Vol 28, No 1 (2024)

In this article, we consider the dynamics of three identical qubits interacting not-resonantly with a thermal field of an ideal resonator with a Kerr medium. We have found the solutions of the Liouville quantum equation for the total density matrix of a system under consideration for the initial separable, biseparable, and genuine entangled states of the qubits and the thermal initial state of the resonator field. By averaging the total density matrix over the variables of the resonator field and the variables of one of the qubits, we found the reduced density matrix of the pair of remaining qubits. Two-qubit density matrices were used to calculate the qubit-qubit negativity. The results showed that detuning and Kerr nonlinearity can greatly enhance the amout of entanglement for initial separable state of a pair of qubits. It is also shown that detuning and a Kerr medium can inhibit the sudden death of entanglement.

In this study, a modified Cauchy problem was examined for an inhomogeneous equation of degenerate hyperbolic type of the second kind in a characteristic triangle. It is known that degenerate hyperbolic equations have a singularity, meaning that the well-posedness of the Cauchy problem with initial data on the line of parabolic degeneracy does not always hold for them. Therefore, in such cases, it is necessary to consider the problem with initial conditions in a modified form.
In present paper, modified Cauchy problems with initial conditions were formulated on the line of parabolic degeneracy for an inhomogeneous equation of degenerate hyperbolic type of the second kind. The considered problem is reduced to a modified Cauchy problem for a homogeneous equation and to a Cauchy problem for an inhomogeneous equation with zero initial conditions. The solutions of the modified Cauchy problem for a homogeneous equation are derived from the general solution of the considered equation. The explicit solutions of the modified Cauchy problem with homogeneous conditions for the inhomogeneous equation are found using the Riemann
method.
It is proven that the discovered solutions indeed satisfy the equation and the initial conditions.

The paper presents a method for determining the stress-strain state of transversely isotropic bodies of revolution under the action of non-axisymmetric stationary volumetric forces. This problem involves the use of boundary state method definitions. The basis of the space of internal states is formed using fundamental polynomials. The polynomial is placed in any position of the displacement vector of the plane auxiliary state, and the spatial state is determined by the transition formulaes. The set of such states forms a finite-dimensional basis according to which, after orthogonalization, the desired state is expanded into Fourier series with the same coefficients. Series coefficients are scalar products of vectors of given and basic volumetric forces. Finally, the search for an elastic state is reduced to solving quadratures.
The solutions of problems of the theory of elasticity for a transversely isotropic circular cylinder from the action of volumetric forces given by various cyclic laws (sine and cosine) are analyzed. Recommendations are given for constructing the basis of internal states depending on the form of the function of given volumetric forces. The analysis of the series convergence and the estimation of the solution accuracy in graphical form are given.

The article presents new mathematical models for the dynamics of nonlinear nano/micro/macro-scale functionally graded porous closed cylindrical shells. The Kirchhoff–Love hypothesis is chosen as the kinematic model for the shells. Geometric nonlinearity is considered according to the von Karman model. Nanoeffects are accounted for using by a modified moment theory of elasticity. Variational and differential equations, as well as boundary and initial conditions, are derived from Hamilton’s principle. A proof of the existence of a solution is conducted based on the theory of generalized solutions to differential equations (using methods of Hilbert spaces and variational methods).
As examples, nano/micro/macro-scale closed cylindrical shells are considered as systems with "almost" an infinite number of degrees of freedom subjected to banded transverse alternating loading. The Bubnov–Galerkin method in higher approximations is adopted as the method for reducing partial differential equations to the Cauchy problem. Its convergence is investigated.
The Cauchy problem is solved using Runge–Kutta methods of fourth to eighth order accuracy and the Newmark method. The application of several numerical methods at each stage of modeling is necessary to ensure the reliability of the obtained results. The study of complex oscillation characteristics of the closed cylindrical nano/micro/macro-scale shell is conducted using nonlinear dynamics methods, which involve constructing signals, phase portraits, applying Fourier analysis, and various wavelet transformations,
among which the Morlet wavelet proved to be the most informative.
An analysis of the type of chaotic oscillations is carried out based on the spectrum of Lyapunov exponents using the Sano–Sawada method and the dominant exponent through several methods: Kanca, Rosenstein, and Wolf. It is shown that the size-dependent parameter and the consideration of porosity have a significant impact on the nature of the oscillations of cylindrical shells. The phenomenon of hyper-chaos has been discovered.

A new version of the implicit iterative scheme is proposed for the implementation of which only matrix-vector computational procedures are required. This makes the proposed computational scheme potentially highly efficient for solving a wide class of high-dimensional problems on modern high-performance computing platforms, such as Nvidia Cuda. It is shown that the proposed algorithms can be used to solve ill-conditioned linear systems and least squares problems, as well as to construct iterative regularization algorithms. The results of computational experiments are presented, confirming the effectiveness of the proposed computational algorithms.

Research the properties of iron-based solid solutions by Mössbauer spectroscopy has the problem of interpreting the results of processing experimental data within the traditional mathematical model. Since the disordered solid solutions, for example, as a result of mechanical activation, are consisted of a set of the different local atomic configurations, the corresponding Mössbauer spectra contain a large number of the shifted relative to each other spectral components with close values of the hyperfine interaction parameters. The magnitude and sign of these shift are determined by many factors: the quantitative distribution of atoms of each type in the coordination spheres, the symmetry of their distribution relative to the quantization axis, the possible local shift relative to the average statistical position in the crystallographic structure, etc. In the mathematical model, as a rule, it’s not possible to taken into account all these effects of the shift by analytically.
The proposed extended mathematical model for describing the Mössbauer spectra of solid solutions makes it possible to take into account the shifts in the spectral components by using Gaussian normal distribution function, as a function of statistical set of local distortions. The width of the Gaussian distribution makes it possible to estimate the degree of local distortions of the crystal lattice that arise due to differences in the sizes of atoms of the mixed components, local distortions of the structure and
symmetry of the environment of the resonant atom.
The inverse problem of nuclear gamma-resonance is formulated by the Fredholm integral equation of the first kind and is an ill-posed problem with a priori constraints on the desired solution. The introduction of two Gaussian functions with a priori unknown linewidths into the kernel of the integral equation leads to the problem of solving the equation by classical methods. Algorithm for obtaining a reliable solution based on the Tikhonov regularization method with correction of the parameters of the kernel of the
integral equation is proposed in this paper. On the examples of the study of real objects, the reliability and informative application of the extended mathematical model of the inverse problem of nuclear gamma-resonance is proved.

In the paper we consider the second boundary value problem in a rectangular domain for an inhomogeneous partial differential equation of the third
order with multiple characteristics with variable coefficients. The uniqueness
of the solution of the posed problem is proved by the method of energy integrals. The uniqueness theorem is proved. A counter-example is constructed
in the case of violation of the conditions of the uniqueness theorem. By the
method of separation of variables, the solution of the problem is sought as
a product of two functions X (x) and Y (y). In order to determine Y (y),
we generate an ordinary differential equation of the second order with two
boundary conditions at the boundaries of the segment [0, q]. For this problem, the eigenvalues and the corresponding eigenfunctions, when n = 0 and
n ∈ N, are found. For determining X (x), we generate an ordinary differential
equation of the third order with three boundary conditions at the boundaries
of the segment [0, p]. A new function, which makes the boundary conditions
homogeneous, is introduced. The solution of the given problem is constructed
by means of the Green function. For n = 0 and for n ∈ N, the Green function
is individually constructed. It is verified that the obtained Green’s functions
satisfy the boundary conditions and properties of the Green function. The
solution X (x) is written by virtue of the constructed Green functions. After
some transformations, the Fredholm integral equation of the second kind is
generated and its solution is written in terms of the resolvent. Estimates
for the resolvent and the Green function are obtained. The uniform convergence of both solution and its possible partial derivatives are shown under the conditions on the given functions. The convergence of the third order derivative of the solution with respect to the variable x is proved using the Cauchy-Bunyakovsky and Bessel inequalities. When justifying the uniform
convergence of the solution, the absence of a “small denominator” is proved.