Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование @vestnik-susu-mmp
Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Все статьи: 767
Inverse problem for a linearized quasi-stationary phase field model with degeneracy
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The inverse problem for a linearized quasi-stationary phase field model is considered. The inverse problem is reduced to a linear inverse problem for the first order differential equation in a Banach space with a degenerate operator at the derivative and an overdetermination condition on the degeneracy subspace. The unknown parameter in the problem dependens on the source time function. The theorem of existence and uniqueness of classical solutions is proved by methods of degenerate operator semigroup theory at some additional conditions on the operator. General results are applied to the original inverse problem.
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Статья научная
We consider inverse problems of evolution type for mathematical models of quasistationary electromagnetic waves. It is assumed in the model that the wave length is small as compared with space inhomogeneities. In this case the electric and magnetic potential satisfy elliptic equations of second order in the space variables comprising integral summands of convolution type in time. After differentiation with respect to time the equation is reduced to a composite type equation with an integral summand. The boundary conditions are supplemented with the overdetermination conditions which are a collection of functionals of a solution (integrals of a solution with weight, the values of a solution at separate points, etc.). The unknowns are a solution to the equation and unknown coefficients in the integral operator. Global (in time) existence and uniqueness theorems of this problem and stability estimates are established.
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Inverse problems for some Sobolev-type mathematical models
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The present article is devoted to the study of mathematical models based the Sobolev-type equations and systems arising in dynamics of a stratified fluid, elasticity theory, hydrodynamics, electrodynamics, etc. Along with a solution we determine an unknown right-hand side and coefficients in a Sobolev-type equations of the forth order. The overdetermination conditions are the values of a solution in a collection of points of a spatial domain. The problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. The existence and uniqueness theorems of solutions for the linear and nonlinear cases are proven. In the linear case the result is global in time and it is local in the nonlinear case. The main spaces in question are the Sobolev spaces.
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Inverse spectral problems and mathematical models of continuum mechanics
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The article contains results in the field of spectral problems for mathematical models with discrete semi-bounded operator. The theory is based on linear formulas for calculating the eigenvalues of a discrete operator. The main idea is to reduce spectral problem to the Fredholm integral equation of the first kind. A computationally efficient numerical method for solving inverse spectral problems is developed. The method is based on the Galerkin method for discrete semi-bounded operators. This method allows to reconstruct the coefficient functions of boundary value problems with a high accuracy. The results obtained in the article are applicable to the study of problems for differential operators of any order. The results of a numerical solution of the inverse spectral problem for a fourth-order perturbed differential operator are presented. We study some mathematical models of continuum mechanics based on spectral problems for a discrete semi-bounded operator.
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In this paper, we study the problem of boundary control and final observation for one degenerate mathematical model of motion speed potentials distribution of filtered liquid free surface with the Showalter-Sidorov initial condition. The mathematical model is based on the degenerate Boussinesq equation with an inhomogeneous Dirichlet condition. This model belongs to the class of semilinear Sobolev-type models in which the nonlinear operator is p-coercive and s-monotone. In the paper, the problem of boundary control and final observation for a semilinear Sobolev-type model is considered and conditions for the existence of a control-state pair of the problem are found. In applied studies of a research problem, it is allowed to find such a potentials distributionof filtered liquid free surface, at which the system transitions from the initial condition to a given final state within a certain period of time T.
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Investigation of the transient responses of a beam on an elastic polymeric foundation
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The negative impact of vibrations on various devices and mechanisms can be significant, so it is important to take this factor into account when designing, operating and maintaining various equipment and engineering systems. Various methods and technologies can be used to protect against the negative effects of vibrations. Special damping materials are often used. This research paper is devoted to the analysis of the effectiveness of vibration reduction taking into account the physical parameters of elastic polymeric materials. To conduct the study, a mathematical model describing motion of the beam resting on an elastic polymeric foundation is constructed. The model is based on a system of nonlinear differential equations. An algorithm was developed and applied for the numerical solution of this system of equations. Numerical experiments were carried out for the study of the system reaction to different cases of accelerations. As a result, the deflection structure for materials with different physical characteristics were obtained. These results can serve as a starting point for a deeper study of materials and creation of more complex structures.
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Статья научная
The study of the phase space morphology of the mathematical model deformation of an I-beam, which lies on smooth Banach manifolds with singularities (k-Whitney assembly) depending on the parameters of the problem, is devoted to the paper. The mathematical model is studied in the case when the operator at time derivative is degenerate. The study of the question of non-uniqueness of the solution of the Showalter-Sidorov problem for the Hoff model in the two-dimensional domain is carried out on the basis of the phase space method, which was developed by G.A. Sviridyuk. The conditions of non-uniqueness of the solution in the case when the dimension of the operator kernel at time derivative is equal to 1 or 2 are found. Two approaches for revealing the number of solutions of the Showalter-Sidorov problem in the case when the dimension of the operator kernel at time derivative is equal to 2 are presented. Examples illustrating the non-uniqueness of the solution of the problem on a rectangle are given.
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Analysis of complex hydraulic networks, electric circuits, electronic schemes, chemical processes etc. often results in a system of interconnected differential and algebraic equations. If the process under study has after-effect, then the system includes integral equations. This paper addresses simulation of hydraulic networks by means of the theory for singular systems of integral differential equations. We present theoretical tools that help investigate qualitative properties of such systems and search for effective methods of solution. A mathematical model for the straight through boiler circuit has been developed and a numerical method for its solution has been constructed. Experimental results showed that the theory for singular systems of integral differential equations performs well when applied to simulation of the hydraulic networks.
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In the design of iterative learning control (ILC) algorithm for stochastic nonlinear networked systems, the underlying assumption is differentiability of the system dynamics. In many cases, in reality, stochastic nonlinear networked systems have non-differentiable dynamics, but their dynamics functions after discretization by using conventional methods have global Lipschits’ continuous (GLC) condition. In this paper, we apply an ILC algorithm for stochastic nonlinear networked systems that have the GLC condition. We demonstrate that to design the ILC algorithm, differentiability of the system dynamics is not necessary, and the GLC condition is sufficient for designing the ILC algorithm for stochastic nonlinear networked systems with non-differentiable dynamics. We investigate the analysis of convergence and the tracking performance of the proposed update law for stochastic nonlinear networked systems with GLC condition. We show that there exists no limited condition for the stochastic data dropout probabilities in the convergence investigation of the input error. Then, the results are reviewed and confirmed with a numerical example.
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Constant structure closed semantic systems are the systems each element of which receives its definition through the correspondent unchangeable set of other elements (neighbors) of the system. The definitions of the elements change iteratively and simultaneously based on the neighbor portraits from the previous iteration. In this paper, I study the behavior of such model systems, starting from the zero state, where all the system's elements are equal. The development of constant-structure discrete time closed semantic systems may be modelled as a discrete time coloring process on a connected graph. Basically, I consider the iterative redefinition process on the vertices only, assuming that the edges are plain connectors, which do not have their own colors and do not participate in the definition of the incident vertices. However, the iterative coloring process for both vertices and edges may be converted to the vertices-only coloring case by the addition of virtual vertices corresponding to the edges assuming the colors for the vertices and for the edges are taken from the same palette and assigned in accordance with the same laws. I prove that the iterative coloring (redefinition) process in the described model will quickly degenerate into a series of pairwise isomorphic states and discuss some directions of further research.
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Kolmogorov-Arnold neural networks technique for the state of charge estimation for Li-ion batteries
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Kolmogorov-Arnold Network (KAN) is an advanced type of neural network developed based on the Kolmogorov-Arnold representation theorem, offering a new approach in the field of machine learning. Unlike traditional neural networks that use linear weights, KAN applies univariate functions parameterized by splines, allowing it to flexibly capture and learn complex activation patterns more effectively. This flexibility not only enhances the model's predictive capability but also helps it handle complex issues more effectively. In this study, we propose KAN as a potential method to accurately estimate the state of charge (SoC) in energy storage devices. Experimental results show that KAN has a lower maximum error compared to traditional neural networks such as LSTM and FNN, demonstrating that KAN can predict more accurately in complex situations. Maintaining a low maximum error not only reflects KAN's stability but also shows its potential in applying deep learning technology to estimate SoC more accurately, thereby providing a more robust approach for energy management in energy storage systems.
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L-stability of nonlinear systems represented by state models
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Stability theory plays a key role in systems theory and engineering. The stability of equilibrium points is usually considered within the framework of the stability theory developed by the Russian mathematician and mechanic A.M. Lyapunov (1857-1918), who laid its foundations and gave it its name. Nowadays, the point of view on stability has become very widespread, as stability in relation to disturbance of the input signal. The research is based on the space-state approach for modelling nonlinear dynamic systems and an alternative "input-output'' approach. The input-output model is implemented without explicit knowledge of the internal structure determined by the equation of state. The system is considered as a "black box'', which is accessed only through the input and output terminals ports. The concept of stability in terms of "input-output'' is based on the definition of L-stability of a nonlinear system, the method of Lyapunov functions and its generalization to the case of nonlinear dynamical systems. The interpretation of the problem on accumulation of perturbations is reduced to the problem on finding the norm of an operator, which makes it possible to expand the range of models under study, depending on the space in which the input and output signals act.
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Modelling of various natural and technical processes often results in systems that comprise ordinary differential equations and algebraic equations This paper studies systems of quasi-linear integral-differential equations with a singular matrix multiplying the higher derivative of the desired vector-function. Such systems can be treated as differential algebraic equations perturbed by the Volterra operators. We obtained solvability conditions for such systems and their initial problems and consider possible ways of linearization for them on the basis of the Newton method. Applications that arise in the area of thermal engineering are discussed and as an example we consider a hydraulic circuit presented as a system comprising an interconnected set of discrete components that transport liquid. Numerical experiments that employed the implicit Euler scheme showed that the mathematical model of the straight-through boiler with a turbine and a regeneration system has a solution and this solution tends to the stationary mode preset by regulators.
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Lord Kelvin and Andrey Andreyevich Markov in a queue with single server
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We use Lord Kelvin's method of images to show that a certain infinite system of equations with interesting boundary conditions leads to a Markovian dynamics in an L1-type space. This system originates from the queuing theory.
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The electronic structure and magnetic properties of FeRhSbZ ( = 0, 0,25, 0,5, 0,75, 1) alloys with = P, As, Sn, Si, Ge, Ga, In, Al are studied by first-principles methods. For all compounds, three cubic phases with different atomic arrangement (, , and ) are considered. It is shown that the -phase is energetically favorable for FeRhSbP( = 0,75, 1), FeRhAs and FeRhSi alloys. For the remaining 29 alloys, the phase is more energetically stable. The values of equilibrium lattice parameters and magnetic moments of stoichiometric ternary alloys are in good agreement with the literature values collected from other theoretical studies. The half-metallic ferromagnetic behavior is predicted for FeRhSbSn, FeRhGe, FeRhSn, and FeRhSbAl. It has been found that the replacement of the element with another element allows for the creation of new four-component alloys that exhibit 100 % spin polarization.
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The problem of three-dimensional mathematical modelling of the effect of air flow on an optical-mechanical unit (OMU) located in the unpressurised compartment of the aircraft, is considered. To solve this problem, a mathematical model of gas dynamics based on the solution of a complete system of Navier - Stokes equations that describe the dynamics of a turbulent, spatially unsteady flow of a viscous gas is developed. The software for simulating the process of flow past a WMU model in the aircraft compartment was created. The effect of the air flow on the OMU is described by the torque acting on the OMU from the airflow side. A numerical method for solving the three-dimensional gasdynamic problem is presented. The numerical method is based on the numerical high order Godunov scheme, realized on an irregular grid with arbitrary cells (tetrahedral, prismatic shape). Flows of conservative variables are calculated by solving the Riemann problem with an approximate AUSM method. The system of equations is supplemented by a two-parameter k-model of turbulence, modified for the calculation of high-speed compressible flows. To significantly reduce the cost of computing resources, it is suggested to use stochastic models of the effect of air flow on WMU. A general simulation algorithm is described.
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Mathematical model of a successful stock market game
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All available predictive models of stock market trade (like regression or statistical analysis, for instance) are based on studying of price fluctuation. This article proposes a new model of a successful stock market strategy based on studying of the behavior of the largest successful players. The main point of this model is that a relatively weak player repeats the actions of stronger players in the same fashion as in a race after leader a cyclist following a motorbike reaches greater velocity. We represent the leader as a vector in the nonnegative orthant Rn+ depending on the most successful traders (hedge funds). When buying and selling stocks, we should always keep the vector of own resources collinear to the leader's. This strategy will not yield significant profit, but it prevents considerable loss.
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Mathematical model of a wide class memory oscillators
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A mathematical model is proposed for describing a wide class of radiating or memory oscillators. As a basic equation in this model is an integro-differential equation of Voltaire type with difference kernels - memory functions, which were chosen by power functions. This choice is due, on the one hand, to broad applications of power law and fractal properties of processes in nature, and on the other hand it makes it possible to apply the mathematical apparatus of fractional calculus. Next, the model integro-differential equation was written in terms of derivatives of fractional Gerasimov - Caputo orders. Using approximations of operators of fractional orders, a non-local explicit finite-difference scheme was compiled that gives a numerical solution to the proposed model. With the help of lemmas and theorems, the conditions for stability and convergence of the resulting scheme are formulated. Examples of the work of a numerical algorithm for some hereditary oscillators such as Duffing, Airy and others are given, their oscillograms and phase trajectories are constructed.
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The mathematical model of liquid hydrogen sulfide injection into the semi-infinite porous layer saturated with the oil and water accompanied by H2S gas hydrate formation is presented here. We considered the case when the hydrate formation occurs at the frontal border and the oil displacement's front by hydrogen sulfide is ahead of this boundary. Solutions for pressure and temperature in every layer's area are built by help of the self-similar variable formation method. The values of the parameters of the moving interphase boundaries are found as the result of the iteration procedure. The coordinate dependence of phase boundaries on the injection pressure was studied on the basis of the obtained solutions. We have established that for the existence of solution with two different interphase boundaries, the injection pressure must be above a certain limiting value. The dependence of the limiting value of pressure on the initial temperature of the layer at different temperatures of the injected hydrogen sulphide is constructed. The results of the calculations showed that the constructed mathematical model with three areas in the reservoir gives an adequate description of the process at high injection pressures, the temperature of the injected hydrogen sulfide and the initial temperature of the layer.
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