Математическое моделирование. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Inverse problem for Sobolev type mathematical models
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The work is devoted to the study of an inverse problem for the linear Sobolev type equation of higher order with an unknown coefficient depending on time. Since the equation might be degenerate the phase space method is used. It consists in construction of projectors splitting initial spaces into a direct sum of subspaces. Actions of operators also split. Therefore, the initial problem is reduced to two problems: regular and singular. The regular one is reduced to the first order nondegenerate problem which is solved via approximations. The needed smoothness of the solution is obtained. Then it is substituted into the singular problem which is solved using the methods of relatively polynomially bounded operator pencils theory. The main result of the work contains sufficient conditions for the existence and uniqueness of the solution to the inverse problem for a complete Sobolev type model of the second order. This technique can be used to investigate inverse problems of the considered type for Boussinesq-Love mathematical model.
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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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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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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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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 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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Mathematical model of the downward two-phase flow of a heat-transfer agent in an injection well
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At the present time, the main method of developing highly viscous and bituminous oil reservoirs is the injection of hot water or steam into such reservoirs. When injecting heat-transfer agent into a porous reservoir, its characteristics at the wellhead are known. It is important to know the parameters of a heat-transfer agent (pressure, temperature, mass content of steam in a two-phase mixture "water-steam", etc.) directly at the reservoir entrance. In order to calculate various parameters of a heat-transfer agent along the injection well depth (including the bottomhole), we propose a mathematical model of the downward flow of a hot "water-steam" mixture in a vertical channel. The model takes into account phase transitions occurring in a two-phase "water-steam" mixture, and external heat exchange of the well product with surrounding rocks (including permafrost). Based on the proposed mathematical model, we develop an algorithm to solve a quasistationary problem. In this case, we use the Runge-Kutta method in order to solve the system of differential equations describing the stationary flow of a heat-transfer agent in a well. Also, in order to solve the non-stationary problem of temperature distribution in the rocks that surround the well (including permafrost), we use the author enthalpy method with implicit scheme. For each time moment, the developed software allows to find the distributions along the well depth of various parameters of the downward two-phase flow, taking into account external heat exchange, as well as the temperature distribution in the rocks that surround the well and the permafrost thawing radius.
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Mathematical modelling of a transport system with minimal maintenance costs
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We suggest a mathematical model of a transport system. The model describes the delivery of products from several suppliers to different points of consumption. It is assumed that consumer demands are random. The model is a two-stage stochastic programming problem. At the first stage suppliers make the commodity stocks. At the second stage we consider the product distribution to the points of consumption while minimizing compensation expenses for the goods shortage caused by the random demand. The model takes into account a random loss that occurs during the transportation of goods and the detection of defective products. The total cost of the transport system operation is minimized. The algorithm for solving the problem is proposed. This algorithm is based on reduction of the original problem to an equivalent mixed-integer linear programming problem after discretization. An example is considered.
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Development of periodic disturbances on free surface of water film falling down vertical plane for Reynolds' number Reє[5;10] is investigated. The investigation is implemented in a scope of the nonlinear differential equation for evolution of free surface of falling down liquid film. The equation is solved by a finite differencies method at rectangular uniformly spaced grid. By researching the growth of unit inaccuracy, the conditions on parameters of computation grid for inaccuracies to be not increasing are obtained. As a result, waveforms of water film, time spent to form the regular wave mode and amplitudes of periodic disturbances are calculated. Calculated amplitudes and experimental ones are compared.
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Mathematical terrain modelling with the help of modified Gaussian functions
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Based on a fundamentally new approach, we present a complete mathematical model for estimating the mass of water in the flooded coastal relief, taking into account the water in the basin of the reservoir in a given region. Taking into account stochastic studies, we construct an approximate model of the relief of the reservoir basin bottom, as well as the relief of a possible section of the flooding of this basin coastline. The modelling is based on the empirical data of measurements of the reservoir depths, as well as on the study on the architecture of the lines of the coastal maps of the possible flooding zone. Based on the measurements of the depths and bumps of the bottom surface, we verify the hypothesis that the use of the two-dimensional Gauss distribution is adequate. Numerous confirmation of this hypothesis on the basis of empirical measurements allows to use localized elliptic Gauss surfaces as a model function in order to construct an approximate model of hillocks and valleys. At the same time, the coordinates of local extremes of the depths, as well as the values of these extremes are constant. In order to simulate the surfaces of the underwater slopes, we construct planes according to depth measurements. This simulation is not a real copy, but is stochastic in nature and allows to take into account the main goal of the model, i.e. a full adequate estimation of the water mass of the flooded coastal relief included the water in the basin of the reservoir in the region. The equation of the model of the entire flooded region includes all local functions constructed for the mounds and troughs of the reservoir, as well as the functions of the planes of the slope models. For an approximate construction of the surface equations of the coastal zone, we use maps with detailed level lines as empirical data.
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Model of conveyer with the regulable speed
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This article is devoted to mathematical modelling of the production flow lines of the conveyor-type. Here is an analytical method for calculating the parameters of a production line with a regulated speed of the movement of the subjects of the labour along the conveyor developed. The description of the parameters of the state of the production line is made in the one-moment approximation using partial differential equations. There has been derived a solution that determines the state of the parameters of the production line for a given technological position as a function of the time. The transitional period during which the initial condition of the distributing of the subjects of the labour along a conveyor has the influence on the state parameters of a production line is certain. The developed method of the calculation of the flow parameters of the production line allows designing control systems of the production line of the conveyor-type with a regulated rate of the movement of the subjects of the labour.
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The paper considers a boundary-value problem for a hybrid system of differential equations, which represents a generalized mathematical model for a system of interconnected rigid bodies attached to the rod by elastic-damping links. A hybrid system of differential equations is understood as a system of differential equations composed of ordinary differential equations and partial differential equations. For the theoretical foundations of our approach to investigation of the boundary value problem for the hybrid system of differential equations we propose a method of finding eigenvalues for the boundary-value problem. The comparative analysis of the results of numerical computations conducted with the use of the proposed method and the results obtained by other techniques known from the literature have proved the validity and the universal character of the proposed approach.
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Modelling of the axisymmetric precision electrochemical shaping
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The problem on modelling of a precision shaping and boundary conditions are formulated according to Faraday's law and with applying of stepwise dependence current efficiency on current density. The problem is reduced to the solution of a boundary problem for definition of two analytical functions of the complex variable. The first function is a conformal mapping of region of parametrical variable on the physical plane. In order to determine this function we use the Schwartz's integral and a spline interpolation. Unlike a plane problem for determination of potential and stream function of an axisymmetric field, the integration transformations of the second analytical function are used. The analytical function is defined in the form of a sum of two addends. The first addend takes into account the singularities of the function so that the second addend has no singularities. The second function is defined by the Schwartz's integral. Interpolation by spline functions is carried out, where the spline coefficients are derivatives of these functions by means of which the intensity vector components are calculated. We propose the method to solve the axisymmetric stationary problems, which differs from the known methods by the accuracy. By means of the method, we obtain the numerical results, describing the workpiece form. The error estimation of the obtained results is carried out. Also, we show qualitative coincidence with results of plane problem solution.
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Modelling the flow of character recognition results in video stream
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The paper considers problems of developing stochastic models consistent with results of character image recognition in video stream. A set of assumptions that define the models structure and properties is stated. A class of distributions, namely the Dirichlet distribution and its generalizations, that set a description of the model components is pointed out; and methods for statistical estimation of the distribution parameters are given. To rank the models, the Akaike information criterion is used. The proposed theoretical distributions are verified vs sample data.
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Modelling the human papilloma virus transmission in a bisexually active host community
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In this article, we construct a mathematical model describing the transmission dynamics of Human Papilloma Virus (HPV) in a bisexually active host community. Comprehensive mathematical techniques are used to qualitatively and quantitatively analyze the model. We analyze the local and global stabilities of the model's equilibria and show that if the basic reproduction number is less than unity, then the model is locally and globally asymptotically stable at the HPV-free static states. Also, if the basic reproduction number is less than unity, then the HPV-endemic static state is globally asymptotically stable. Numerical simulations are carried out and graphical illustrations are presented to validate the theoretical results.
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Modelling the process of non-equilibrium hydrate formation in a porous reservoir
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This paper presents a solution to the flat-dimensional problem of gas hydrate formation in a porous medium. Highly permeable reservoirs are considered, as a result of which it is assumed that the process accompanied by the transition of gas into the hydrate composition is nonequilibrium. Based on the numerical solution, the influence of injection pressure and formation permeability on the peculiarities of phase transitions process has been studied. It is shown that with an increase in the injection pressure, both the maximum possible temperature of the system and the length of the hydrate-containing region increase. It has been found that the maximum temperature realized in the system, depending on the permeability of the reservoir, has a non-monotonic character. The influence of the initial temperature of the porous reservoir on the dynamics of phase transitions has been studied.
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At the application of bifurcation theory methods to nonlinear boundary value problems for ordinary differential equations of the fourth and higher order there usually arise technical difficulties, connected with determination of bifurcation manifolds, spectral investigation of the direct and conjugate linearized problems and the proof of their Fredholm property. For overcoming of this difficulty here the roots separation method is applied to the relevant characteristic equations with subsequent presentation of critical manifolds, that allows to investigate nonlinear problems in the precise statement. Such approach is applied here to two-point boundary value problem for the nonlinear ODE of the fourth order describing the buckling (divergence) of an elongated plate in a supersonic flow of gas, subjected to compressed or extended boundary stresses at the various boundary fastenings.
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