Mixed initial-boundary value problem for telegraph equation in domain with variable borders is considered. On one part of domain’s border are the boundary conditions of the first type, on other part of the boundary are set boundary conditions of the second type. Besides, the sizes of area are variable. The solution of such problem demands development of special methods. With the help of consecutive application of procedure of construction waves reflected from borders of domain, it is possible to obtain the solution of this problem in quadratures. In addition, for construction of the waves reflected from mobile border, it is necessary to apply the procedure specially developed for these purposes.

Initial boundary value problems are considered for rods that change length over time. These problems are transformed to problems about rods with mobile ends. A special method which allows one to obtain exact solutions of such problems is developed. This method is a generalization of a method of reflections for rods and strings of constant length. As an application of this class of problems, the hoisting of ropes in mining lifts, including their problems, are considered. Exact expressions for displacements in rods of variable length are obtained. The same results can be applied to cross oscillations of strings of variable length.

We consider the mathematical model of interaction of a vibrating surface with the load placed on it. With the purpose of accounting for influence on behavior of not only system interaction of a blade with a load but also internal interaction of particles of a material, the load is submitted as a finite number of strips with zero thickness. The carrying blade is represented as a vibrating membrane. It is supposed that the weight of the material is comparable to or considerably surpasses the weight of the blade. Therefore, the model takes into account the inertia of the material. In the model with joint movement of the blade and the load, the separation opportunity of the load from the blade is provided. Therefore, there is a phase of separate movement of the blade and the load, with their subsequent connection accompanied with impact. The process of system movement is represented as alternating sequences of joint and separate movements of the load and the blade. The modeling of the process of the interaction of the load and the blade is represented as an initial-boundary value problem. The method of solution is developed and the exact solution of the set problem is obtained in a class of generalized functions.

The problem about construction of the external electromagnetic field created by MWNT antennas is considered. For the description of such field Lorentz -Maxwell average microscopic equations are used. Each element of the multilayered antenna is described with the help of the generalized functions concentrated on smooth surfaces. Such approach allows to present behavior of an electromagnetic field by means of one differential equation which factors are the generalized functions. The special method is developed, allowing obtaining the general solution of the differential equations of the given class. In an obvious kind the exact solution representing external electromagnetic field of MWNT antennas is obtained.

No abstract