Berestov A.V., Kolychev V.D., Kudryavtsev Ye. M.
The goal of research and development is to increase the reliability, environmental friendliness, and economic efficiency of water-based nuclear power reactors of the new generation with promising types of nuclear fuel. The efficiency of the organization of high-tech system production for industrial and operational quality control of nuclear fuel elements and assemblies of water–water reactors of the new generation. Within the creation of small-scale production of parameter control systems, the characteristics affecting the time, cost, and resource project parameters of product sample manufacturing are analyzed. Attention is paid to the economic aspects of exploitation of systems for parameter monitoring of high-tech products in order to reduce costs, both at the stage of functioning and at the stage of production of nuclear fuel elements and assemblies. Taking into account the operating conditions of the nuclear fuel elements, the features of the controlled defects are revealed in order to present additional requirements that must be taken into account in the design of high-tech products. The economic model for estimating the efficiency of small-scale production of quality control systems using methods of the queuing theory is considered. The schedule of the project, the plan of attraction of borrowed financial resources, the production plan are built taking into account factors of the taxation and inflation and costs of attraction of personnel. Using the method of discounted cash flows, the integral indicators of the efficiency of making decisions are calculated. The risks and uncertainties of project implementation are analyzed. Application of methods for analyzing the sensitivity of integrated financial indicators for change prediction in uncertain data, as well as the Monte Carlo method, allows adjusting the strategy of the project in order to minimize future losses. The results obtained make it possible to formulate conclusions about the feasibility of the project with varying time, cost and resource parameters affecting the integral economic efficiency of developed solutions.
Sorokin V.G., Polyanin A.D.
Nonlinear reaction–diffusion models with a constant delay τ have been considered. Specific qualitative features of numerical integration of initial boundary value problems for partial differential equations with delay by the method of lines have been described. The method is based on the approximation of spatial derivatives by corresponding finite differences; as a result, the initial equation is replaced by an approximate system of ordinary differential equations with delay. The resulting system is then solved by methods of two classes—Runge–Kutta and BDF (Gear)—which are built into the Wolfram Mathematica system. An extensive comparison of the numerical and exact solutions of the model test problems has been given. The τ stability of a stationary solution of a model initial boundary value problem has been analyzed in the linear approximation and the instability of the solution at small τ values has been proven. It has been shown that singularity can be suppressed by introducing a delay in a blow-up problem. Some illustrative blow-up problems are presented.
Gaiur I.Y., Kudryashov N.A.
A travelling-wave reduction of the fourth-order partial differential equation, which is a continuous model corresponding to the generalization of both the Fermi–Pasta–Ulam and the Frenkel–Kontorova models, is considered. It is shown that the resulting equation can be considered as mechanical system with a polynomial Hamiltonian and the standard Poisson bracket. Using the Lax representation of the continuous model, we obtain a Lax pair for the travelling-wave reduction and the considered Hamiltonian system. The first integrals of the considered system have been obtained from the trace of L matrix squared as the coefficients of the power expansion in the spectral parameter. It has been shown that the first integrals obtained are independent. Consequently, the considered system is integrable in the Arnold–Liouville sense. The results make it possible to construct the phase space of the considered system.
The local dynamics of a quite general logistic equation with state-dependent delay has been studied in detail. It has been found that this equation has a wide range of bifurcation properties. The regions of stability and instability of a nontrivial equilibrium state are identified in the plane of bifurcation parameters, and critical cases are determined before studying these properties. Thus, all relations between the parameters are revealed, under which qualitative changes take place in the local dynamics of the considered equation. The types of arising bifurcations are studied, and the criteria for the occurrence of each of them are formulated for all critical cases using the method of normal forms. These bifurcations are transcritical and pitchfork, as well as the Andronov–Hopf bifurcation. Asymptotic approximations are constructed for solutions that acquire stability because of the arising bifurcations in each of those cases. The condition for the appearance of the Andronov—Hopf bifurcation in the considered equation is rather cumbersome and inconvenient for manual verification. However, it can be verified by computer calculations because the exact formula for the first Lyapunov value has been obtained. Some special cases are considered and compared to similar equations with constant delay. This comparison indicates that the introduction of state-dependent delay can add bifurcation properties.
Aksenov A.V., Druzhkov K.P.
A system of equations of two-dimensional shallow water above the rough bottom is considered. An overdetermined system of equations for determining the functions forming the conservation laws of the system of shallow water equations is obtained. The general form of the solution of the overdetermined system is found. The general classification equation is given. The system of equations of two-dimensional shallow water above the rough bottom for any profile of the bottom is shown to have no more than the nine-dimensional space of the hydrodynamic conservation laws. The new conservation law, supplementary to the basic conservation law, is obtained. All of the hydrodynamic conservation laws have found for all possible bottom profiles. The corresponding classification equations are presented.
Kudryashov N.A., Kutukov A.A.
The Fermi–Pasta–Ulam model is a dynamic system consisting of a chain of interacting masses. The quadratic and cubic terms in the interaction potential between the particles have been taken into account in this work. The Frenkel–Kontorova model makes it possible to describe the dynamics of dislocations in a crystal lattice including an external periodic potential and a linear interaction potential between the particles. A fourth-order nonlinear partial differential equation obtained by generalizing the Fermi–Pasta–Ulam and Frenkel–Kontorova models and using the continuous limit approximation has been considered. This model allows one to describe the propagation of nonlinear waves in the crystal lattice. The integrability of the equation is studied for various values of the parameters using the Painlevé test. The case where the equation has the Painlevé property, as well as the cases where the Painlevé property test is not performed at the stage of finding the arbitrary constants in the expansion of the solution of the equation in the Laurent series, has been analyzed. The Conte–Fordy–Pickering algorithm has been used to determine the number of arbitrary constants in the expansion of the solution in the case of negative Fuchs indices.
Muminov Kh. Kh., Shokirov F. Sh.
The results of numerical modeling of the processes of phased annihilation of interacting topological vortices of the (2+1)-dimensional supersymmetric O(3) nonlinear sigma model in reversed time are presented. Numerical calculations are carried out in a stratified space using methods of the theory of finite difference schemes and the properties of the stereographic projection. In the first group of experiments, the models of head-on collisions of topological vortices are constructed, where a gradual annihilation process is observed during their interaction. In these models, the process of annihilation of the field of interacting vortices occurs through periodic emission of a fixed amount of energy in the form of localized perturbation waves equivalent to the unit value of the Hopf index. Using the numerical data of the models obtained, initial conditions for simulating the processes of phased annihilation of vortices in reversed time are developed. Models showing the processes of formation of the initial state of the field of two topological vortices because of the concentration of radiation waves toward the center of the resonance zone are obtained. The experiments have been carried out for various values of the Hopf index of topological vortices. Thus, the T-invariance property of the field-theoretic model under study is confirmed. A complex program module that implements a special algorithm for the numerical calculation of the evolution of the interaction of spacetime topological structures in reversed time is proposed.
Kudryashov N.A., Muratov R.V., Ryabov P.N.
The process of plastic flow localization in nonpolar and dipolar materials subjected to high-speed shear deformations is considered. A mathematical model of plastic flow localization, taking into account the process of strain hardening and thermal softening of the material, is formulated. Applying the method of splitting by physical processes, a numerical algorithm is developed and used to develop a software package for the mathematical modeling of the process of plastic flow localization in various materials. Stability conditions for this algorithm are obtained. The developed software package has been verified on known test problems and its efficiency has been demonstrated. The influence of dipolar effects on the processes of plastic flow localization in an OFHC copper sample is considered. It is shown that the inclusion of dipolar effects leads both to an increase in the time necessary for plastic flow localization and to an increase in the width of the localization region. The self-organization of shear bands is considered. The distributions of the width of shear bands, the distance between them, and shear bands along the sample are studied. It is shown that the bands in a dipolar material are uniformly distributed along the sample, whereas the bands in a nonpolar material are formed mainly at its boundary.
Krastelev E.G., Maslennikov S.P.
The triggering dynamics of small controlled vacuum switches at commutated current rise rates of 109–1010 A/s has been studied. The experiments have been performed in the voltage range of 2–5 kV for nanosecond pulses of commutated current with an amplitude up to 1.5 kA. The characteristics of the experimental switches with coaxial design electrode systems with interelectrode gaps of about 1 mm have been studied. The results obtained show that the parameters of the initial phase of the ignition current pulses, i.e., the current of initiating spark discharge at the control gap, have a significant effect on the vacuum arc formation conditions in small switches with short interelectrode gaps and, correspondingly, with the fast dynamics of transient processes. Without taking special measures to increase the initiating discharge power, the switch triggering at commutated current rise rates above 109 A/s occurs in the unstable vacuum arc forming mode, which manifests itself in the instability of the recorded signals with short-term (~10–8 s) surges of the voltage on the spark gap and synchronous dips of the current. The delay times of the triggering and energy losses in the discharge for the stable and unstable vacuum arc formation modes are measured. It is shown that an increase in the steepness of the initial part of the ignition current by means of an additional capacitance connected directly to the control gap terminals make it possible to increase the speed of the switches while maintaining a stable triggering mode with a monotonic rise of the commutated current at a rate above 1010 A/s without an increase in the stored energy and power of the control unit.
Savyolova T.I., Shcherbakova A.
Two normal distributions on the circle SO(2)—coiled normal distribution and Mises distribution—have been considered. When developing statistical methods on the circle, a number of specific properties are identified that distinguish them from statistical methods on the straight line. First of all, this difference is manifested with allowance for periodicity naturally arising on the circle. The compactness of the circle simplifies the study of the convergence of distributions. Two methods for calculating the coiled normal distribution on the circle SO(2) are developed by the Monte Carlo method. The first method consists in modeling the values of a normal random variable that satisfies the Lindeberg–Levy central limit theorem for a sequence of uniformly distributed quantities on the (0, 1) line segment. This method is widely known on R1, but in its application, it is necessary to use the property of SO(2) and to take mod (2π) values. The second method uses the group property of the circle SO(2) and is based on the use of Parthasarathy’s central limit theorem for the group SO(m), m ≥ 2. This method is developed including the properties of the rotation group SO(2) and does not require corrections. Examples of numerical calculations are given. The computations have been processed using the agreement criterion. The answer is presented within an appropriate confidence interval.
Kryanev A.V., Sliva D.E., Udumyan D.K.
A method and algorithm for the formation of efficient portfolios under restrictions on the share of investment of resources, which have the form of group inequalities, have been developed and implemented in the form of a computer program. Uncertainty in the values of the effectiveness of resource development in distributed objects is described in terms of fuzzy sets, which allows us to consider the formulation of the task of forming efficient portfolios with two linear criteria and apply an efficient algorithm to calculate the composition of efficient portfolios corresponding to Pareto solutions. The model of formation of an efficient portfolio based on the priority characteristics has been proposed. This model has no shortcomings of the classical schemes of efficient portfolio formation—Markowitz and VaR schemes. The developed model for the formation of efficient portfolios, as well as the Markowitz scheme, has the property of diversifying resources, implementing a certain rule under conditions of uncertainty : “do not put all eggs in one basket.” However, unlike the Markowitz and VaR schemes, the scheme based on the characteristic of priority is aimed at obtaining the maximum value of the realization of efficiency.