Correctness Of Boundary Value Problems Research Articles

NONHOMOGENEOUS BOUNDARY VALUE PROBLEM WITH NONLOCAL CONDITIONS FOR A PARTIAL DIFFERENTIAL EQUATION WITH THE OPERATOR OF THE GENERALIZED DIFFERENTIATION

The article is devoted to investigation of nonlocal boundary value problem for nonhomogeneous partial differential equation with the operator of the generalized differentiation $B=z\frac{\partial}{\partial z}$, which operate on function of scalar complex variable $z$. Problems with nonlocal conditions for partial differential equations represent an important part of the present-day theory of differential equations. Particularly, this is due with the fact that these problems are models of the propagation of heat, process of moisture transfer in capillary-porous environments, diffusion of particles in the plasma, inverse problems, and also problems of mathematical biology. One of the most important question of the general theory of partial differential equations is the establishment of conditions for the correctness of boundary value problems. However, the investigation of problems with nonlocal conditions for partial differential equations in bounded domains connected with the problem of small denominators. This problem connected with the fact, that the denominators of coefficients of the series, which represented the solutions of nonlocal problems may be arbitrary small. Specific of the present work is the investigation of a nonlocal boundary-value problem for nonhomogeneous partial differential equation with the operator of the generalized differentiation $B=z\frac{\partial}{\partial z}$, which operate on functions of one scalar complex variable $z$. The considered problem in the case of many generalized differentiation operators is incorrect in Hadamard sense, and its solvability depends on the small denominators that arise in the constructing of a solution. In the case of one scalar complex variable we showed, that the problem is Hadamard correct. The conditions of correct solvability of the nonlocal boundary value problem in Sobolev spaces are established. The uniqueness theorem and existence theorem of the solution of problem in these spaces are proved.

On the Correctness of Boundary Value Problems for the Two-Dimensional Loaded Parabolic Equation

The paper studies the problems of the correctness of setting boundary value problems for a loaded parabolic equation. The a feature of the problems is that the order of the derivative in the loaded term is less than or equal to the order of the differential part of the equation, and the load point moves according to a nonlinear law. At the same time, the distinctive characteristic is that the line, on which the loaded term is set, is at the zero point. On the basis of the study the authors proved the theorems about correctness of the studied boundary value problems.

ARCHITECTONICS OF CALCULATED MATHEMATICAL MODELS UNDER UNCERTAINTY

This article concerns the improvement of calculated mathematical models of technological, biotechnological, and economic systems. It is necessary to increase the number of considered parameters to increase the accuracy of calculating the parameters of complex systems during mathematical modeling. This leads to the need to solve nonlocal boundary value problems with non-stationary differential equations, to prove the correctness of which it is impossible to apply the traditional theory of existence and unity of solution. Note that after the architecture of boundary value problems assumes the existence of their solution, it is only necessary to prove its uniqueness. To prove the correctness of calculated mathematical models requires neither generalizing the parameters of the goal function and using approximate constraints, which, in turn, will reduce the boundary value problem to a standard form and its correctness will not be in doubt, nor propose a method to prove the correctness of boundary value certain differential equations, which will consider the specific features of the modeled processes. A separate technique must substantiate the correctness of boundary value problems depending on the type of differential equation that describes the physical and economic processes in the simulated systems. This article studied the conditions for the correctness of boundary value problems for differential equations with constant coefficients. It is proved that there is a corresponding boundary value problem for arbitrary homogeneous differential equations. It is defined the parabolic boundary value problems in terms that use constraints from above on the fundamental solution function. The conditions were obtained under which the parabolic boundary value problem exists and cannot exist, respectively. The obtained results will increase the accuracy of the main optimization task of improving the quality of simulated processes.

СМЕШАННАЯ ЗАДАЧА ДЛЯ ВЫРОЖДАЮЩЕГОСЯ МНОГОМЕРНОГО ЭЛЛИПТИЧЕСКОГО УРАВНЕНИЯ

This article shows the unique solvability and obtains an explicit form of the classical solution of the mixed prob-lem in a cylindrical domain for a model degenerate multidimensional elliptic equation. The correctness of boundary value problems in the plane for elliptic equations by the method of the theory of ana-lytic functions of a complex variable has been well studied. The first boundary value problem or the Dirichlet problem for multidimensional elliptic equations with degeneration on the boundary has been sufficiently analyzed. However, as we know, the mixed problem for the indicated equations has been studied very little.

Реалізація математичної моделі в умовах невизначеності і можливих ризиків

The article presents the theoretical and methodological principles for forecasting and mathematical modeling of possible risks in technological and biotechnological systems. The authors investigated in details the possible approach to the calculation of the goal function and its parameters. Considerable attention is paid to substantiating the correctness of boundary value problems and Cauchy problems. In mechanics, engineering, and biology, Cauchy problems and boundary value problems of differential equations are used to model physical processes. It is important that differential equations have a single physically sound solution. The authors of this article investigate the specific features of boundary value problems and Cauchy problems with boundary conditions in a two-point medium, and determine the conditions for the correctness of such problems in the spaces of power growth functions. The theory of pseudo-differential operators in the space of generalized functions was used to prove the correctness of boundary value problems. The application of the obtained results will make it possible to guarantee the correctness of mathematical models built in conditions of uncertainty and possible risks. As an example of a computational mathematical model that describes the state of the studied object of non-standard shape, the authors considered the boundary value problem of the system of differential equations of thermal conductivity for the embryo under the action of a laser beam. For such a boundary value problem, it is impossible to guarantee the existence and uniqueness of the solution of the system of differential equations. To be sure of the existence of a single solution, it is necessary either not to take into account the three-layer structure of the microbiological object, or to determine the conditions for the correctness of the boundary value problem. Applying the results obtained by the authors, the correctness of the boundary value problem of systems of differential equations of thermal conductivity for the embryo is proved taking into account the three-layer structure of the microbiological object. This makes it possible to increase the accuracy and speed of its implementation on the computer. Key words: forecasting, risk, correctness, boundary value problems, conditions of uncertainty

Умови коректності крайових задач

The article deals with the issues of mathematical modeling of technological systems that contain physical fields’ sources. It is believed that in the case of a simple spatial form of the object under study, the boundary value problems will be correct. The interest lies in mathematical models for nonlinear, multilayer objects under the influence of load sources, for which, using the traditional theory of existence and unity, it is impossible to guarantee the correctness of boundary value problems. The author considers boundary value problems for systems of differential and pseudo differential equations in a multilayer medium which describe the state of the studied systems under the action of discrete load sources. The correctness of such problems is proven using the theory of distributions over the space of generalized functions. The object of research is boundary value problems for systems of differential and pseudo differential equations in a multilayer medium. The aim of the research is to build correct boundary value problems, which underlie the calculated mathematical models of the process of action of physical fields on multilayer objects. The necessary and sufficient conditions for the correctness of the parabolic boundary value problem in the space of generalized functions are obtained in the article. It is shown that its solution is infinitely differentiated by a spatial variable. The results of the research can be used to obtain the conditions for the correctness of the boundary value problem for differential equations with variable coefficients. Note that, in some cases, the correctness of the calculated mathematical models determines the correctness of applied optimization mathematical models. The application of the author's research is possible when proving the correctness of boundary value problems for a number of technological processes. The universality of the research allows to widely usage of the results obtained in this work to improve the quality of technological processes.

THE CORRECTNESS OF THE DIRICHLET PROBLEM IN A CYLINDRICAL DOMAIN FOR DEGENERATE MULTIDIMENSIONAL ELLIPTIC-PARABOLIC EQUATIONS

The correctness of boundary value problems on the plane for elliptic equations by the method of the theory of analytic functions of a complex variable has been well studied. When investigating similar questions, when the number of independent variables is greater than two, problems of a fundamental nature arise. A very attractive and convenient method of singular integral equations loses their validity due to the absence of any full theory of multidimensional singular integral equations. Boundary value problems for second-order elliptic equations in domains with edges have been studied in detail. In the authors papers explicit forms of classical solutions of Dirichlet problems in cylindrical domains for multidimensional elliptic equations are found. In this paper we use the method proposed in the authors works, we show the unique solvability and obtain an explicit form of classical solution of the Dirichlet problem in a cylindrical domain for degenerate multidimensional elliptico-parabolic equations.

Boundary value problems for parabolic equations of high order with a changing time direction

We analyze the solvability of boundary value problems for differential equations of high order with a changing direction of time. It is shown that a generalized solution exists in Sobolev spaces. Correctness of boundary value problems with a complete matrix of conjugation conditions is investigated. For such problems, it is shown that the Hölder classes of their solutions essentially depend on both the type of gluing conditions and on the non-integer index Hölder space.

CORRECTNESS OF THE LOCAL BOUNDARY VALUE PROBLEM IN A CYLINDRICAL DOMAIN FOR ONE CLASS OF MULTIDIMENSIONAL ELLIPTIC EQUATIONS

Correctness of boundary value problems in a plane for elliptical equations has been studied properly using the method of the theory of analytic functions. At investigation of analogous problems, when the number of independent variables is more than two, there arise principle difficulties. Quite good and convenient method of singular integral equations has to be abandoned because there is no complete theory of multidimensional singular integral equations. Boundary value problems for second-order elliptical equations in domains with edges have been studied properly earlier. Explicit classical solutions to Dirichlet and Poincare problems in cylindrical domains for one class of multidimensional elliptical equations can be found in the authors works. In this article,the author proved that the local boundary value problem, which is the generalization of Dirichet and Poincare problem, has only solution. Besides, the criterion of uniqueness of regular solution is obtained.

О корректности краевых задач и непрерывной зависимости периодических решений управляемых систем от параметров

О корректности краевых задач и непрерывной зависимости периодических решений управляемых систем от параметров

Two-dimensional steady-state oscillation problems of anisotropic elasticity

The paper deals with the two-dimensional exterior bound- ary value problems of the steady-state oscillation theory for anisotro- pic elastic bodies. By means of the limiting absorption principle the fundamental matrix of the oscillation equations is constructed and the generalized radiation conditions of Sommerfeld-Kupradze type are established. Uniqueness theorems of the basic and mixed type boundary value problems are proved. In the paper we treat the uniqueness theorems of basic and mixed type exterior boundary value problems (BVPs) for equations of two-dimensional steady-state elastic oscillations of anisotropic bodies. In this case questions regarding the correctness of BVPs have not been investigated so far. Here any analogy with the isotropic case is completely violated, since the ge- ometry of the characteristic surface becomes highly complicated and the fundamental matrix cannot be written explicitly in terms of elementary functions. This in turn creates another diculty in obtaining asymptotic estimates. As is well known, even for the metaharmonic equation v(x) + k 2 v(x) = 0; k 2 > 0; x 2R 2 ;

The correctness of boundary value problems in the theory of momentless thin elastic shells of positive curvature and their solution by the mesh method

THREE types of boundary value problems of the theory of momentless thin elastic shells of positive curvature are considered, and for each problem sufficient conditions for its correctness are indicated. Using inequalities proving the stability of the problems, we demonstrate the possibility of solving them by the mesh method. A difference system of equations is obtained by minimization of a quadratic functional (of the least-squares functional type) in a class of functions defined on a net.