Математическое моделирование. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
On one mathematical model described by boundary value problem for the biharmonic equation
Статья научная
In this paper mathematical model described by a generalized third boundary value problem for the homogeneous biharmonic equation in the unit ball with boundary operators up to the third order containing normal derivatives and Laplacian is investigated. Particular cases of the considered mathematical model are the classical models described by Dirichlet, Riquier, and Robin problems, and the Steklov spectral problem, as well as many other mathematical models generated by these boundary conditions. Two existence theorems for the solution of the problem are proved. Existence conditions are obtained in the form of orthogonality on the boundary of some linear combination of boundary functions to homogeneous harmonic polynomials of a particular order. The obtained results are illustrated by some special cases of the general problem.
Бесплатно
On one mathematical model of the extraction process of polydisperse porous material
Статья научная
We consider a mathematical model which represents the extraction process of a target component from the polydispersed porous material. The suggested model is demonstrated by the example of a flat solid material with bidispersed pores of different size in the form of a system of channels of macropores with micropores facing their walls. The macropores and the micropores in the material have homogeneous size. We model a case when micropores of the solid material (dispersed medium) are initially filled with an oil (dispersion phase), which is our target component. The macropores are filled in with a pure solvent. In the process of extraction the oil diffuses from the micropore to the macropore, and then from the micropores to the external solvent volume, wherein the ratio of concentrations in the macropore and the micropore is taken in accordance with the linear law of adsorption. The well-posedness of the formulated mathematical model has been justified.
Бесплатно
On perturbation method for the first kind equations: regularization and application
Статья научная
One of the most common problems of scientific applications is computation of the derivative of a function specified by possibly noisy or imprecise experimental data. Application of conventional techniques for numerically calculating derivatives will amplify the noise making the result useless. We address this typical ill-posed problem by application of perturbation method to linear first kind equations Ax=f with bounded operator A. We assume that we know the operator A and source function f only such as ||A-A||
Бесплатно
Статья научная
We examine inverse problems of recovering coefficients in a linear pseudoparabolic equation arising in the filtration theory. Boundary conditions of the Neumann type are supplemented with the overtermination conditions which are the values of the solution at some interior points of a domain. We expose existence and uniqueness theorems in the Sobolev spaces. The solution is regular, i.e., it possesses all generalized derivatives occurring in the equation containing in some Lebesgue space. The method of the proof is constructive. The problem is reduced to a nonlinear operator equation with a contraction operator whenever the time interval is sufficiently small. Involving the method of the proof, we construct a numerical algorithm, the corresponding software bundle, and describe the results of numerical experiments in the two-dimensional case in the space variables. The unknowns are a solution to the equation and the piezo-conductivity coefficient of a fissured rock. The main method of numerical solving the problem is the finite element method together with a difference scheme for solving of the corresponding system of ordinary differential equations. Finally, the problem is reduced to a system of nonlinear algebraic equations which solution is found by the iteration procedure. The results show a good convergence of the algorithms.
Бесплатно
On some mathematical models of filtration theory
Статья научная
The article is devoted to the study of some mathematical models arising in filtration theory. We examine an inverse problem of determining an unknown right-hand side and coefficients in a pseudoparabolic equation of the third order. Equations of this type and more general Sobolev-type equations arise in filtration theory, heat and mass transfer, plasma physics, and in many other fields. We reduce the problem to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. Together with the natural smoothness conditions for the data, we require also some well-posedness condition to be fulfilled which is actually reduced to the condition of nondegeneracy of some matrix constructed with the use of the data of the problem. Theorems on existence and uniqueness of solutions to this problem are stated and proven. Stability estimates are exposed. In the linear case the result is global in time, while in the nonlinear case it is local. The main function spaces used are the Sobolev spaces.
Бесплатно
On some methods to solve integrodifferential inverse problems of parabolic type
Статья научная
In this paper we give an overview on some methods that are useful to solve a class of integrodifferential inverse problems. Precisely, we present some methods to solve integrodifferential inverse problems of parabolic type that are based on the theory of analytic semigroups, optimal regularity results and fixed point arguments. A large class of physical models can be treated with this procedure, for example phase-field models, combustion models and the strongly damped wave equation with memory to mention some of them.
Бесплатно
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.
Бесплатно
Статья научная
The paper considers the issues of numerical modeling of the limit reachable sets for linear discrete-time systems with convex control constraints. The method based on the principle of contraction mappings has been developed. This method is designed to construct an external estimate of the limit reachable set, which is a significant problem in control theory and analysis of dynamical systems. The application of the principle of contraction mappings makes it possible to obtain an estimate with an arbitrary order of accuracy in the sense of the Hausdorff distance. Moreover, the limit point up to the closure must coincide with the limit reachable set. The value of the compression ratio depends on the choice of the norm in the vector space, which, accordingly, influences the Hausdorff distance in the compact space, as well as the operator norm of the system matrix. To demonstrate the capabilities of the proposed method, a three-dimensional system with real eigenvalues is presented as an example. Additionally, an example for constructing the limit reachable set in the damping system of a high-rise structure located in a seismic zone is provided.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
Performance bounds and suboptimal policies for multi-class queue
Статья научная
In this paper, we consider a general class of a queuing system with multiple job types and flexible service facility. We use a stochastic control policy to determine the performance loss in multi-class M/M/1 queue. The considered system is originally a Markov decision processes (MDP). The author showed how to compute performance bounds for the stochastic control policy of MDP with an average cost criteria. In practice, many authors used heuristic control policies due to some hardness in computing and running mathematically optimal policies. The authors found bounds on performance in order to an optimal policy where the goal of this job is to compute the difference of optimality and a specific policy. In other words, this study shows that, the optimal bounds of the average queue length for any non-idling policies can be found by a factor of service rates.
Бесплатно
Статья научная
In this note we consider the population the model of which, derived on the basis of ethnographical accounts, includes a projection matrix with both positive and negative entries. Interpreting the eventually negative trajectories as representing the collapse of the population, we use some classical tools from convex analysis to determine a cone containing the initial conditions that give rise to the persistence of both the population and its social structure.
Бесплатно
Positive solutions to Sobolev type equations with relatively p-sectorial operators
Статья научная
The article describes sufficient conditions for the existence of positive solutions to both the Cauchy problem and the Showalter-Sidorov problem for an abstract linear Sobolev type equation. A distinctive feature of such equations is the phenomenon of non-existence and non-uniqueness of solutions. The research is based on the theory of positive semigroups of operators and the theory of degenerate holomorphic semigroups of operators. The merger of these theories leads to a new theory of degenerate positive holomorphic semigroups of operators. In spaces of sequences, which are analogues of Sobolev function spaces, the constructed abstract theory is used to study a mathematical model. The results can be used to study economic and engineering problems.
Бесплатно
