Vol 23, No 4 (2017)

In this paper, we consider an initial-boundary problem with dynamical nonlocal boundary condition for a pseudohyperbolic fourth-order equation in a rectangular. Dynamical nonlocal boundary condition represents a relation between values of a required solution, its derivatives with respect of spacial variables, second-order derivatives with respect of time-variables and an integral term. This problem may be used as a mathematical model of longitudinal vibration in a thick short bar and illustrates a nonlocal approach to such processes. The main result lies in justification of solvability of this problem. Existence and uniqueness of a generalized solution are proved. The proof is based on the a priori estimates obtained in this paper, Galerkin’s procedure and the properties of the Sobolev spaces.

A frame of a finite-dimensional Euclidean space composed of vectors and their sums is considered. The operator proof of frame properties of the constructed system is presented, the eigenvalues for the matrix of the frame operator are found, which are also frame boundaries. The property of alternative completeness of the constructed system is proved. This property is the cause of interest in the constructed frame, since in real Euclidean space it is equivalent to injectivity of the measurement operator, which maps the signal vector into a sequence of measurement modules. The investigated frame underlies the fast algorithm of signal reconstruction, proposed by M. Shapiro and stated in [1]. An operator that translates the constructed frame into the Parseval-Steklov frame closest to it, is found.

In the proposed cycle of work, we study the equations of the motion of dynamically symmetric fixed n-dimensional rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of the motion of a free n-dimensional rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint. In this work, we study the case of independence of force fields on the tensor of angular velocity.