Hyperbolic Systems with Multiple Characteristics and Some Applications

Abstract: We consider a class of hyperbolic systems of linear inhomogeneous partial differentialequations with one spatial variable. As a rule, in the case of systems of partial differentialequations, when solving particular problems, additional conditions are immediately used to ensurethe uniqueness of the solution. However, this greatly complicates the construction of a solution inthe case of additional conditions of nonstandard form. For a similar situation in the case ofordinary differential equations, one tries to find a general solution, for which one can then try touse the given additional conditions. However, for systems of partial differential equations, thisapproach is difficult, since, as a rule, in this case it is not possible to construct the generalsolution. For the class of systems of linear inhomogeneous partial differential equations consideredin the present paper, it was possible to find an algorithm for constructing a general solution. Adistinctive feature of the considered systems of equations is the multiplicity of the correspondingcharacteristics. As an application of the proposed algorithm, a general solution of the Kolmogorovsystem of equations for the probabilities of the states of a process is obtained, which describes thebehavior of a popular model of a stochastic system of the type k-out-of- n: F with a general distribution of the repair time forfailing components. The indicated system of Kolmogorov equations is a system of partialdifferential equations of the mentioned class. Therefore, it is possible to construct a generalsolution for this system. © 2021, Pleiades Publishing, Ltd.