Математическое моделирование. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Статья научная
The quasi-stationary problem for modelling the process of electrochemical cutting with a plate electrode-tool is formulated. The formulation of the problem is based on the use of a stepwise function of current efficiency from the current density. Thus three areas with various conditions are formed on the machined surface. The usual stationarity condition is used in the area of high current densities. In the area of low current densities the dissolution is absent and the initial form of the boundaries remains. In the intermediate zone, the current density at each point is equal to the critical value. The presence of boundary conditions on each section of the machined surface allows to formulate a boundary problem for the analytical function of the complex variable and to find the shape of the boundary at any moment, regardless of the background. The solutions of quasi-stationary and non-stationary problems are compared, and the range of existence of quasi-stationary solutions is found.
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Construction of approximate mathematical models on results of numerical experiments
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A mathematical model of an artillery shot is represented as a system of non-stationary one- and two-dimensional differential equations of the multiphase gas dynamics and heat transfer. Conjunction Euler-Lagrange method is used for the numerical solution of gas-dynamic equations. The initial mathematical model is approximated by a system of ordinary differential equations using a vector of correction functions. Correction functions are found from solutions of multiobjective optimal control problem. Multiobjective optimization is carried out using a hybrid genetic algorithm. The resulting model is adequate and allows doing more processing series of calculations the main process parameters (projectile velocity and maximum pressure) depending on the input parameters. Comparative analysis of different approximators (linear multiple regression, support vector machines, multi-layer neural network, radial network, the method of fuzzy decision trees) showed that an acceptable accuracy 0,4-0,5 % is provided by only non-linear approximation methods, such as multi-layer and radial neural networks. Constructed approximate models are not require much computing time and can be implemented in the control systems.
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Continuous and generalized solutions of singular integro-differential equations in Banach spaces
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Continuous and generalized solutions of singular equations in Banach spaces are studied. We apply Lyapunov-Schmidt’s ideas and the generalized Jordan sets techniques and reduce partial differential-operator equations with the Fredholm operator in the main expression to regular problems. In addition the left and right regularizators of singular operators in Banach spaces and fundamental operators in the theory of generalized solutions of singular equations are constructed.
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Статья научная
Delay differential-algebraic equations (DDAEs) can be used for modelling real-life phenomena that involve simultaneously time-delay effect and constraints. It is also known that solving delay DAEs is more complicated than solving non-delay ones because approximation of solutions in the past time is usually needed and discontinuity in higher derivatives of the solutions is typical. Recently, we have proposed and investigated linear multistep (LM) methods for strangeness-free DAEs (without delay). In this paper, we extend the use of LM methods to a class of structured strangeness-free DAEs with constant delay. For the approximation of solutions at delayed time we use polynomial interpolation. Convergence analysis for LM methods is presented. It is shown that, similarly to the non-delay case, the strict stability of the second characteristic polynomial associated with the methods is not required for the convergence if we discretize an appropriately reformulated DDAE instead of the original one. Numerical experiments are also given for illustration.
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Cooperation in a conflict of persons under uncertainty
Статья научная
The paper considers a model of a conflict system with N active participants with their own interests when exposed to an uncertain factor. At the same time, decision-makers do not have any statistical information about the possible implementation of an uncertain factor i.e. they only know the many possible realizations of this factor - uncertainties. Under the assumption that the participants of to the conflict can coordinate their actions in the decision-making process the model is formalized as a cooperative N-person game without side payments and under uncertainty. In this paper, we introduce a new principle of coalitional equilibration (CE). The integration of individual and collective rationality (from theory of cooperative games without side payments) and this principle allows us to formalize the corresponding concept of CE for a conflict of N persons under uncertainty. At the same time, uncertainty is taken into account along with using the concept of the "analogue of maximin'' proposed earlier in the our works and the "strong guarantees'' constracted on its basis. Next, we establish sufficient conditions for existence of coalitional equilibrium, which are reduced to saddle point design for the Germeier convolution of guaranteed payoffs. Following the above-mentioned approach of E. Borel, J. von Neumann and J. Nash, we also prove existence of coalitional equilibrium in the class of mixed strategies under standard assumptions of mathematical game theory (compact uncertainties, compact strategy sets, and continuous payoff functions). At the end of the paper, some directions or further research are given.
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Статья научная
In this article, we give new results on the study of elliptic abstract second order differential equation with variable operators coefficients under the general Robin homogeneous boundary value conditions, in the framework of UMD spaces. Here, we do not assume the differentiability of the resolvent operators. However, we suppose that the family of variable operators verifies the Labbas-Terreni assumption inspired by the sum theory and similar to the Acquistapace-Terreni one. We use Dunford calculus, interpolation spaces and semigroup theory in order to obtain existence, uniqueness and maximal regularity results for the classical solution to the problem.
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Differential-algebraic equations with regular local matrix pencils
Статья научная
In the projector based framework, any regular linear DAE features several continuous time-varying characteristic subspaces that are independent of construction technicalities, among them the so-called sum-subspaces. As it is well-known, the local matrix pencils of a higher-index time-varying linear DAE do not reflect the global structure of the DAE in general. We show that, on the given interval, the local matrix pencils of the DAE are regular and reflect the global DAE structure if several of these characteristic subspaces are time-invariant. We discuss practicable methods to check the time-invariance of these subspaces. The corresponding class of DAEs is related to the class of DAEs formerly introduced and discussed by Yuri E. Boyarintsev.
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Discrete model of paired relay-race
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The case of the active and passive team relay-race, in which an active team operates in accordance with rigid schedule and a passive team overcome the stage of its distance at randomly selected alternative routs during occasional time intervals is considered. Due to high complexity of classical relay-race analysis, method of simulation, based on representation of time intervals densities of passing stages routs with discrete distributions is proposed. It is shown, that after transformation of time intervals densities into discrete distributions the problem of a relay race analysis reduces to the task of analysis of two-team system with rigid schedules. The method of sampling of densities composition with estimation a sampling error, and recursive procedure of rigid schedule relay-race analysis with calculation of forfeit are worked out. It is shown, that forfeit depends on the difference of stages, teams overcome at current time and a strategy, which active team realizes during relay-race.
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Статья научная
We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schrödinger equation in a bounded domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated face. We establish a log log stability estimate for the L2-norm (resp. the H-1-norm) of Ht, for t>0, and bounded (resp. L2) potentials.
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Статья научная
In this paper we prove some new results on complete operational second order differential equations of elliptic type with coefficient-operator conditions, in the framework of the space Lp(0,1;X) with general pϵ(1,+∞), X being a UMD Banach space. Existence, uniqueness and optimal regularity of the classical solution are proved. This paper improves and completes naturally our last two works on this problematic.
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Exact solutions of beta-fractional Fokas-Lenells equation via sine-cosine method
Статья научная
In nonlinear plasma physics, photonics and optics, the space-time fractional nonlinear Fokas-Lenells equation associated with beta derivative has significiant applications. This equation is used in this study to provide precise solutions using the Sine-Cosine method. Furthermore, using computer software, we plot the 2D-3D figures of the obtained solutions based on the appropriate parameters. The findings indicate that the suggested technique is simple, efficient and capable of producing complete solutions to nonlinear models due to mathematical physics.
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Exact solutions of the (2+1)-dimensional Kundu-Mukherjee-Naskar model via IBSEFM
Статья научная
The aim of this study is to construct the exact solutions of the (2+1)-dimensional Kundu-Mukherjee-Naskar (KMN) equation via Improved Bernoulli Sub-Equation Function Method (IBSEFM). The physics of this model describes optical dromions in (2+1)-dimensional case. It is also studied in fluid dynamics. Applying the proposed method, we obtain new exact solutions of (2+1)-dimensional KMN equation. Moreover, we plot the 2D-3D figures and contour surfaces according to the suitable parameters by the aid of computer software. The results confirm that IBSEFM is powerful, effective and straightforward for solving nonlinear partial differential equations arising in mathematical physics.
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Статья научная
We prove existence of upper and lower solutions in reverse order with respect a part of the variables in a system of nonlinear ordinary differential equations modelling acidogenesis in anaerobic digestion. The corresponding existence theorems are established. The upper and lower solutions are constructed analytically, by defining semi-trivial solutions for each of the variables in the model. We introduce the concept of indicator semi-trivial solutions. Finally, we numerically solve the system supported by the Matlab software and matching the graphs of the numerical solutions with analytical solutions is found.
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Статья научная
We investigate stability of solutions in linear stochastic Sobolev type models with the relatively bounded operator in spaces of smooth differential forms defined on smooth compact oriented Riemannian manifolds without boundary. To this end, in the space of differential forms, we use the pseudo-differential Laplace-Beltrami operator instead of the usual Laplace operator. The Cauchy condition and the Showalter-Sidorov condition are used as the initial conditions. Since "white noise'' of the model is non-differentiable in the usual sense, we use the derivative of stochastic process in the sense of Nelson-Gliklikh. In order to investigate stability of solutions, we establish existence of exponential dichotomies dividing the space of solutions into stable and unstable invariant subspaces. As an example, we use a stochastic version of the Barenblatt-Zheltov-Kochina equation in the space of differential forms defined on a smooth compact oriented Riemannian manifold without boundary.
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Finite non-commutative associative algebras as carriers of hidden discrete logarithm problem
Статья научная
The article introduces new finite algebras attractive as carriers of the discrete logarithm problem in a hidden group. In particular new 4-dimensional and 6-dimensional finite non-commutative algebras with associative multiplication operation and their properties are described. It is also proposed a general method for defining finite non-commutative associative algebras of arbitrary even dimension m≥2. Some of the considered algebras contain a global unit, but the other ones include no global unit element. In the last case the elements of the algebra are invertible locally relatively local bi-side units that act in the frame of some subsets of elements of algebra. For algebras of the last type there have been derived formulas describing the sets of the (right-side, left-side, and bi-side) local units. Algebras containing a large set of the global single-side (left-side and right-side) units and no global bi-side unit are also introduced. Since the known form of defining the hidden discrete logarithm problem uses invertibility of the elements of algebra relatively global unit, there are introduced new forms of defining this computationally difficult problem. The results of the article can be applied for designing public-key cryptographic algorithms and protocols, including the post-quantum ones. For the first time it is proposed a digital signature scheme based on the hidden discrete logarithm problem.
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Forecasting the return of the loan portfolio on the basis of Markov model
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We consider the problem of mathematical modelling of flows of loan portfolio payments. We assume that the change in the quality of each loan is described by a simple Markov chain with a finite number of states. In this case, the flow of loan payments is a random process, which depends on the Markov chain. On the basis of the proposed model and known relations of the stochastic systems theory, we describe the expected flows of payments of the entire loan portfolio and construct a method to forecast the expected return (net present value) of the portfolio. We analyze an accuracy of the obtained model and a sensitivity of net present value of the portfolio to a change in the transition probabilities in the Markov chain.
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Статья научная
This study aims to apply a novel technique devised by the authors to process the results of thermal physics experiments. The paper describes a two-stage technique for identifying coefficients of mathematical models from observed experimental data. The technique is based on the maximum likelihood method and is informed by the errors of all sensors used to obtain parameter measurements. Stage 1 of the technique minimizes the maximum relative error over all measured parameters, which allows gross measurement errors to be identified in qualitative terms and reduces the maximum relative error down to acceptable values. At Stage 2, we propose to use the method of weighted least absolute values to minimize the sum of absolute values of relative errors of all measured parameters. The technique was applied to process the results of thermal physics experiments aimed at generalizing the size of vapor bubbles of various types during unsteady heating of a vertical steel cylindrical heater surrounded by an upward flow of water subcooled to the saturation temperature. The numerical simulations reported in this study attest to the high quality of the proposed two-stage technique for identifying coefficients of mathematical models. The study also presents a comparative analysis of the results obtained by the classical least squares method and the novel two-stage technique.
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Global Schumpeterian dynamics with structural variations
Статья научная
In this paper, we present the investigations developing the schumpeterian theory of endogenous evolution of economic systems. The proposed approach allows to simulate the emergence and propagation of new technologies. We develop a mathematical model of dynamics of sector capital distribution over efficiency levels on the base of the system of nonlinear differential equations. In order to take into account the boundedness of the economic growth conditioned by the boundedness of the markets, the resource base and other factors, we introduce the notion of economical niche volume. The scenario of the emergence of the new highest efficiency level is proposed. In order to simulate the process of the emergence of the new highest efficiency level, the notion of intellectual capital is proposed. According to the proposed scenario, the new level emerges when the intellectual capital achieves the threshold value. Herewith, the dimension of the dynamic system is varied. The necessary condition for the functioning of the new level is formulated. The invariant set of the dynamic system is defined. The local stability of the equilibria is investigated. The global stability of the dynamic system is established on the base of a geometrical method. The proposed models allow to evaluate and predict the dynamics of the technological levels of the economic sector firms development.
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Gravitational model of population dynamics
Статья научная
We consider the author's mathematical model of population dynamics of territories, taking into account the migration between the territories. The model is a system of ordinary differential equations with constant coefficients. The main idea of the presented modelling is to take into account the migration interactions of territories in the form of nonlinear terms (they are the pair products of phase variables of the territories population). On the one hand, the approach allows to consider the model as a variant of expanding the gravitational approach in migration studies. On the other hand, this approach allows to apply the approaches of mathematical biology, which are successfully used in econophysics and sociodynamics. In order to verify the model, we use statistical data on population and migration between federal districts of the Russian Federation. The results of the modelling show the significance of the "repulsion" of migrants arriving in the Central and North-Western federal districts, mainly in the nearby regions (Southern, North Caucasian and Volga federal districts). Model evaluations of the migration balance are obtained. The evaluations exceed statistical ones by dozens of times and, to all appearances, describe the "latent" migration of the population, covering both long-term and short-term movements. An analysis of the change in the stationary values of the population for a linear change in the parameters is carried out. It is shown that there are such values of the parameters of migration attractiveness of the Russian Federation federal districts, under which the population increases both in the whole in the Russian Federation and in individual districts. It is established that such changes can occur due to significant differences in the opportunities, which are "provided" by different federal districts for migrants (e.g., living and working conditions, upbringing and education of children, etc.), and intra-Russian migration.
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Статья научная
The motion of a hydrodynamic flow in a chemical reactor described by a one-dimensional one-parameter diffusion model is considered. Within the framework of this model, the task is set to identify the boundary condition at the reactor outlet containing an unknown concentration of the reagent under study leaving the reactor in a stream. In this case, the law of change in the concentration of the reagent over time at the reactor inlet is additionally set. After the introduction of dimensionless variables, a discrete analogue of the transformed inverse problem in the form of a system of linear algebraic equations is constructed by the method of difference approximation. The discrete analogue of the additional condition is written as a functional and the solution of a system of linear algebraic equations is presented as a variational problem with local regularization. A special representation is proposed for the numerical solution of the constructed variational problem. As a result, the system of linear equations for each discrete value of a dimensionless time splits into two independent linear subsystems, each of which is solved independently of each other. As a result of minimizing the functional, an explicit formula was obtained for determining the approximate concentration of the reagent under study in the flow leaving the reactor at each discrete value of the dimensionless time. The proposed computational algorithm has been tested on the data of a model chemical reactor.
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