Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Все статьи: 767
Causal relations in support of implicit evolution equations
Статья научная
This is a brief exposition of dynamic systems approaches that form the basis for linear implicit evolution equations with some indication of interesting applications. Examples in infinite-dimensional dissipative systems and stochastic processes illustrate the fundamental notions underlying the use of double families of evolution equations intertwined by the empathy relation. Kisynski's equivalent formulation of the Hille-Yosida theorem highlights the essential differences between semigroup theory and the theory of empathy. The notion of K-bounded semigroups, a more direct approach to implicit equations, and related to empathy in a different way, is included in the survey.
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Choosing average values when determining characteristics of the unsteady boiling of liquid
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This paper presents an analysis of the issues associated with constructing mathematical models for processes of intense phase transformations and, in particular, focuses on the aspect of using closing relations of empirical origin. The main trend in the implementation of modern numerical algorithms for practical problems is aimed at improving the accuracy of calculation results. The latter is usually achieved by refining a certain set of coefficients in mathematical models. These refinements are carried out both on the basis of the modernization of existing approaches, and with the involvement of new empirical information obtained for a limited number of regime conditions. Predictive models for describing the dynamics of phase transformations, as one of the most difficult in the mathematical formulations, refer to a particularly striking manifestation of the problem under study. In this research, we discuss the existing and widely used experimental work devoted to the extraction of primary information about the dynamics of vapor bubbles on the surface of metal heaters. Their example reveals the presence of a simplified approach in the existing development methodology, and shows a way to determine the correct generalization of empirical information that has a pseudo-stochastic nature.
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Circular shift of loop body - programme transformation, promoting parallelism
Статья научная
The article deals with the programme transformation executing the circular shift of loop body statements. It can be used for vectorizing or parallelizing. This becomes possible due to the fact that when the order of loop body statements is changed, some of the bottom-up arcs become top-down arcs. Besides, sometimes loop carried dependence arcs are substituted by loop independent ones. It should be pointed out that in executing the circular shift the number of loop iterations is reduced by one. The transformation can be used both independently and in conjunction with other transformations promoting parallelism. These could be "forward substitution", "scalar expansion", "privatization", "array expansion", etc. The transformation under consideration in this article can be used both in hand parallelization and added to a paralleling (optimizing) compiler. Moreover, the application of the transformation results in the equivalent code only for the loops where loop unrolling is the equivalent transformation. Thus, they can contain nested loops, if statements and other programming language statements.
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Статья научная
Inverse problems of identification of the fractional diffusivity and the order of fractional differentiation are considered for linear fractional anomalous diffusion equations with the Riemann - Liouville and Caputo fractional derivatives. As an additional information about the anomalous diffusion process, the concentration functions are assumed to be known at several arbitrary inner points of calculation domain. Numerically-analytical algorithms are constructed for identification of two required parameters of the fractional diffusion equations by approximately known initial data. These algorithms are based on the method of time integral characteristics and use the Laplace transform in time. The Laplace variable can be considered as a regularization parameter in these algorithms. It is shown that the inverse problems under consideration are reduced to the identification problem for a new single parameter which is formed by the fractional diffusivity, the order of fractional differentiation and the Laplace variable. Estimations of the upper error bound for this parameter are derived. A technique of optimal Laplace variable determination based on minimization of these estimations is described. The proposed algorithms are implemented in the AD-TIC package for the Maple software. A brief discussion of this package is also presented.
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Commutative encryption method based on hidden logarithm problem
Статья научная
A candidate for post-quantum commutative encryption algorithm is proposed, which is based on the hidden discrete logarithm problem defined in a new 6-dimensional finite non-commutative associative algebra. The properties of the algebra are investigated in detail and used in the design of the proposed commutative cipher. The formulas describing the set of p2 different global right-sided units contained in the algebra and local left-sided units are derived. Homomorphisms of two different types are considered and used in the commutative cipher. The encrypted message is represented in the form of a locally invertible element T of the algebra and encryption procedure includes performing the exponentiation operation and homomorphism map followed by the left-sided multiplication by a randomly selected local right-sided unit. The introduced commutative cipher is secure to the known-plaintext attacks and has been used to develop the post-quantum no-key encryption protocol providing possibility to send securely a secret message via a public channel without using any pre-agreed key. The proposed commutative encryption algorithm is characterized in using the single-use keys that are selected at random directly during the encryption process.
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Статья научная
The paper presents a concept for comparing the solvers for the mixed integer linear programming problems and the software environments that call them. This concept involves multiple repetition of solving mathematical programming problems with the same initial data to take into account the fact that the computer operations time can be considered as random. It is also assumed to solve the mathematical programming problem with the same structure by varying the initial data to compare the solvers. The comparison is carried out for a number of practical mathematical programming problems. For example we consider the portfolio optimization problem with the probability criterion. Solvers CPLEX, Gurobi, MATLAB, SCIP are used in testing. The features of calling solvers in various software environments are described. In particular, a modification of the source codes for calling the CPLEX solver through the Opti Toolbox add-on in Matlab environment is provided. The components of the time required to obtain a solution for various solvers and software environments are described and studied in detail. It is shown that the operating time of the solver itself can be comparable to the time of reading data from files and the time of forming constraints in a mathematical programming problem.
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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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Complex approach to assessment of investment attractiveness of power generating company
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Present approaches based on the qualitative analysis methods are not effective enough for a comprehensive evaluation of the investment attractiveness of the power generating company (PGC). It resolves the urgency of the complex deterministic method of accounting for aggregated risk. The article presents the diagnostics of power generating company risks' and the assessment of the actual aggregated risk as the integral indicator of investment attractiveness of the PGC. The proposed authors' approach to ranking the risk taking into account the level of hazard is based on the calculation of individual limits of risk states variation and risk relative value. The individual risk assessment is based on the Bayes method complemented by a two-step normalization to account for the specificity of PGC. The Merton - Vasicek method and basic principles of the economic capital theory are used in developing the method of the final evaluation of the PGC investment attractiveness. Research veracity is confirmed by the practical implementation. The research results are recommended for use in assessing the current level of the PGC investment attractiveness and development strategy of its increase.
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Computational experiment for a class of mathematical models of magnetohydrodynamics
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The first initial-boundary value problem for the system modelling the motion of the incompressible viscoelastic Kelvin - Voigt fluid in the magnetic field of the Earth is investigated considering that the fluid is under external influence. The problem is studied under the assumption that the fluid is under different external influences depending not only on the coordinates of the point in space but on time too. In the framework of the theory of semi-linear Sobolev type equations the theorem of existence and uniqueness of the solution of the stated problem is proved. The solution itself is a quasi-stationary semi-trajectory. The description of the problem's extended phase space is obtained. The results of the computainal experiment are presented.
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Computational experiment for one class of evolution mathematical models in quasi-Sobolev spaces
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In the article the mathematical model representing one class of evolution equations in quasi-Banach spaces is studied. A theorem on the unique solvability of the Cauchy problem is stated. The conditions for the phase space existence are presented. We also give the conditions for exponential dichotomies of solutions. Based on the theoretical results there was developed an algorithm for the numerical solution of the problem. The algorithm is implemented in Maple. The article includes description of the algorithm which is illustrated by variety of model examples showing the work of the developed program and represent the main properties of solutions.
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Computational experiment for one mathematical model of ion-acoustic waves
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In the article the mathematical model of ion-acoustic waves in a plasma in an external magnetic field is studied. This model can be reduced to a Cauchy problem for a Sobolev type equation of the fourth order with polynomially (A,p)-bounded operator pencil. Therefore abstract results on solvability of the Cauchy problem for such equation can be used. In the article a theorem on the unique solvability of the Cauchy - Dirichlet problem is mentioned. Based on the theoretical results there was developed an algorithm for the numerical solution of the problem, using a modified Galerkin method. The algorithm is implemented in Maple. The article includes description of this algorithm. It is illustrated by model examples showing the work of the developed program.
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We investigate of a special dam optimal location at the Volga river in the area of the Akhtuba left sleeve beginning (7 km to the south of the Volga Hydroelectric Power Station dam). We claim that a new water-retaining dam can resolve the key problem of the Volga-Akhtuba floodplain related to insufficient water amount during spring floodings due to the overregulation of the Lower Volga. Using a numerical integration of Saint-Vanant equations we study the water dynamics across the northern part of the Volga-Akhtuba floodplain taking into account its actual topography. As the result we found an amount of water VA passing to the Akhtuba during spring period for a given water flow through the Volga Hydroelectric Power Station (so-called hydrograph which characterises the water flow per unit of time). By varying the location of the water-retaining dam xd,yd we obtained various values of VA(xd,yd) as well as various flow spatial structure on the territory during the flood period. Gradient descent method provides the dam coordinated with the maximum value of VA. Such approach to the dam location choice let us find the best solution, that the value VA increases by a factor of 2. Our analysis demonstrates a good potential of the numerical simulations in the field of hydraulic works.
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Consistency and Lyapunov stability of linear singular time delay systems: a geometric approach
Статья научная
When we consider the control design of practical systems (chemical engineering systems, lossless transmission lines, large-scale electric network control, aircraft attitude control, flexible arm control of robots, etc.), time-delay often appears in many situations. Singular time delayed systems are the dynamic systems described by a mixture of algebraic and differential equations with retarded argument. This paper investigates the geometric description of initial conditions that generate smooth solutions to such problems and the construction of the Lyapunov stability theory to bound the rates of decay of such solutions. The new delay dependent conditions for asymptotic stability for the class of systems under consideration were derived. Moreover, the result is expressed in terms of only systems matrices that naturally occur in the model, therefore avoiding the need to introduce algebraic transformations into the statement of the main theorem.
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Constructing of OE-postman path for a planar graph
Статья научная
With automated preparation of the cutting process, the cutting plan can be represented as a flat graph. The purpose of this simulation is to determine the shortest path of the cutting tool, provided that the part cut from the sheet would not require additional cuts. The article deals with the task of building the path of the Chinese postman in a flat graph, which is a model of the cutting plan. An orderly coverage condition is imposed on this path (i.e., the cycle of traversed edges does not encompass those not yet traversed). Such a path will also be called an OE path. This limitation means the absence of additional cuts for parts. The article discusses the recursive algorithm for constructing such chains. It is proved that the algorithm has polynomial complexity. The developed software allows solving the problem for an arbitrary flat graph. The program has been tested for various types of flat graphs.
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Construction of approximate mathematical models on results of numerical experiments
Статья научная
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
Статья научная
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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Convergence analysis of the finite difference solution for coupled Drinfeld-Sokolov-Wilson system
Статья научная
This paper is devoted to drive the matrix algebraic equation for the coupled Drinfeld-Sokolov-Wilson (DSW) system using the implicit finite difference (IMFD) method. The convergence analysis of the finite difference solution is proved. Numerical experiment is presented with initial conditions describing the generation and evolution. The numerical results were being compared on the basis of calculating the absolute error (ABSE) and the mean square error (MSE). The numerical results proved that the numerical solution was close to the real solution at different values of time.
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This paper considers the parameter estimation problem for models of one-dimensional chaotic systems. The guaranteed algorithm is proposed in the context of set-membership approach, which assumes that only intervals of possible values are known for the uncertain variables in the model (initial condition, parameter and measurement errors). The algorithm recursively computes the interval estimates of the parameter at every time step. If the prior information is correct, found interval estimates always contain the true value of the parameter. For certain models of measurement errors the result of the algorithm is the exact value of the parameter (the final interval estimate contains a single point). The goal of this study is to derive conditions under which the guaranteed algorithm improves the interval estimate of the parameter.
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Convergence modelling in international integration associations
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The article considers mathematical tools for modelling economic policy as a whole, as well as convergence in the field of labor, foreign economic activity, monetary and debt policy. Convergence was estimated using the σ-convergence model, which characterizes the decrease in time spread in the levels of development of countries and regions, reflecting the negative relationship between economic growth rates and the initial level of development of countries and regions. The σ-convergence was estimated by the coefficient of variation and by the dispersion-based model. To assess β-convergence, we used the Barro and Sala-i-Martin models, as well as the Baumol, Solow-Svan, and Quadrado-Rour models. The use of this mathematical toolkit allows to explore the presence and speed of convergence before and after joining international integration associations. The proposed mathematical modelling tools are recommended to be used in order to analyze convergence processes, study the dynamics of convergence or divergence, and also to adjust the directions and methods of state and regional economic policies of countries included in the integration association.
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