It is investigated an unique solvability of the boundary value problem for the cases in the presence of differential operation in the order of degeneration of the Hilbert space, by using operator theory and its special case.

Keywords: ГИЛЬБЕРТОВО ПРОСТРАНСТВО, HILBERT'S SPACE, ОБОБЩЕННОЕ РЕШЕНИЕ, КОМПАКТНОЕ МНОЖЕСТВО, COMPACT SET, СПЕКТР ОПЕРАТОРА, SPECTRUM OF THE OPERATOR, ЛИНЕЙНЫЙ НЕПРЕРЫВНЫЙ ФУНКЦИОНАЛ, LINEAR CONTINUOUS FUNCTIONAL, THE GENERALIZED DECISION

In a cylindrical domain of the space $R^{n+1}$ for mixed type equation of the second order boundary value problem is considered, which for the first time was investigated by A.N. Terekhov. Under certain conditions on the equation coefficients is proved the existence and the uniqueness of the generalized solutions, the Fredholm property of boundary value problem in a weighted Sobolev space.

Keywords: УРАВНЕНИЕ СМЕШАННОГО ТИПА, EQUATION OF MIXED TYPE, СУЩЕСТВОВАНИЕ, EXISTENCE, ЕДИНСТВЕННОСТЬ, UNIQUENESS, ОБОБЩЕННОЕ РЕШЕНИЕ, GENERALIZED SOLUTION, НЕРАВЕНСТВО, INEQUALITY, ОЦЕНКА, ESTIMATION, ФРЕДГОЛЬМОВОСТЬ, FREDHOLM PROPERTY

The model of waves propagation on a shallow taking into account the capillarity was studied in terms of the relatively $p$-bounded operator theory. The algorithm of a numerical solution the Showalter-Sidorov-Dirichlet problem for the IMBq equation was suggested.

Keywords: ФАЗОВОЕ ПРОСТРАНСТВО, УРАВНЕНИЕ СОБОЛЕВСКОГО ТИПА, ОТНОСИТЕЛЬНО p-ОГРАНИЧЕННЫЙ ОПЕРАТОР, ЗАДАЧА ШОУОЛТЕРА—СИДОРОВА, PHASE SPACE, SOBOLEV TYPE EQUATION, RELATIVELY p-BOUNDED OPERATOR, SHOWALTER-SIDOROV PROBLEM

We study solvability of some boundary value problems for linear nonclassical equations including linearized equations of small longitudinal perturbations of solitary waves. For these problems, we prove existence and uniqueness theorems of regular solutions.

Keywords: КРАЕВАЯ ЗАДАЧА, BOUNDARY VALUE PROBLEM, ЛИНЕАРИЗОВАННОЕ УРАВНЕНИЕ, LINEARIZED EQUATION, АПРИОРНАЯ ОЦЕНКА, СУЩЕСТВОВАНИЕ, EXISTENCE, ЕДИНСТВЕННОСТЬ, UNIQUENESS, APRIORY ESTIMATE

The work is devoted to the study of Diophantine equation $x^2-y^2(k^2m^2-4m)=4t$, Where numbers $k,m$ are odd, and the right side $4t$ of the equation is enough small natural number. We find necessary solvability conditions of the Diophantine equation.

Keywords: ДИОФАНТОВО УРАВНЕНИЕ, DIOPHANTINE EQUATION, РЕШЕНИЯ В ЦЕЛЫХ ЧИСЛАХ, INTEGER SOLUTIONS, ОБОБЩЕННОЕ УРАВНЕНИЕ ПЕЛЛЯ, GENERALIZED PELL'S EQUATION, КВАДРАТИЧНЫЕ ПОЛЯ, QUADRATIC FIELDS, ГРУППА ЕДИНИЦ, UNIT GROUP, ДИОФАНТОВЫ ПРИБЛИЖЕНИЯ, DIOPHANTINE APPROXIMATIONS

In this paper the rods calculation by the spline of the fifth degree of the defect 2 method tasks and solve the nest problems: are formulated theoretical basis of the spline of the fifth degree of the defect 2; the spline algorithm of the fifth degree of defect 2 for the numerical solution of the problem of bending of an elastic rod - differential equation of fourth order.

Keywords: СПЛАЙН ПЯТОЙ СТЕПЕНИ ДЕФЕКТА 2, ДИФФЕРЕНЦИАЛЬНОЕ УРАВНЕНИЕ ЧЕТВЕРТОГО ПОРЯДКА, ИЗГИБ УПРУГОГО СТЕРЖНЯ, THE SPLINE ALGORITHM OF THE FIFTH DEGREE OF DEFECT 2, DIFFERENTIAL EQUATION OF FOURTH ORDER, PROBLEM OF BENDING OF AN ELASTIC ROD

In this paper we prove the existence theorem for regular decisions, and also proved that the necessary for a possible Laplace transform inversion of a priori estimates, depending on parameter.

Keywords: КРАЕВАЯ ЗАДАЧА, THE FIRST BOUNDARY VALUE PROBLEM, УРАВНЕНИЕ СОБОЛЕВСКОГО ТИПА, EQUATIONS OF SOBOLEV TYPE, РЕГУЛЯРНЫЕ РЕШЕНИЯ, REGULAR SOLUTIONS, АПРИОРНЫЕ ОЦЕНКИ

This paper investigates the problem of equilibrium of three-dimensional viscoelastic body with a rigid inclusion. The cases without delamination and partially delaminated inclusion are considered. The corresponding formulations of problems in the form of the boundary value problem, and also in the variational form are studied, we prove the equivalence of these formulations. Existence and uniqueness of solutions of the problems are proved.

Keywords: ВЯЗКОУПРУГОЕ ТЕЛО, VISCOELASTIC BODY, ЖЕСТКОЕ ВКЛЮЧЕНИЕ, ВАРИАЦИОННОЕ НЕРАВЕНСТВО, VARIATIONAL INEQUALITY, КРАЕВЫЕ УСЛОВИЯ ТИПА НЕРАВЕНСТВ, INEQUALITY TYPE BOUNDARY CONDITION, ТРЕЩИНА, CRACK, RIGID INCLUSION

In this paper the solvability of a nonlinear inverse problem for a degenerate parabolic equation is considered. The solution of degenerate parabolic equation also should satisfy to a first initial-boundary problem condition and to a redefinition condition. Conditions are found for solvability of the problem, the existence theorem of generalized solution of degenerate parabolic equation formulated and proved.

Keywords: ОБРАТНАЯ ЗАДАЧА, NON-LINEAR INVERSE PROBLEM, УСЛОВИЯ ПЕРЕОПРЕДЕЛЕНИЯ, ЭЛЛИПТИКО-ПАРАБОЛИЧЕСКОЕ УРАВНЕНИЕ, МЕТОД РЕГУЛЯРИЗАЦИИ, METHOD OF REGULARIZATION, АПРИОРНАЯ ОЦЕНКА, THE A PRIORI EVALUATION, DEGENERATE EQUATION, PARABOLIC EQUATION, GENERALIZED SOLVABILITY

We consider the variational free boundary problem describing contact problem for two elastic plates located at a given angle to each other. One of the plates is deformed in its plane with the other one being vertically deformed. Both plates contain rigid inclusion crossing the contact area. Inequality-type boundary condition is imposed providing a mutual nonpenetration between their surfaces. The problem solvability is stated using a variational methods. Assuming that the solution is smooth, the differential statement being equivalent to the variations formulation is justified.

Keywords: ЗАДАЧА О КОНТАКТЕ, CONTACT PROBLEM, ВАРИАЦИОННОЕ НЕРАВЕНСТВО, VARIATIONAL INEQUALITY, ЖЕСТКОЕ ВКЛЮЧЕНИЕ, ПЛАСТИНА КИРКГОФА-ЛЯВА, KIRCHHOFF-LOVE PLATE, УПРУГИЕ ПЛАСТИНЫ, ELASTIC PLATES, RIGID INCLUSION

УIn this paper we establish the connection of Liouville’s theorem for a nonautonomous system of ordinary differential equations (ODE) with resistance movement in the Lyapunov sense. In particular, the sufficient conditions for instability with a strong Lyapunov family of trajectories $\Omega_t=\{x(x^0,s,t)=T(t,s)x^0:\, x^0\in\Omega_s\}$, ODE system relatively compact set $\Omega_s\subset R^n$ of initial states. Introduced in review and lower bounds of the function describing the local divergence of trajectories and unlimited condensability nonautonomous ODE system. An interesting example.

Keywords: СИСТЕМА ОБЫКНОВЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ, SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS, ТЕОРЕМА ЛИУВИЛЛЯ, LIOUVILLE'S THEOREM, ОПЕРАТОР СДВИГА, SHIFT OPERATOR, ЛОКАЛЬНАЯ РАСХОДИМОСТЬ, THE LOCAL DIVERGENCE, НЕОГРАНИЧЕННАЯ СГУЩАЕМОСТЬ, UNBOUNDED CONDENSABILITY

The aim of the present paper is to investigate basic geometrical properties of an Euclidean conic manifold whose singular set is the trefoil knot with a bridge and the underlying space is the three sphere. Existence theorem is established. The lengths of singular geodesics are calculated. The volume formula is given.

Keywords: ЕВКЛИДОВО КОНИЧЕСКОЕ МНОГООБРАЗИЕ, УЗЕЛ ТРИЛИСТНИК С МОСТОМ, EUCLIDEAN CONIC MANIFOLD, TREFOIL KNOT WITH A BRIDGE

We investigated the periodical homogeneous Gaussian infinite systems of linear algebraic equations in terms of the general theory of infinite system. We have used the fundamental theorem about the necessary and sufficient conditions for the existence of nontrivial solutions of the homogeneous Gaussian infinite systems.

Keywords: БЕСКОНЕЧНЫЕ, ГАУССОВЫ, GAUSSIAN, ПЕРИОДИЧЕСКИЕ, СИСТЕМЫ, ФУНДАМЕНТАЛЬНЫЕ РЕШЕНИЯ, INFINITE, PERIODIC SYSTEMS, BASIC DECISIONS

We consider a continuous logic $\mathfrak{L}$ with truth values in a compact space $X$. For logic $\mathfrak{L}$ for which $||\mathfrak{L}||=\omega$ is given a new proof of compactness theorem. We study also a notion of elementary theory and a notion of D-limit.

Keywords: НЕПРЕРЫВНАЯ ЛОГИКА, CONTINUOUS LOGIC, ЭЛЕМЕНТАРНАЯ ТЕОРИЯ, ТЕОРЕМА О КОМПАКТНОСТИ, COMPACTNESS THEOREM, D-ПРЕДЕЛ, D-LIMITS

It is investigated the resolvability of some space nonlocal boundary problems for parabolic equations with discontinuous coefficients. Uniqueness and existence theorems are proofed.

Keywords: НЕЛОКАЛЬНЫЕ КРАЕВЫЕ ЗАДАЧИ, NONLOCAL BOUNDARY CONDITIONS, ИНТЕГРАЛЬНЫЕ УСЛОВИЯ, INTEGRAL CONDITIONS, УРАВНЕНИЯ С РАЗРЫВНЫМИ КОЭФФИЦИЕНТАМИ, EQUATIONS WITH DISCONTINUOUS COEFFICIENTS, РЕГУЛЯРНЫЕ РЕШЕНИЯ, REGULAR SOLUTIONS, АПРИОРНАЯ ОЦЕНКА, PRIORI ESTIMATE, СУЩЕСТВОВАНИЕ, EXISTENCE, ЕДИНСТВЕННОСТЬ, UNIQUENESS

In order to determine the influence of real gas properties on the dynamics of pressure and temperature fields of the latter, a comparative analysis of the results computational experiment of two-dimensional mathematical model of the selection of ideal and real gases through a single hole, provided heat to the surrounding rocks. The experiment was carried out within the framework of modified mathematical model of non-isothermal gas filtration, obtained from the energy and mass conservation laws and the Darcy law. The physical and caloric equations of state together with the Newton-Rihman law of heat exchange of gas-bearing reservoir with surrounding rocks, are used as closing relations. It is shown that the inclusion of real gas properties leads only to small quantitative changes in pressure, but despite this, involves a significant redistribution of temperature field of gas-bearing reservoir.

Keywords: МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ, MATHEMATICAL MODELING, НЕИЗОТЕРМИЧЕСКАЯ ФИЛЬТРАЦИЯ, NON-ISOTHERMAL FILTRATION, ИДЕАЛЬНЫЙ ГАЗ, IDEAL GAS, РЕАЛЬНЫЙ ГАЗ, REAL GAS, ЧИСЛЕННЫЕ МЕТОДЫ, NUMERICAL METHODS

We consider the process of filtering grounds freezing. The mathematical model includes the equation for the pore water, Darcy’s law and the law of conservation of energy. Numerical implementation of the model based on the finite element method using a software package FEniCS. The computational algorithm is implemented on high-performance computing systems. The numerical results are reduced in graphs of temperature and pressure around of a freezing column.

Keywords: ТЕПЛОПЕРЕНОС, HEAT TRANSFER, ФИЛЬТРАЦИЯ, FILTRATION, ФАЗОВЫЙ ПЕРЕХОД, PHASE TRANSFER, МЕТОД ФИКТИВНЫХ ОБЛАСТЕЙ, FICTITIOUS DOMAIN METHOD, МЕТОД КОНЕЧНЫХ ЭЛЕМЕНТОВ, FINITE ELEMENT METHOD, ВЫЧИСЛИТЕЛЬНЫЙ КЛАСТЕР, COMPUTING CLUSTER, DIFFERENCE SCHEMES

A two-dimensional mathematical model of the Earth’s interior moving by tidal deformations is developed. In this model the condition of partial slip on the boundary between solid and liquid cores is used. In the proposed boundary condition a viscosity of the liquid core of the Earth, one dimensional and one dimensionless parameters are used. Computational realization of the model was made by small parameter method. It is shown that in the first order of approximation a viscosity of the liquid core of the Earth does not effect on the angular velocity of inner core differential rotation. An estimate of the value of the differential rotation of the inner core of the Earth model with the condition of partial slip was 0.39 min/year to eastward.

Keywords: THE EARTH'S TIDAL DEFORMATION, INNER CORE, DIFFERENTIAL ROTATION, MATHEMATICAL MODEL

Algorithms and program on a pc implementation of solutions of systems of linear algebraic equations of any order, up to infinite are proposed. Conversions of gauss for an infinite system and strictly private solution of non-homogeneous are used.

Keywords: БЕСКОНЕЧНАЯ СИСТЕМА ЛИНЕЙНЫХ УРАВНЕНИЙ, INFINITE SYSTEMS, МЕТОД ИСКЛЮЧЕНИЯ ГАУССА, METHOD OF ELIMINATION OF GAUSS, СТРОГО ЧАСТНОЕ РЕШЕНИЕ, STRICTLY PRIVATE DECISION, LINEAR EQUATIONS

In this article is considered the question on representation of one class of plurisubharmonic functions and defines pluriharmonic extension from sphere in the ball.

Keywords: ПЛЮРИСУБГАРМОНИЧЕСКИЕ ФУНКЦИИ, ПЛЮРИГАРМОНИЧЕСКОЕ ПРОДОЛЖЕНИЕ СО СФЕРЫ В ШАР, PLURISUBHARMONIC FUNCTIONS, PLURIHARMONIC EXTENSION OF FUNCTIONS FROM SPHERE IN BALL

Under consideration is a mixed problem for the multi-dimensional wave equation in a quarter of the real space. The boundary condition is given as a linear combination of the first derivatives, and the uniform Lopatinskii condition is assumed to be satisfied. In this case, we constructed all possible dissipative energy integrals and parametrized them by the points of the upper part of a bodily cone of the second order. We characterize the cone location and its geometric parameters by means of the coefficients of the boundary condition of the problem.

Keywords: ВОЛНОВОЕ УРАВНЕНИЕ, WAVE EQUATION, СМЕШАННАЯ ЗАДАЧА, MIXED PROBLEM, РАВНОМЕРНОЕ УСЛОВИЕ ЛОПАТИНСКОГО, UNIFORM LOPATINSKII CONDITION, ДИССИПАТИВНЫЙ ИНТЕГРАЛ ЭНЕРГИИ, DISSIPATIVE ENERGY INTEGRAL

The notion is considered of a monotone linear operator acting from a Riesz space into a normed space. It is shown that every monotone operator can be represented as the composition of a Riesz homomorphism and a linear isometry. The results are applied to the study of continuous and measurable bundles of Banach lattices.

Keywords: ВЕКТОРНАЯ РЕШЕТКА, NORMED LATTICE, НОРМИРОВАННАЯ РЕШЕТКА, РЕШЕТОЧНЫЙ ГОМОМОРФИЗМ, RIESZ HOMOMORPHISM, ЛИНЕЙНАЯ ИЗОМЕТРИЯ, LINEAR ISOMETRY, РЕШЕТОЧНО НОРМИРОВАННОЕ ПРОСТРАНСТВО, LATTICE-NORMED SPACE, ПРОСТРАНСТВО БАНАХА — КАНТОРОВИЧА, BANACH BUNDLE, БАНАХОВО РАССЛОЕНИЕ, ЛИФТИНГ, LIFTING, RIESZ SPACE, BANACH-KANTOROVICH SPACE

We proved a theorem on the weak limit of a sequence of mappings satisfied the differential inequality constructed by a quasiconvex function and a Lagrangian.

Keywords: СЛАБЫЙ ПРЕДЕЛ, КВАЗИВЫПУКЛОСТЬ, НУЛЬ-ЛАГРАНЖИАН, WEAK LIMIT, QUASICONVEXITY, NULL LAGRANGIAN

In this paper, we consider the first boundary value problem for an equation of mixed type of even order in a cylindrical domain. Under certain conditions on the coefficients of the equation proved its dense solvability, uniqueness of the generalized solutions and Fredholm solvability of the first boundary value problem in a weighted Sobolev space.

Keywords: УРАВНЕНИЕ СМЕШАННОГО ТИПА, EQUATION OF MIXED TYPE, ФРЕДГОЛЬМОВА РАЗРЕШИМОСТЬ, FREDHOLM SOLVABILITY OF GENERALIZED SOLUTION, ОБОБЩЕННОЕ РЕШЕНИЕ, НЕРАВЕНСТВО, INEQUALITY, ОЦЕНКА, EVALUATION, ОПЕРАТОР, OPERATOR

The Sobolev type equations theory, experiencing an epoch of blossoming, is very useful for investigation of linearized Benney-Luke model. In this article the theory of relatively p-sectorial operators is developed to the case of higher order equations. The propagators of homogeneous equation are constructed. The sufficient conditions for the unique solvability of abstract Cauchy problem are given. On the basis of abstract results the analytical investigation of linearized Benney-Luke model is held.

Keywords: ОТНОСИТЕЛЬНО P-СЕКТОРИАЛЬНЫЙ ОПЕРАТОР, RELATIVELY P-SECTORIAL OPERATOR, УРАВНЕНИЕ СОБОЛЕВСКОГО ТИПА ВЫСОКОГО ПОРЯДКА, HIGHER ORDER SOBOLEV TYPE EQUATION, ПРОПАГАТОРЫ, PROPAGATORS

In 1940, Lebesgue described the vicinities of vertices of degree 5 in plane triangulations with minimum degree 5, presenting only an idea of the proof but not the details. We give a complete proof of Lebesgue’s description, correct several inaccuracies happened in this description, and improve one of its parameters without worsening the others.

Keywords: ПЛОСКИЙ ГРАФ, PLANE GRAPH, СТРОЕНИЕ, STRUCTURE, ТРИАНГУЛЯЦИЯ, TRIANGULATION, ОКРЕСТНОСТЬ, VICINITY

We study solvability of a nonlinear inverse problem for a hyperbolic equation. Existence and uniqueness theorems of regular solution are proved.

Keywords: ОБРАТНАЯ КОЭФФИЦИЕНТНАЯ ЗАДАЧА, INVERSE COEFFICIENT PROBLEM, УСЛОВИЕ ИНТЕГРАЛЬНОГО ПЕРЕОПРЕДЕЛЕНИЯ, INTEGRAL OVERDETERMINATION CONDITION, НАГРУЖЕННОЕ УРАВНЕНИЕ, LOADED EQUATION, МЕТОД НЕПОДВИЖНОЙ ТОЧКИ, FIXED POINT METHOD

In this article we prove that wreath products of some permutation groups have finite width. We prove in addition that the wreath product of some permutation groups cannot be embedded to a skew linear group.

Keywords: ДЕКАРТОВО СПЛЕТЕНИЕ ГРУПП ПРЕОБРАЗОВАНИЙ, КОНЕЧНАЯ ШИРИНА, СВОЙСТВО БЕРГМАНА, ГРУППА АВТОМОРФИЗМОВ КОНЕЧНОМЕРНОГО ВЕКТОРНОГО ПРОСТРАНСТВА НАД ТЕЛОМ, THE WREATH PRODUCT OF PERMUTATION GROUP, FINITE WIDTH, BERGMAN PROPERTY, SKEW LINEAR GROUP

Let I be a unit segment in R. W. M. Priestley contracts an example of sequentially closed countable dense subset in the topological space $I^I$ of all functions from I to I endowed with Tykhonov topology. Construction of Priestley’s example essentially uses the axiom of choice. In the present article we give another, more elementary solution of this problem, in which the axiom of choice was not applied (even in the weakened form, known as the axiom of countable choice).

Keywords: ТИХОНОВСКАЯ ТОПОЛОГИЯ, TYKHONOV TOPOLOGY, СЕКВЕНЦИАЛЬНО ЗАМКНУТОЕ МНОЖЕСТВО, SEQUENTIALLY CLOSED SET, ПЛОТНОЕ МНОЖЕСТВО, ФУНКЦИИ РАДЕМАХЕРА, RADEMACHER FUNCTIONS, АКСИОМА ВЫБОРА, AXIOM OF CHOICE, АКСИОМА СЧЕТНОГО ВЫБОРА, AXIOM OF COUNTABLE CHOICE, DENSE SET

We study solvability of linear inverse problems for pseudo-parabolic equations with one and two unknown coefficients depending on the time variable t. Auxiliary non-local problems have boundary conditions of overdetermination. For these problems, we prove existence theorems of regular solutions.

Keywords: ПСЕВДОПАРАБОЛИЧЕСКОЕ УРАВНЕНИЕ, ОБРАТНАЯ ЗАДАЧА, INVERSE PROBLEM, АПРИОРНАЯ ОЦЕНКА, УСЛОВИЯ ПЕРЕОПРЕДЕЛЕНИЯ, CONDITIONS OF OVERDETERMINATION, СУЩЕСТВОВАНИЕ, EXISTENCE, PSEUDO-PARABOLIC EQUATIONS, APRIORY ESTIMATE

Consider parabolic equations 2n-th order changing the direction of time to complete the matrix of conditions bonding associated with the use of the theory of singular integral equations. The solvability of boundary value problems in Holder spaces. It is shown that the Holder classes of decisions in some cases depend on the conditions of adhesion of noninteger Holder exponent, if the necessary and sufficient conditions on the data.

Keywords: ПАРАБОЛИЧЕСКОЕ УРАВНЕНИЕ С МЕНЯЮЩИМСЯ НАПРАВЛЕНИЕМ ВРЕМЕНИ, PARABOLIC EQUATIONS WITH CHANGING TIME DIRECTION, ПОЛНАЯ МАТРИЦА УСЛОВИЙ СКЛЕИВАНИЯ, THE COMPLETE MATRIX OF BONDING CONDITIONS, УРАВНЕНИЕ 2N-ГО ПОРЯДКА, THE EQUATIONS OF THE 2N-TH ORDER, КОРРЕКТНОСТЬ, ПРОСТРАНСТВО ГЁЛЬДЕРА, HOLDER SPACE, СИНГУЛЯРНОЕ ИНТЕГРАЛЬНОЕ УРАВНЕНИЕ, SINGULAR INTEGRAL EQUATIONS, PROPRIETY

The solvability of the initial boundary value problems for linear pseudohyperbolic third- order equations with spatially nonlocal boundary conditions A. A. Samarsky with variables coefficients. Proving theorems of existence and uniqueness of regular solutions.

Keywords: ПСЕВДОГИПЕРБОЛИЧЕСКОЕ УРАВНЕНИЕ, PSEUDOHYPERBOLIC EQUATION, ПРОСТРАНСТВО СОБОЛЕВА, SOBOLEV SPACE, НАЧАЛЬНО-КРАЕВАЯ ЗАДАЧА, REGIONAL PROBLEM, МЕТОД ПРОДОЛЖЕНИЯ ПО ПАРАМЕТРАМ, CONTINUATION METHOD ON PARAMETERS, АПРИОРНЫЕ ОЦЕНКИ, РЕГУЛЯРНОЕ РЕШЕНИЕ, THE REGULAR DECISION, APRIORISTIC ESTIMATIONS

Of concern are problems of the optimal and hard control of the solutions for the operator- differential equation, unsolved with respect to the derivative by time, with Showalter-Sidorov condition. In this case, one of the operators in the equation is a operator-function of the variable t, i.e. operator depends of time. The strong solution is define in paper. The existence and uniqueness of strong solution for Showalter-Sidorov's problem for the nonstationary equation are proved. The existence of a unique optimal control and hard control for the solutions of this problem is proved using these results.

Keywords: ЗАДАЧА ОПТИМАЛЬНОГО УПРАВЛЕНИЯ, ЗАДАЧА ЖЕСТКОГО УПРАВЛЕНИЯ, НЕСТАЦИОНАРНЫЕ УРАВНЕНИЯ, УРАВНЕНИЯ СОБОЛЕВСКОГО ТИПА, OPTIMAL CONTROL PROBLEM, HARD CONTROL PROBLEM, NONSTATIONARY EQUATIONS, SOBOLEV TYPE EQUATIONS

Splitting theorem of two Banach spaces and action pair of operators moved in the quasi-Banach spaces. Abstract results are illustrated by examples from theory of Quasi- Sobolev spaces and Laplas’ Quasi-operator.

Keywords: ТЕОРЕМА О РАСЩЕПЛЕНИИ, КВАЗИБАНАХОВЫ ПРОСТРАНСТВА, КВАЗИСОБОЛЕВСКИЕ ПРОСТРАНСТВА, КВАЗИОПЕРАТОР ЛАПЛАСА, SPLITTING THEOREM, QUASI-BANACH SPACES, QUASI-SOBOLEV SPACES, LAPLAS' QUASI-OPERATOR

Under study is the solvability of the inverse problem of determining the right-hand side for a fourth order parabolic equation with overdetermination boundary condition. In addition, the existence of solution to a nonlocal problem is shown for a fourth order parabolic equation. The latter is of interest in its own right.

Keywords: ПАРАБОЛИЧЕСКИЕ УРАВНЕНИЯ ВЫСОКОГО ПОРЯДКА, PARABOLIC EQUATIONS OF HIGHER ORDER, ОБРАТНАЯ ЗАДАЧА, INVERSE PROBLEM, НЕЛОКАЛЬНАЯ ЗАДАЧА, СУЩЕСТВОВАНИЕ РЕШЕНИЯ, EXISTENCE, NONLOCAL PROBLEM

It is investigated the so-called n-periodic infinite systems of linear algebraic equations. The author used the fundamental theorem of Gaussian Infinite Systems.

Keywords: БЕСКОНЕЧНЫЕ ГАУССОВЫ ПЕРИОДИЧЕСКИЕ СИСТЕМЫ, ПОЧТИ ПЕРИОДИЧЕСКИЕ СИСТЕМЫ, INFINITE, GAUSSIAN, PERIODIC SYSTEMS, ALMOST PERIODIC SYSTEM

This work is considered to S. N. Bernstein’s modification of Lagrange interpolation formula. Here were acquired new matrices of interpolation modification of which brings to uniform convergence of the process for any continuous function on the whole interval of interpolation. This work is also shows that it is impossible to greatly improve the known valuation of H. L. Lebesgue function in the case of the usage of such modification on the matrix of equidistant nodes.

Keywords: ИНТЕРПОЛЯЦИЯ, INTERPOLATION, СХОДИМОСТЬ, CONVERGENCE, МАТРИЦЫ ИНТЕРПОЛИРОВАНИЯ, MATRICES OF INTERPOLATION, ОПЕРАТОР БЕРНШТЕЙНА, BERNSTEIN OPERATOR, ФУНКЦИЯ ЛЕБЕГА, FUNCTION OF LEBESGUE

The paper proposes a numerical algorithm for solving the problem of recovering dynamically distorted signal with the inertia and resonances of the measuring device. The basis for the construction of the algorithm is a mathematical model of the measuring device. The model considered as an optimal control for the Leontief type system with initial conditions Showalter-Sidorova (or model Shestakov-Sviridyuk) include numerical methods for solving such problems. The article explains the convergence of approximate solutions obtained in the implementation of the algorithm.

Keywords: ОПТИМАЛЬНОЕ ИЗМЕРЕНИЕ, СИСТЕМЫ ЛЕОНТЬЕВСКОГО ТИПА, ЧИСЛЕННЫЙ МЕТОД, ОПТИМАЛЬНОЕ УПРАВЛЕНИЕ, OPTIMAL MEASUREMENT, LEONTIEF TYPE SYSTEMS, NUMERICAL METHOD, OPTIMAL CONTROL

Ecological-economic model of the protected population is investigated. This model contains a system of two differential equations of population dynamics and hierarchical two-person game. Phase portraits of the trajectories are constructed. Bifurcation parameters of the system of differential equations are defined. Optimal strategies of players for the hierarchical game are constructed. Density distributions of random variables of the payoff functions are found.

Keywords: СИСТЕМА ДВУХ ОБЫКНОВЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ, ODE SYSTEM, СЛУЧАЙНЫЕ ПАРАМЕТРЫ, RANDOM PARAMETERS, ФАЗОВЫЕ ПОРТРЕТЫ ТРАЕКТОРИЙ, ИЕРАРХИЧЕСКАЯ ИГРА, HIERARCHICAL GAME, ПЛОТНОСТИ РАСПРЕДЕЛЕНИЯ, DENSITY DISTRIBUTION, DYNAMIC OF POPULATION, STABILITY PROBLEM, BIFURCATION

This paper is dedicated to a numerical simulation of filtration model in cavity porous media. We consider classical double porosity models which more precisely reflect filtration process taking into the availability of cavities system. We derive pseudoparabolic model using basic Barenblatt model. Calculations were performed on the basis of finite element method.

Keywords: МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ, МОДЕЛЬ ДВОЙНОЙ ПОРИСТОСТИ, DOUBLE POROSITY MODEL, ПСЕВДОПАРАБОЛИЧЕСКИЕ УРАВНЕНИЯ, PSEUDOPARABOLIC EQUATION, ТРЕЩИНОВАТО-ПОРИСТАЯ СРЕДА, МЕТОД КОНЕЧНЫХ ЭЛЕМЕНТОВ, FINITE ELEMENT METHOD, NUMERICAL SIMULATION, CAVITY POROUS MEDIA

In this work a program module have been developed to modelling simplified structural 3D reservoir for it further implementation in reservoir simulation. Borehole data located at the area interested are used in the capacity of input data. Borehole data are the well’s coordinates absolute vertical depth subsea of reservoir’s top and net productive section at the well. The output data are map of reservoir’s top depth and 3D reservoir grid as text files recognizable by other reservoir simulators. Method of simple kriging is used to interpolate depth and net productive section for geological modelling of reservoir.

Keywords: СТРУКТУРНАЯ 3D МОДЕЛЬ ПЛАСТА, ОБЫКНОВЕННЫЙ КРИГИНГ, АБСОЛЮТНАЯ ГЛУБИНА, КРОВЛЯ КОЛЛЕКТОРА, ЭФФЕКТИВНАЯ МОЩНОСТЬ, STRUCTURAL 3D RESERVOIR, SIMPLE KRIGING, ABSOLUTE VERTICAL DEPTH SUBSEA, RESERVOIR'S TOP, NET PRODUCTIVE SECTION

Domain decomposition methods used for the approximate solutions of boundary value problems for partial differential equations on parallel computing systems. For time- dependent problems are most suitable iteration-free domain decomposition schemes. We consider the factorized regionally-additive scheme for the solution of the initial-boundary value problem for a parabolic equation. The parallel solution is based on domain division (domain decomposition without overlap) and on 3-stage algorithm, which consists of prediction, solution and correction. We give a theoretical estimate of the error for implemented factorized scheme, which is supplemented by practical experiments.

Keywords: ПАРАЛЛЕЛЬНЫЙ АЛГОРИТМ, PARALLEL ALGORITHMS, МЕТОДЫ ДЕКОМПОЗИЦИИ ОБЛАСТИ, DOMAIN DECOMPOSITION METHODS, ФАКТОРИЗОВАННАЯ СХЕМА, FACTORIZED SCHEMES

Under study are a two-dimensional kinetic model of CO oxidation on $PdO$/$Al_2O_3$ catalyst and its parametric portrait in the plane of the two parameters.

Keywords: ДИНАМИЧЕСКАЯ СИСТЕМА, DYNAMICAL SYSTEM, СТАЦИОНАРНОЕ СОСТОЯНИЕ, МНОЖЕСТВЕННОСТЬ, БИФУРКАЦИЯ, BIFURCATION, STEADY STATES, SELF-OSCILLATIONS, MULTIPLICITY

Under study are some problems of multipeak oscillations in a dynamical system that models the kinetic self-oscillations in catalytic CO oxidation reaction.

Keywords: ДИНАМИЧЕСКАЯ СИСТЕМА, DYNAMICAL SYSTEM, МНОГОПИКОВЫЕ АВТОКОЛЕБАНИЯ, MULTIPEAK OSCILLATIONS, БИФУРКАЦИЯ, BIFURCATION

We consider the two-grid iterative method of solution of the system of equations approximating third boundary value problem for the Laplace equation. Numerical results demonstrate the efficiency of the proposed method.

Keywords: МНОГОСЕТОЧНЫЙ МЕТОД, MULTIGRID METHOD, ИТЕРАЦИОННЫЙ МЕТОД, ITERATIVE METHOD