Открытый доступ
Доступ предоставлен
Только для подписчиков
Том 223, № 1 (2017)
- Год: 2017
- Статей: 7
- URL: https://ogarev-online.ru/1072-3374/issue/view/14825
Article
Problem for an Inhomogeneous Second-Order Evolutionary Equation with Homogeneous Integral Conditions
Аннотация
We propose a method for the solution of the problem with homogeneous integral conditions for an inhomogeneous evolutionary equation with abstract operator in a Banach space H . The right-hand side of the evolutionary equation that belongs, for the fixed time variable, to a special subspace N ⊆ H can be represented as a Stieltjes integral with respect to a certain measure. The solution of this problem is also represented as a Stieltjes integral with respect to the same measure. We present examples of application of the method to the solution of the problem with integral conditions for the second-order partial differential equation in the time variable and, in general, an infinite-order partial differential equation in the space variable.
1-17
Pseudocompactness, Products, and Topological Brandt λ0 -Extensions of Semitopological Monoids
Аннотация
In the present paper, we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, ω -boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) topological Brandt \( {\lambda}_i^0 \) -extensions of semitopological monoids with zero. In particular, we show that if \( \left\{\left({B}_{\uplambda_i}^0\left({S}_i\right),\kern0.5em {\uptau}_{B\left({S}_i\right)}^0\right):i\in \mathrm{\mathcal{I}}\right\} \) is a family of Hausdorff pseudocompact topological Brandt \( {\uplambda}_i^0 \) -extensions of pseudocompact semitopological monoids with zero such that the Tychonoff product \( \prod \left\{{S}_i:i\in \mathrm{\mathcal{I}}\right\} \) is a pseudocompact space, then the direct product \( \prod \left\{\left({B}_{\uplambda_i}^0\left({S}_i\right),\kern0.5em {\uptau}_{B\left({S}_i\right)}^0\right):i\in \mathrm{\mathcal{I}}\right\} \) endowed with the Tychonoff topology is a Hausdorff pseudocompact semitopological semigroup.
18-38
39-49
Integer Solutions of Matrix Linear Unilateral and Bilateral Equations over Quadratic Rings
Аннотация
For matrix linear equations AX + BY = C and AX + YB = C over quadratic rings \( \mathbb{Z}\left[\sqrt{k}\right] \), we establish necessary and sufficient conditions for the existence of integer solutions, i.e., solutions X and Y over the ring of integers \( \mathbb{Z} \). We also present the criteria of uniqueness of the integer solutions of these equations and the method for their construction.
50-59
Boundary-Value Problem for Parabolic Equations with Impulsive Conditions and Degenerations
Аннотация
By using the maximum principle and a priori estimates, we study the first boundary-value problem for a linear parabolic equation with power singularities in coefficients with respect to the space variables and impulsive conditions with respect to the time variable. In Hölder spaces with power weights, we establish the existence and uniqueness of the solution of the posed problem.
60-71
Free Axisymmetric Vibrations of a Hollow Cylinder of Finite Length Made of a Functionally Graded Material
Аннотация
On the basis of the three-dimensional elasticity theory, we study a problem of free axisymmetric vibrations of inhomogeneous hollow cylinders of finite length made of functionally graded materials under different boundary conditions imposed on their end faces. The elastic properties of the material continuously vary in the radial direction. We propose a numerical-analytic approach for the solution of this problem. The original problem of the elasticity theory containing partial differential equations is reduced to a boundary-value problem for the systems of ordinary differential equations of high order in the radial coordinate by using spline approximations and collocation methods. The obtained one-dimensional problem is solved by using a stable numerical method of discrete orthogonalization together with the method of step-by-step search. We also present the results of determination of the frequencies and modes of vibrations of the cylinder made of a functionally graded material obtained as a composition of stainless steel and nickel for different types of boundary conditions imposed at the end faces and different values of temperature.
72-86
Dynamic Analysis of Composite Rods Under Thermal and Force Loads
Аннотация
We propose a general approach to the solution of direct dynamic problems of the numerical analysis of composite rods subjected to thermal and force loads for a large variety of their materials and loading conditions. We take into account the shear effects and interactions between the elements of structures and supporting media. The proposed inhomogeneous rods are regarded as elements of rod systems and characterized by higher parameters of strength and stiffness and lower costs of their production as compared with homogeneous elements. In the equations of motion and physical relations, we introduce four functional characteristics of stiffness and viscosity and three mass functional characteristics equivalently reflecting the dynamic deformation of composite rods with the help of a one-dimensional model. With the help of trigonometric Fourier series, the dynamic loads, displacements, and temperature are represented as products of functions of the coordinate and time. The solution of the homogeneous problem is obtained via the matrizant of the system of equations of the first order. Partial solutions for loads of various types and generalized temperature loads are obtained on the basis of approximation of given quantities and the required displacements by trigonometric series.
87-102