Vol 25, No 4 (2019)

In mathematical modeling of electromagnetic fields in space, the nature of electromagnetic process is determined by the properties of the medium. If the medium is non-conducting, we obtain degenerate three-dimensional hyperbolic equations. If the medium has a high conductivity, then we come to degenerate
three-dimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the medium’s conductivity changes) is reduced to degenerate three-dimensional hyperbolic-parabolic equations. The mixed problem for multidimensional hyperbolic equations is well studied and has been previously considered in the works of various authors. In the articles of Professor S.A. Aldashev, the unique solvability of the mixed problem for degenerate multidimensional hyperbolic equations is proved. It is known that mixed problems for multidimensional hyperbolic-parabolic equations have not been studied much. The paper finds a new class of degenerate three-dimensional hyperbolic-parabolic equations for which the mixed problem has a unique solution and gives an explicit representation of its classical solution.

In this paper we consider a nonlocal problem with integral conditions of the II kind for one-dimensional hyperbolic equation. Nonlocal conditions of the second kind differ in type of non-integral terms, that may contain traces of required solution and traces of derivatives. This difference turns out to be significant for choosing a method for investigating the solvability of the problem. In this work we consider the case when
nonintegral terms are traces of required solution on boundary of the domain. To investigate the solvability of the problem we use method of reduction to the boundary problem for loaded equation. This method allowed us to define a generalized solution, to obtaim apriori estimates and to prove existence of unique generalized solution of the given problem.

Proposed work is the third in the cycle, therefore, we explain such notions as checking sphere, checking ellipsoid and checking tubes. The checking problem is stated and the algorithms for solving it are formulated. The criterion for a malfunction in a controlled system whose motion is described by ordinary differential equations is taken to be the attainment of a checking surface by the checking vector. We first propose the methods for solving the checking problems in which the checking surfaces are chosen in the form of a checking sphere, checking ellipsoid or checking tube. Then we consider the general techniques for constructing the checking surface by using the statistical testing method. We also give the extended statement of the checking problem. And we also prepare the material for the consideration of the problem of diagnostics.