Vol 23, No 3 (2017)

In this paper, we study an inverse problem for hyperbolic equation. This problem arises when we consider vibration of a thin bar if one endpoint is fixed but behavior of the other is unknown and is the subject to find. Overdetermination is given in the form of integral with respect to spacial variable. The problem is reduced to the second kind Volterra integral equation. Special case is considered.

In this paper we consider an initial-boundary problem with nonlocal boundary condition for one-dimensional hyperbolic equation. Nonlocal condition is dynamic so as represents a relation between values of derivatives with respect of spacial variables of a required solution, first-order derivatives with respect to time variable and an integral of a required solution of spacial variable. We prove the existence and uniqueness of a generalized solution, which belongs to the Sobolev space. To prove uniquely solvability of the problem techniques developed specifically for research nonlocal problems are used. The application of these methods allowed us to obtain a priori estimates, through which the uniqueness of the solution is proved. The proof is based on the a priori estimates obtained in this paper and Galyorkin’s procedure.

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 thit work, we derive the general multi-dimensional dynamic equations of the systems under study.

Analytical investigating of the patterns of relaxation (volume-charge) polarization in dielectric materials class hydrogen bonded crystals (HBC) in the wide range of temperature (1–1500 K) and polarizing field strengths (100 kV/m–100 MV/m) in alternating field at frequencies of about 1 kHz–10 MHz is made. The generalized nonlinear by the polarizing field the semi-classical kinetic equation of proton relaxation, having (in this model) sense the protons current continuity equation solving by method of successive approximation by decomposition in infinite power series in comparison parameter is built. It is established that in the range of low fields (100–1000 kV/m) and high temperatures (100–250 K) the generalized kinetic equation is converted to the linearized Fokker-Planck equation and at low (70–100 K) and sufficiently high (250–450 K) temperatures are showed the nonlinear polarization effects caused respectively by proton tunneling and volume charge relaxation. With ultra - low (1–10 K) and ultra-high (500–1500 K) temperatures in the range of high fields (10 MV/m–100 MV/m) the contribution of such effects to the polarization is amplified. The influence of the non-linearities to relaxation times for microscopic acts of transitions protons through the potential barrier is studied.