Asymptotic equivalence of differential equations and asymptotically almost periodic solutions

In this paper we use Rab's lemma [M. Rab, Uber lineare perturbationen eines systems von linearen differentialgleichungen, Czechoslovak Math. J. 83 (1958) 222-229; M. Rab, Note sur les formules asymptotiques pour les solutions d'un systeme d'equations differentielles lineaires, Czechoslovak Math. J. 91 (1966) 127-129] to obtain new sufficient conditions for the asymptotic equivalence of linear and quasilinear systems of ordinary differential equations. Yakubovich's result [V.V. Nemytskii, VX Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1966; V.A. Yakubovich, On the asymptotic behavior of systems of differential equations, Mat. Sb. 28 (1951) 217-240] on the asymptotic equivalence of a linear and a quasilinear system is developed. On the basis of the equivalence, the existence of asymptotically almost periodic solutions of the systems is investigated. The definitions of biasymptotic equivalence for the equations and biasymptotically almost periodic solutions are introduced. Theorems on the sufficient conditions for the systems to be biasymptotically equivalent and for the existence of biasymptotically almost periodic solutions are obtained. Appropriate examples are constructed

APPLICATION OF THE EQUIVALENCE THEOREM FOR SOLVING OPERATOR EQUATIONS AND FINDING THE MINIMUM OF A QUADRATIC FUNCTIONAL ELEMENT

Law of transfer physical substances described using differential equations, where the unknown functions are subject to the given boundary conditions. These conditions depend on the position in which the object under study. Such equations can be regarded as the operator equations acting in specific functional areas. Information obtained from this formulation can be used to build an efficient numerical methods for solving problems with great practical importance. In this paper the relationship between the operator equations are considered in a Banach space with a quadratic functional and its application for a modification postreniyavariational method. Considered in the work of the physical problem: transverse bending beams with constant stiffness, which lies on an elastic foundation. A mathematical model of this process is described by a differential equation of the fourth order with the given boundary conditions, regarded as as an operator equation with a positive operator on the left side. A solution of this equation is studied by using a special form of a quadratic functional. It is proved that the problem of finding solutions of this equation with the given boundary conditions, equivalent to the problem of finding the minimum of the functional. Further, the variational method is based on minimization of a quadratic functional, numerical calculation which gives solutions of the original physical problem. The practical significance of the work is that the numerical calculation of finding the minimum of the functional is easily accomplished rapidly convergent numerical algorithm based minimization of a quadratic functiona

Poincaré chaos for a hyperbolic quasilinear system

The existence of unpredictable motions in systems of quasilinear differential equations with hyperbolic linear part is rigorously proved. We make use of the topology of uniform convergence on compact sets and the contraction mapping principle to prove the existence of unpredictable motions. Appropriate examples with simulations that support the theoretical results are provided

Asymptotic behavior of linear impulsive integro-differential equations

Asymptotic equilibria of linear integro-differential equations and asymptotic relations between solutions of linear homogeneous impulsive differential equations and those of linear integro-differential equations are established. A new Gronwall-Bellman type lemma for integro-differential inequalities is proved. An example is given to demonstrate the validity of one of the results. © 2008 Elsevier Ltd. All rights reserved

Control and optimal response problems for quasilinear impulsive integrodifferential equations

In various real-world applications, there is a necessity given to steer processes in time. More and more it becomes acknowledged in science and engineering, that these processes exhibit discontinuities

Control and optimal response problems for quasilinear impulsive integrodifferential

equation

Исследование аэродинамических параметров парусной лопасти

This article studies the aerodynamic characteristics of a triangular sail blade of various parameters. For this purpose, we made a triangular sail blade with a dynamically changing surface shape. The airflow velocity varied from 3 to 12 m/s. The dependences of the aerodynamic forces of the sail blade on the flow velocity were investigated at various angles of the apex of the triangular blade. The experiments were carried out at different vertices of the angles: 00; 300; 600; 900. As a result of the experiment, it was revealed that at the vertex angle γ = 900, the triangular sail blade has optimal aerodynamic parameters. The dependences of the aerodynamic coefficients on the dimensionless angle of attack are obtained. It is found that the optimal number of triangular blades for a wind power plant with sailing blades is 6. It is established that at the angle of attack α = 00, the maximum value of the middle section of the wind wheel to the streamlined airflow will introduce a decrease in the value of the drag coefficient with an increase in attack α. The analysis of the experiment results on the change in α from the speed of the airflow of the sail blade is carried out. When the blade position changes, drag changes relatively to the airflow. The wind wheel will change its position relative to the stream with an increase in the attack angle. With an angular position change, the area of the middle section of the wind wheel begins to decrease relative to the incoming flow. With a decrease in the middle section of the wind wheel, the drag force decreases, and the drag coefficient decreases accordingly. Thus, the total result of pressure changes on the leeward and windward surfaces of the sail can be represented as one resultant aerodynamic force directed at an angle to the line perpendicular to the wind direction