Vol 27, No 2 (2023)

March 25, 2023 marks the 70th anniversary of the famous Russian scientist, honorary worker of higher and professional education of the Russian Federation, teacher, administrator, organizer of science and higher education in Russia, Doctor of Physical and Mathematical Sciences, Professor Leonid Alexandrovich Saraev.
The article presents the key bibliographic data of Leonid A. Saraev, presents the main scientific directions and results of scientific activity on the fundamental problems of predicting the nonlinear properties of composite materials and the development of mathematical and stochastic methods and models for economic analysis.

The residual power series method is effective for obtaining approximate analytical solutions to fractional-order differential equations. This method, however, requires the derivative to compute the coefficients of terms in a series solution. Other well-known methods, such as the homotopy perturbation, the Adomian decomposition, and the variational iteration methods, need integration. We are all aware of how difficult it is to calculate the fractional derivative and integration of a function. As a result, the use of the methods mentioned above is somewhat constrained. In this research work, approximate and exact analytical solutions to time-fractional partial differential equations with variable coefficients are obtained using the Laplace residual power series method in the sense of the Gerasimov–Caputo fractional derivative. This method helped us overcome the limitations of the various methods. The Laplace residual power series method performs exceptionally well in computing the coefficients of terms in a series solution by applying the straightforward limit principle at infinity, and it is also more effective than various series solution methods due to the avoidance of Adomian and He polynomials to solve nonlinear problems. The relative, recurrence, and absolute errors of the three problems are investigated in order to evaluate the validity of our method. The results show that the proposed method can be a suitable alternative to the various series solution methods when solving time-fractional partial differential equations.

A system consisting of two identical qubits not-resonantly interacting with a thermal quantum field of a lossless resonator with a Kerr media via degenerate two-photon transition is considered. An exact solution of the quantum Liouville equation for the total density matrix of the considered system is obtained. To solve the quantum evolution equation we used the dressed states representation. The complete set of dressed states is found. The exact solution of the quantum Liouville equation is used to calculate the time dependencies of qubit-qubit entanglement parameter (negativity) for Bell type entangled qubits states. The results showed that Kerr nonlinearity can diminish the amplitudes of the Rabi oscillations of entanglement parameter and suppress the effect of sudden death of entanglement.

A method for predicting creep and long-term strength in conditions of viscous failure mechanism has been proposed and implemented. It is assumed that when the material is loaded, there is no instant plastic deformation or the first stage of creep, and the hypothesis of incompressibility is satisfied. In the developed method, it is shown that if the creep curve under constant stress and the time to failure are known for a pre-tested sample (leader sample), then to obtain the rheological deformation diagram and long-term strength of the material at other stress levels, it is sufficient to know only the initial minimum creep deformation rate (at the initial moment of time) for the samples at these stress levels.
The adequacy of the developed method to experimental data for a range of alloys under conditions of tension and torsion of samples has been tested. It has been shown that the prediction results do not depend on the choice of a leader sample from the series of samples tested at different stress levels.
The research results demonstrate that the developed method allows not only predicting creep curves and long-term strength (in the asymptotic formulation), but also optimizing the planning of experimental studies to obtain a series of steady-state creep curves under constant stresses.

Evaporating droplets and films are used in applications from different fields. Various methods of evaporative self-assembly are of particular interest. The paper describes a mathematical model of mass transfer in a droplet drying on a substrate based on the lubrication approximation. The model takes into account the transfer of a dissolved or suspended substance by a capillary flow, the diffusion of this substance, the evaporation of liquid, the formation of solid deposit, the dependence of the viscosity and the vapor flux density on the admixture concentration.
The case with pinning of the three-phase boundary (“liquid–substrate–air”) is considered here. Explicit and implicit finite-difference schemes have been developed for the model equations. A modification of the numerical method is proposed, in which splitting by physical processes, the iterative method of explicit relaxation and Thomas algorithm are combined. A practical recipe for suppressing sawtooth oscillations is described using the example of a specific problem.
A software module in C++ has been developed, which can be used for evaporative lithography problems in the future. With the help of this module, numerical calculations were carried out, the results of which were compared with the results obtained in the Maple package.
Numerical simulation predicted the case in which the direction of the capillary flow changes to the opposite over time due to a change in the sign of the gradient of the vapor flux density. This can lead to a slowdown in the transfer of the substance to the periphery, which as a result will contribute to the formation of a more or less uniform precipitation over the entire contact area of the droplet with the substrate. This observation is useful for improving methods of annular deposit suppression associated with the coffee-ring effect and undesirable for some applications, such as inkjet printing or coating.

The paper outlines a method for numerical analysis of sets of solutions for linear systems of ordinary differential equations that contain perturbations in the right-hand side. The method determines extreme values of the solutions, which comprise the sets of solutions along the coordinate axes or in a specified direction. The estimations are based on using the Cauchy operator, written with symbolic formulas for variations of arbitrary constants. Additionally, control is implemented over the deviation of solutions when calculating a bundle of trajectories. The paper also is devoted to examples of estimating reachability sets of systems under the influence of control and disturbance effects.

An n-dimensional system of equations with dominant partial derivatives of the nth order is being studied. The distinguishing feature of the considered system compared to other systems with partial derivatives is the presence of a first term in the equations on the right side of the system, representing a dominant derivative, while all other derivatives appearing in the system equations are obtained from it by discarding at least one differentiation with respect to any of the independent variables. The aim of the study is to find conditions for the unique solvability of the Goursat problem for the considered system. The main problem is reduced to a system of integral equations, the solution of which exists and is unique when the requirements of continuity of the kernels and right sides of this system are satisfied in the corresponding closed parallelepipeds of variable ranges. Conditions under which the main problem is uniquely solvable have been obtained. The final result in terms of the coefficients of the original system is formulated as a theorem.