Том 40, № 7 (2019)
- Жылы: 2019
- Мақалалар: 16
- URL: https://journals.rcsi.science/1995-0802/issue/view/12741
Article
Exact Solutions of Three Nonclassical Equations, and Their Construction with Maple System
Аннотация
Since the second half of the twentieth century, wide studies of Sobolev-type equations are undertaken. These equations contain items that are derivatives with respect to time of the second order derivatives of the unknown function with respect to space variables. They can describe nonstationary processes in semiconductors, in plasm, phenomenons in hydrodynamics and other ones. It is important to notice that wide studies of qualitative properties of solutions of Sobolev-type equations exist. Namely, results about existense and uniqueness of solutions, their asymptotics and blow-up are known. But there are few results about exact solutions of Sobolev-type equations. There are books and papers about exact solutions of partial equations, but they are devoted mainly to classical equations, where the first or second order derivative with respect to time or the derivative with respect to time of the first order derivative of the unknown function with respect to the space variable is equal to a stationary expression. Therefore it is interesting to study exact solutions of Sobolev-type equations.
In the present paper, three nonclassical nonlinear partial equations are studied. Some results about existense, uniqueness and blow-up of their solutions are already known. Here, we construct some classes of their exact solutions with the help of Maple System. We use the method of travelling waves, the method of short-cut decompositions and construction of solutions of some special forms. Also, we discuss the way of realizing of these investigations with the help of Maple system of computer mathematics.
Information Model of LibMeta Digital Library
Аннотация
When developing a digital library, heavy emphasis is laid onto the library information model. The content of digital libraries can be featured in different formats and presented in different ways. The library, as defined by the LibMeta system, is a storage of structured and diverse data with the possibility of integrating it with other data sources and specifying its content by analysing the subject area. The semantic content ontology of the library provides basis for further content formalization. The article introduces the basic concepts needed to describe the problem of data integration, as well as concepts required for the definition of an unspecified thesaurus. The ontology’s structure enables the possibility to determine the semantic library within any subject area.
Mathematical Physics Branches: Identifying Mixed Type Equations
Аннотация
The article focuses on the problem of defining paradigmatic relations between definitions of certain fields in mathematical physics. The ultimate goal is to outline the hierarchical relations between the terms that can be used when searching on the mathematical resources along with additional classification parameters set in secondary documents. A thesaurus entry is selected as an information model. The thesaurus was formed by analyzing the original works of classics of mathematical analysis and differential calculus, and a representative list of articles was organized for that purpose. Following the example of thesaurus on the ‘problem of mixed type equations’ domain, a way of employing formulas in a mathematical article search is proposed. The paper covers a work script of a user, who is familiar with the subject domain and deals with papers done with the help of TeX-notation. A natural document indexing mechanism is set by key words in cited secondary documents. Such an approach helps to specify the search query with mathematical notation regardless the source language. The semantic links effect is based on usage of terms from the mathematical subject domain thesaurus stored with formulas that serve as a background for a mathematical search. It results in lower level of information noise and reduced search time.
Optimization of Thermal Protection Panels Subjected to Intense Heating and Mechanical Loading
Аннотация
In this work we solve a higly-nonlinear structural optimization problem for the sandwich panel with external thermal protection layer that can be used in the spacecraft systems. Objective function of the problem is the mass per unit area of the panel. Constraints are formulated based on the simplified analytical solutions of structural mechanics and heat transfer problems, which are suitable for the preliminary design considerations. The set of design variables includes the geometric parameters of the panel and additional microstructural parameter—porosity of the heat protection material. Direct random search and simulated annealing method are applied to solve considered problem. Change of limit states and optimal configurations of the panel are studied for different levels of the mechanical loading.
On Variation Models of the Irreversible Processes in Mechanics of Solids and Generalized Hydrodynamics
Аннотация
It is proposed a procedure for constructing a non-integrable variational form, allowing to extend the procedure of formal construction of variational models of mechanics of deformable media to irreversible deformation processes. We introduce the non-integrable variational forms that determine possible dissipation channels depending on the list of generalized variables for a particular model of media. The problems of filtering of Biot models are considered and it is proved that non-integrable variational forms allow us to construct variational hydrodynamic models of Darcy, Navier-Stokes and Navier-Stokes-Darcy. It is shown that the equations of the Navier—Stokes—Darcy model contain the Helmholtz operator, which will allow to take into account the scale effects associated with the boundary layers.
The Stability Problem for a Dynamic System with the Assimilation of Observational Data
Аннотация
The problem of stability of a dynamic system defined by a system of differential equations to the perturbation of the initial data and with the data assimilation is considered. The assimilation of observational data is realized by the previously published author’s method. The stability condition for this data assimilation method is formulated in the classical sense of Lyapunov, and the solution of the system is corrected using observational data for a given time interval. Necessary and sufficient conditions are proposed under which this system is stable as a function of the observed values. A possible numerical experiment to test and to apply this theory is discussed.
Attractors for Model of Viscoelastic Media with Memory Motion in Non-Autonomous Case
Аннотация
We study attractors existence of weak solutions to viscoelastic media with memory motion model in non-autonomous case. The theory of trajectory attractors for non-invariant trajectory spaces is applied and the existence of uniform trajectory attractor and uniform global attractor for this system is proved. The proof of existence theorems is based on the approximation-topological method.
Application of Hypergeometric Functions of Two Variables in Wireless Communication Theory
Аннотация
Hypergeometric functions of several variables are widely used in applied problems, in particular, in the mathematical theory of communication in the problems of calculating the error probabilities of receiving signal constructions. Error probabilities can often be expressed through various types of special functions. One of the most important functions of this kind is the Marcum function and its generalization ℋ-function.
On the Second Initial-boundary Value Problem for Parabolic Equation, Containing Mixed Derivatives
Аннотация
In this work new analytical solution of the second initial-boundary value problem of the theory of thermal conductivity in a general anisotropic strip has been found considering that heat transfer characteristics are not scalar values but second symmetric tensors. The theorems about the existence and uniqueness of found solution have been proved.
About one Approach to a Solution of Linear Differential Equations with Variable Coefficients
Аннотация
Using the uniform approach, linear differential equations of elliptic, hyperbolic and parabolic types with variable coefficients depending on coordinates and time are considered. It is shown that the solution of the initial equation can be expressed by means of an integral formula through the solution of the accompanying equation of the same type, but with constant coefficients. It is considered that the solution of the accompanying equation is known. From the integral formula, assuming smoothness of the accompanying solution, an equivalent representation of the solution of the initial equation is obtained in the form of series in all possible derivatives of the solution of the accompanying equation. For coefficients at derivatives a system of recurrent equations is obtained, which can be solved analytically at some cases.
Spectral Solution of a Boundary Value Problem for Equation of Mixed type
Аннотация
In this paper we apply spectral method to the Gellerstedt problem for Lavrent’ev-Bitsadze equation in a half-strip. We consider Frankl matching condition at the type change line. Using the Darboux solution formulae for mixed type equationinthe hyperbolic subdomainwe reduce problem to an auxiliary problem for the Laplace operator.
On the Radial Multipliers Method in the Gradient Elastic Fracture Mechanics
Аннотация
A non-singular solution of the gradient elastic fracture mechanics for the cracks of Modes I and II is constructed in this paper. Previously, a similar problem was solved only for cracks of Mode III. A generalized theory of elasticity is used, in which the governing equation in displacements is represented as a product of the Lamé operator and the Helmholtz operator, and the classical boundary value problem for the total stress is completely distinguished in the problems of crack mechanics. As a result, the determination of local stress fields reduces to solving the Helmholtz equations with the known right-hand side of the equations. The Papkovich-Neuber representation in a complex form is used to construct a solution of the mechanics of cracks. We used also the method of radial multipliers, which allows us to construct fundamental solutions of the Helmholtz equations corresponding to analytical functions with a fractional exponent and, as a result, to find solutions that compensate for the singularities.
On the Problem of Eigenvalues of Material Tensor Objects and Wave Velocities
Аннотация
We formulate the eigenvalue problem for a tensor-block matrix of any order and of any even rank. It is well known that the eigenvalues of the tensor and the tensor-block matrix are invariant quantities. Therefore, in this work, our purpose is to find the expression for the wave velocities of some media through the eigenvalues of the material objects. In particular, the classical and micropolar materials with the different anisotropy symbols are considered, and the expressions for the wave velocities through the eigenvalues of the material objects are given.
Approximate Analytical Solution for a Unilateral Contact Problem with Heavy Elastica
Аннотация
The non-linear bending of slender heavy beam resting on rigid plane and subjected to the end point force is studied. Solution of the problem is derived by using series expansion technique. Approximate closed-form analytical solution for the length of the separated segment of elastica is found. It is shown that obtained simple solution can be used as the approximate lower bound for the evaluation of the force that should be applied to the beam’s end to provide the prescribed length of separation, while the known classical solution of linear theory can be used as the upper bound.
Systems of Functions Consistent with Inhomogeneities of Elliptic and Spheroidal Shapes in Problems of Continuum Mechanics
Аннотация
Structure of the fundamental solutions of the Laplace equation in the elliptic and spheroidal coordinate systems is investigated. It was shown that among the solutions separated in these coordinate systems there are such solutions that have the form of harmonic polynomials in the original Cartesian coordinate system. These polynomials constitute basis for constructing with the help of radial multipliers the fundamental systems of functions that are consistent with the inhomogeneities of the elliptic and ellipsoidal shapes. These functions are used in problems of mechanics of inhomogeneous media for accurate modeling of physical processes near such inclusions. On the basis of these functions algorithms are constructed for defining effective thermophysical characteristics of inhomogeneous media with inclusions of elliptic and spheroidal shapes.