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Hyperbolic Systems with Multiple Characteristics and Some Applications

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Abstract

We consider a class of hyperbolic systems of linear inhomogeneous partial differential equations with one spatial variable. As a rule, in the case of systems of partial differential equations, when solving particular problems, additional conditions are immediately used to ensure the uniqueness of the solution. However, this greatly complicates the construction of a solution in the case of additional conditions of nonstandard form. For a similar situation in the case of ordinary differential equations, one tries to find a general solution, for which one 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 the general solution. For the class of systems of linear inhomogeneous partial differential equations considered in the present paper, it was possible 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 is obtained, which describes the behavior of a popular model of a stochastic system of the type \(k \)-out-of-\( n: F \) with a general distribution of the repair time for failing components. The indicated system of Kolmogorov equations is a system of partial differential equations of the mentioned class. Therefore, it is possible to construct a general solution for this system.

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Funding

This work was financially supported by the Russian Foundation for Basic Research, project no. 20-01-00575A.

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Authors and Affiliations

  1. Gubkin University, Moscow, 119991, Russia

    V. V. Rykov

  2. RUDN University, Moscow, 117198, Russia

    V. V. Rykov

  3. Russian University of Transport (MIIT), Moscow, Russia

    A. M. Filimonov

Authors
  1. V. V. Rykov
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  2. A. M. Filimonov
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Correspondence to V. V. Rykov or A. M. Filimonov.

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Translated by V. Potapchouck

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Rykov, V.V., Filimonov, A.M. Hyperbolic Systems with Multiple Characteristics and Some Applications. Autom Remote Control 82, 1262–1270 (2021). https://doi.org/10.1134/S0005117921070092

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  • DOI: https://doi.org/10.1134/S0005117921070092

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