Hyperbolic systems with multiple characteristic and applications

The article considers a certain class of hyperbolic systems of linear partial differential equations with one spatial variable. As a rule, in the case of systems of partial differential equations, when solving problems, additional conditions are immediately used that ensure the uniqueness of the problem. However, this greatly complicates the construction of the solution in the case of additional conditions of a non-standard form. For a similar situation, in the case of ordinary differential equations, they try to find a general solution, for which you can then try to use the given additional conditions. However, for systems of partial differential equations this approach is difficult, since, as a rule, in this case it is not possible to construct a general solution. For the class of systems of linear inhomogeneous partial differential equations considered in the article, we managed to find an algorithm for constructing a general solution. A distinctive feature of the considered systems of equations is the multiplicity of the corresponding characteristics. As an application of the proposed algorithm, a general solution of the Kolmogorov system of equations for the probabilities of the states of a process that describes the behavior of the popular in applications of a model of a stochastic system of type k-fromn: F with a common distribution of repair time of failed components. The specified system of Kolmogorov equations is a system of differential equations in partial derivatives of the mentioned class. Therefore, for her it is possible to build a common solution.