Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Все статьи: 729
On the mean-value property for polyharmonic functions
Статья научная
The mean-value property for normal derivatives of polyharmonic function on the unit sphere is obtained. The value of integral over the unit sphere of normal derivative of mth order of polyharmonic function is expressed through the values of the Laplacian's powers of this function at the origin. In particular, it is established that the integral over the unit sphere of normal derivative of degree not less then 2k-1 of k-harmonic function is equal to zero. The values of polyharmonic function and its Laplacian's powers at the center of the unit ball are found. These values are expressed through the integral over the unit sphere of a linear combination of the normal derivatives up to k-1 degree for the k-harmonic function. Some illustrative examples are given.
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On the one-dimensional harmonic oscillator with a singular perturbation
Статья научная
In this paper we investigate the one-dimensional harmonic oscillator with a left-right boundary condition at zero. This object can be considered as the classical selfadjoint harmonic oscillator with a singular perturbation concentrated at one point. The perturbation involves the delta-function and/or its derivative. We describe all possible selfadjoint realizations of this scheme in terms of the above mentioned boundary conditions. We show that for certain conditions on the perturbation (or, equivalently, on the boundary conditions) exactly one non-positive eigenvalue can arise and we derive an analytic expression for the corresponding eigenfunction. These eigenvalues run through the whole negative semi-line as the perturbation becomes stronger. For certain cases an explicit relation between suitable boundary conditions, the non-positive eigenvalue and the corresponding eigenfunction is given.
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On the regularizability conditions of integral equations
Статья научная
Solving of integral equations of the first kind is an ill-posed problem. It is known that all problems can be divided into three disjoint classes: correct problems, ill-posed regularizable problems and ill-posed not regularizable problems. Problems of the first class are so good that no regularization method for them is needed. Problems of the third class are so bad that no one regularization method is applicable to them. A natural application field of the regularization method is the problems from the second class. But how to know that a particular integral equation belongs to the second class rather than to the third class? For this purpose a large number of sufficient regularizability conditions were constructed. In this article one infinite series of sufficient conditions for regularizability of integral equations constructed with the help of duality theory of Banach spaces is investigated. This method of constructing of sufficient conditions proved to be effective in solving of ill-posed problems. It is proved that these conditions are not pairwise equivalent even if we are restricted by the equations with the smooth symmetric kernels.
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Статья научная
We consider the problem of scheduling cargo transportation on a railway network segment. The railway network is represented by an undirected multigraph. The traffic along the edges of the multigraph is carried out only at certain intervals - using "subthreads". We formulate a new mathematical model of traffic along the edges of the multigraph. A universal criterion of optimality for the scheduling problem is proposed. We propose an algorithm to find a suboptimal solution. A meaningful example is given.
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On the stability of two-dimensional flows close to the shear
Статья научная
We consider the stability problem for two-dimensional spatially periodic flows of general form, close to the shear, assuming that the ratio of the periods tends to zero, and the average of the velocity component corresponding to the "long" period is non-zero. The first terms of the long-wavelength asymptotics are found. The coefficients of the asymptotic expansions are explicitly expressed in terms of some Wronskians and integral operators of Volterra type, as in the case of shear basic flow. The structure of eigenvalues and eigenfunctions for the first terms of asymptotics is identified, a comparison with the case of shear flow is made. We study subclasses of the considered class of flows in which the general properties of the qualitative behavior of eigenvalues and eigenfunctions are found. Plots of neutral curves are constructed. The most dangerous disturbances are numerically found. Fluid particle trajectories in the self-oscillatory regime in the linear approximation are given.
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On the telegraph equation with a small parameter
Другой
This work is an edited translation, by A. Bobrowski and W. Chojnacki, of the original Polish text entitled O rownaniu telegrafistow z malym parametrem which appeared in Prace Naukowe Politechniki Lubelskiej 212, Matematyka (Research Bulletin of the Lublin University of Technology 212, Mathematics), 1991, vol. 12, pp. 17-40.
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On the uniqueness of a nonlocal solution in the Barenblatt - Gilman model
Статья научная
This article deals with the question of uniqueness of a generalized solution to the Dirichlet-Cauchy problem for the Barenblatt-Gilman equation, which describes nonequilibrium countercurrent capillary impregnation. The unknown function corresponds to effective saturation. The main equation of this model is nonlinear and implicit with respect to the time derivative, which makes it quite hard to study. In a suitable functional space, the Dirichlet-Cauchy problem for the Barenblatt-Gilman equation reduces to the Cauchy problem for a quasilinear Sobolev-type equation. Sobolev-type equations constitute a large area of nonclassical equations of mathematical physics. The techniques used in this article originated in the theory of semilinear Sobolev-type equations. For the Cauchy problem we obtain a sufficient condition for the existence of a unique generalized solution. We establish the existence of a unique nonlocal generalized solution to the Dirichlet-Cauchy problem for the Barenblatt-Gilman equation.
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On time selection for track possession assignment at the railway station
Статья научная
We consider the problem on track possession assignment at the railway station. The problem is to determine the time during which the train traffic is interrupted for repair works at several railway tracks. To this end, we use a traffic schedule of the station in order to solve the problem on searching for the time interval having the following two properties. First, during this time interval, all track sections that are necessary to be repaired are vacant simultaneously. Second, this time interval has the maximum length. In addition, we solve two problems to determine the time interval having length that is not less than the length of the specified time interval in the following two cases. First, the desired time interval has the minimum number of occupied track sections that are necessary to be repaired. Second, the desired time interval has the minimum number of delayed (transferred) passenger/freight trains going through the tracks that are necessary to be repaired. All problems are solved by methods of mixed integer linear programming.
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Optimal control in linear Sobolev type mathematical models
Статья научная
The article presents a review of the work of the Chelyabinsk mathematical school on Sobolev type equations in studying the optimal control problems for linear Sobolev type models with initial Cauchy (Showalter-Sidorov) conditions or initial-final conditions. To identify the nonemptiness of the set of feasible solutions to the control problem we use the phase space method, which has already proved itself in solving Sobolev type equations. The method reduces the singular equation to a regular one defined on some subspace of the original space and applies the theory of degenerate (semi)groups of operators to the case of relatively bounded, sectorial and radial operators. Here mathematical models are reduced to initial (initial-final) problems for an abstract Sobolev type equation. Abstract results are applied to the study of control problems for the Barenblatt-Zheltov-Kochina mathematical model, which describes fluid filtration in a fractured-porous medium, the Hoff model on a graph simulating the dynamics of I-beam bulging in a construction, and the Boussinesq-Löve model describing longitudinal vibrations in a thin elastic rod, taking into account inertia and under external load, or the propagation of waves in shallow water.
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We study the problem of optimal control of solutions to an operator-differential equation, which is not solved with respect to the time derivative, together with a multipoint initial-final condition. In this case, one of the operators in the equation is multiplied by a scalar function of time. By the properties of the operators involved, the stationary equation has analytical resolving group. We construct a solution to the multipoint initial-final problem for the nonstationary equation. We show that a unique optimal control of solutions to this problem exists. Apart from the introduction and bibliography, the article consists of three sections. The first section provides the essentials of the theory of relatively p-bounded operators. In the second section we construct a strong solution to the multipoint initial-final problem for nonstationary Sobolev-type equations. The third section contains our proof that there exists a unique optimal control of solutions to the multipoint initial-final problem.
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Optimal measurement of dynamically distorted signals
Статья научная
There has been suggested new approach to measure a signal distorted as by inertial measurement transducer, as by its resonances.
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Optimal solutions for inclusions of geometric Brownian motion type with mean derivatives
Статья научная
The idea of mean derivatives of stochastic processes was suggested by E. Nelson in 60-th years of XX century. Unlike ordinary derivatives, the mean derivatives are well-posed for a very broad class of stochastic processes and equations with mean derivatives naturally arise in many mathematical models of physics (in particular, E. Nelson introduced the mean derivatives for the needs of Stochastic Mechanics, a version of quantum mechanics). Inclusions with mean derivatives is a natural generalization of those equations in the case of feedback control or in motion in complicated media. The paper is devoted to a brief introduction into the theory of equations and inclusions with mean derivatives and to investigation of a special type of such inclusions called inclusions of geometric Brownian motion type. The existence of optimal solutions maximizing a certain cost criterion, is proved.
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Optimization of nozzle layout in continuous casting machine
Статья научная
The article presents the results of the scientific research around development of mathamatical foundation for calculating optimal nozzle layout in secondary cooling zone of continuous casting machine and describes implementation of this foundation in specialized computer-aided design system. Optimization procedure is based on the 2-dimensional mathematical model of the thermal state of the slab, which accounts nozzle layout, and optimization criteria for the cross-cut thermal profile of the slab surface. Optimization problem solution consists of two basic blocks: solving thermal conductivity problem and iterative search of the nozzle positions that provide best solution from the point of proposed criteria. The results of running optimization procedure on input conditions based on existing casting machine showed that the obtained solution enables to provide smoother cross-surface thermal profile of the slab. Results of the work can be used for design of new continouous casting machines, modernization of the secondary cooling system of existing machines, and general numeric analisys of secondary cooling system operation for user-provided conditions.
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Optimization of training modules choice during multipurpose training of specialists
Статья научная
The article is devoted to the problems of mathematical modelling of the processes of organizing multipurpose learning. Under the multipurpose training is understood such an organization of the educational process, in which in one group specialists are trained in several related areas of activity, trajectory of training for which at certain intervals intersect. In order to reduce the total training time, as well as the cost or resources required in the training process, it is expedient to carry out the dynamic grouping of students by subgroups in order to master certain competences. The development of the mathematical apparatus used to optimize the multipurpose learning process has not been completely studied at present. To reduce the dimension of the overall task of optimizing the process of organizing multipurpose training, its step-by-step solution is proposed. The article describes the approach to calculating estimates of the possibility of training specialists in the areas of training and selection of training modules available in the educational organization. The paper considers the options for optimizing the selection of modules for training specialists on the following criteria: minimizing the total duration of training, the cost of training and resources used for training. The algorithm with the help of which it is possible to form an optimal group of students is proposed.
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Статья научная
This paper is concerned with the oscillatory properties of second order non-linear dynamic equation with integro forcing term on an arbitrary time scales. We reduce our original dynamic equation into an alternate equation by introducing a function of forward jump operator. To study oscillations we establish some crucial Lemmas and employ generalized Riccati transformation technique which transforms our second order dynamic equation into the first order dynamic equation on an arbitrary time scales. These results also guarantee that the solution of our equation oscillates. Furthermore, we establish the Kamenev-type oscillation criteria of our system. At the end, we consider a second order dynamic equation on time scales with deviating argument and compare it with our result which gives the sufficient conditions of oscillation of it.
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Oskolkov models and Sobolev-type equations
Статья научная
This article is a review of the works carried out by the author together with her students and devoted to the study of various Oskolkov models. Their distinctive feature is the use of the semigroup approach, which is the basis of the phase space method used widely in the theory of Sobolev-type equations. Various models of an incompressible viscoelastic fluid described by the Oskolkov equations are presented. The degenerate problem of magnetohydrodynamics, the problem of thermal convection, and the Taylor problem are considered as examples. The solvability of the corresponding initial-boundary value problems is investigated within the framework of the theory of Sobolev-type equations based on the theory for -sectorial operators and degenerate semigroups of operators. An existence theorem is proved for a unique solution, which is a quasi-stationary semitrajectory, and a description of the extended phase space is obtained. The foundations of the theory of solvability of Sobolev-type equations were laid by Professor G.A. Sviridyuk. Then this theory, together with various applications, was successfully developed by his followers.
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Статья научная
One of the most interesting and relevant approaches for solving optimization problems are parallel algorithms that work simultaneously with a large number of tasks. The paper presents a new parallel algorithm for NACO that is a hybrid algorithm that consists of the Ant Colony Optimization method combined with the Neighbour Joining method to get accurate and efficient results when solving the Traveling Salesman Problem. Through carrying out comprehensive experiments using a wide variety of real dataset sizes and the multi-core system, the practical results show that the developed program outperforms NACO in terms of execution time and consumed storage space. Availability and implementation: source codes in MATLAB 2017 are publicly available at Internet[i].
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Статья научная
For the analysis of huge problems which are very sensitive to the rounding errors, the software providing rational calculations is developed. Software uses MPI interface for communication in the distributed computational environment. Improved efficiency of such software my be achieved by using heterogeneous computation systems. Local arithmetic operations with long numbers may be done in parallel mode with a lot of processes per one operation. This work introduces the research of increasing of the scalability of basic arithmetic operations.Abilities of the massive parallelism for the heterogeneous computation systems for the efficiency improving are shown. Redundant numerical system with a constant time of the addition operation is introduced. It allows to design well scaled algorithms for all basic arithmetic operations with integer numbers. Scalability of the basic integer arithmetic algorithms is easy applied to rational arithmetic.
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Parallel delivery operations modelling
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Some delivery organization scheme is considered. The key point is the principle of routes parallelization using several carriers at the same time and these auxiliary carriers can be based on the main carrier. An example of such a delivery system is a van carrying several autonomous carriers, which in turn can carry out simultaneous so-called parallel deliveries. Delivery routes are determined based on the coordinates of customers, the determination of acceptable starting points for auxiliary carriers, the technical and energy limitations of the main and auxiliary carriers, and the minimization of the amount of time spent on delivery operations. The developed algorithm for solving the problem on routing of delivery using primary and secondary carriers allows to reduce delivery time and resources. The algorithm is implemented in Python using the libraries for processing and visualization of trajectories and other space-time data, packages for extracting, modelling, analyzing and visualizing street networks on the example of the Yekaterinburg city.
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Parameter identification and control in heat transfer processes
Статья научная
The article is devoted to the study of some mathematical models describing heat transfer processes. We examine an inverse problem of recovering a control parameter providing a prescribed temperature distribution at a given point of the spatial domain. The parameter is a lower order coefficient depending on time in a parabolic equation. This nonlinear problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. Existence and uniqueness theorems of solutions to this problem are stated and proved. Stability estimates are exposed. The main result is the global (in time) existence of solutions under some natural conditions of the data. The proofs rely on the maximum principle. The main functional spaces used are the Sobolev spaces.
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