffraction is proved. Conclusions. The problem of electromagnetic wave diffraction on a dielectric ball coated with graphene has been studied, and a system of equations for numerical solution has been obtained.

Background. The purpose of the paper is the development of a computational algorithm for approximate recovery of derivative order in the generalized wave equation. Urgency of the stated problem is dictated not only by significant need for improvement of mathematical apparatus for solution of inverse and ill-posed problems but also by the growing number of applications of equations with partial derivatives of fractional order to math-ematical modelling in different fields of physical and technical sciences. Materials and methods. The approach for solution of the stated problem is based on its reduction to a nonlinear in the unknown parameter integral equation and subsequent solution of this integral equation with the help of continuous operator method for solution of nonlinear equations in Banach spaces. Results. Application of the continuous operator equation made it possible to develop the numerical algorithm for recovery of fractional derivative order in the generalized wave equation on the extra assumption of that the solution of this equation is additionally known at one arbitrary point. Conclusions. The approach described in this paper appears to be quite efficient for solution of inverse problems for partial differential equations with fractional order derivatives. Extending of the used approach to a wider range of inverse and ill-posed problems for equations with fractional order derivatives is of considerable interest.

Background. The features of determining the parameters of the cubic crystal lattice of clathrate hydrates are considered. Many properties of clathrate hydrates are similar to hexagonal ice, however, the interaction of absorbed molecules with the ice-like crystal lattice has its own characteristics. Materials and methods. The main method used in the work is to obtain the parameters of the functional dependence using the least squares method. A poly-nomial approach to a unified description is proposed in view of the complex nature of the motion of guest molecules with their numerous degrees of freedom and various degrees of coupling of this motion with the host lattice. Results. It is proposed to use the obtained ratio depending on the system temperature and the type of hydrate former. Average discrepancies according to the proposed method for hydrate formers in temperature ranges from 10 to 280 K are 0.04% and do not exceed 0.09%. Conclusions. The developed approach allows to obtain more accurate results in a wide range of conditions.

Background. Currently, methods of contactless control of the dielectric properties of semiconductor nanostructures and the matrix surrounding them are of considerable interest. Optical modulation of permittivity in combination with controlled tunneling processes makes it possible to change the properties of low-dimensional structures in a targeted manner and, as a consequence, optimize the characteristics of semiconductor nanoelectronic devices. In this regard, semiconductor quantum dots tunnel-coupled to the surrounding matrix are of interest, since in such structures the formation of impurity complexes A⁺ + e is possible, the photoexcitation of which can lead to the photodielectric effect (PDE). The purpose of the work is to theoretically study the influence of tunnel transparency of the potential barrier on the PDE associated with the excitation of impurity complexes A⁺ + e in quasi-zerodimensional structures in an external magnetic field. Materials and methods. The relative change in permittivity (RCDP) is calculated in the dipole approximation. The field dependence curves of RCDP are plotted for an InSb quantum dot. Numerical calculations and plotting were performed using the numerical mathematics systems Mathcad 14.0 and Wolfram Mathematica 10.2. Results. The dependence of the RCDP in a quasi-zerodimensional semiconductor nanostructure on the magnitude of the external magnetic field induction and the parameters of 1D dissipative tunneling was investigated in the dipole approximation. Dichroism of PDE associated with the presence of an external magnetic field was detected. It is shown that an external magnetic field suppresses PDE, which is associated with an increase in the localization of the electron wave function in the magnetic field, as well as with a modification of the electron adiabatic potential. It is shown that the magnitude of the RCDP depends on the parameters of dissipative 1D tunneling. Conclusions. In a magnetic field, effective control of the PDE is possible by modifying the electron adiabatic potential and the electron wave function by varying the parameters of dissipative tunneling.