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Evaluation of the Index and Singular Points of Linear Differential-Algebraic Equations of Higher Order

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We study linear systems of ordinary differential equations of higher order with an identically singular matrix at the higher derivative of the required vector-valued function in the domain of definition. We give definitions of the index and singular points for these systems, formulate the conditions of solvability, and deduce a formula for the general solution. The algorithms for finding the index and singular points are presented.

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References

  1. E. I. Ushakov, Static Stability of Electric Systems [in Russian], Nauka, Novosibirsk (1988).

    Google Scholar 

  2. J. Vlach and K. Singhal, Computer Methods for Circuit Analysis and Design, Van Nostrand Reinhold Company, New York (1983).

    Google Scholar 

  3. Yu. E. Boyarintsev, Regular and Singular Systems of Ordinary Linear Differential Equations [in Russian], Nauka, Novosibirsk (1980).

    MATH  Google Scholar 

  4. K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Society for Industrial and Applied Mathematics, Philadelphia (1996).

    MATH  Google Scholar 

  5. V. F. Chistyakov, Algebraic-Differential Operators with Finite-Dimensional Kernel [in Russian], Nauka, Novosibirsk (1996).

    Google Scholar 

  6. L. A. Vlasenko, Evolutionary Models with Implicit and Degenerate Differential Equations [in Russian], Sistemnye Tekhnologii, Dnepropetrovsk (2006).

    Google Scholar 

  7. Yu. E. Boyarintsev and V. F. Chistyakov, Algebraic-Differential Systems. Methods of Solution and Investigation [in Russian], Nauka, Novosibirsk (1998).

    MATH  Google Scholar 

  8. Yu. E. Boyarintsev, Methods of Solution of Continuous and Discrete Problems for Singular Systems of Equations [in Russian], Nauka, Novosibirsk (1996).

    MATH  Google Scholar 

  9. A. M. Samoilenko, M. I. Shkil’, and V. P. Yakovets’, Linear Systems of Differential Equations with Degenerations [in Ukrainian], Vyshcha Shkola, Kyiv (2000).

    Google Scholar 

  10. P. Kunkel and V. Mehrmann, Differential-Algebraic Equations. Analysis and Numerical Solution, European Mathematical Society Publ. House, Zúrich (2006).

    Book  MATH  Google Scholar 

  11. R. Lamour, R. Marz, and C. Tischendorf, Differential-Algebraic Equations: a Projector Based Analysis, Springer, Berlin (2013).

    Book  MATH  Google Scholar 

  12. N. N. Luzin, “On the investigation of a matrix system in the theory of differential equations,” Avtomat. Telemekh., No. 5, 4–66 (1940).

  13. V. F. Chistyakov, “Algebraic-differential operators with finite-dimensional kernels,” in: Algebraic-Differential Systems and Methods for Their Solution. Collection of Works [in Russian], Nauka, Novosibirsk (1993).

  14. M. V. Bulatov and M.-G. Lee, “Application of matrix polynomials to the analysis of linear differential-algebraic equations,” Differents. Equat., 44, No. 10, 1353–1360 (2008).

    Article  MATH  Google Scholar 

  15. S. P. Pafyk and V. P. Yakovets’, “On the structure of the general solution and conditions of solvability of the Cauchy problem for degenerate linear systems of higher-order differential equations,” Ukr. Mat. Zh., 65, No. 2, 296–306 (2013); English translation: Ukr. Math. J., 65, No. 2, 328–340 (2013).

  16. V. Mehrmann and Ch. Shi, “Transformation of high-order linear differential-algebraic systems to first order,” Numer. Algorithms, 42, 281–307 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  17. V. F. Chistyakov, “One theorem on the existence of solutions of a singular linear system of ordinary differential equations,” Chisl. Met. Mekh. Splosh. Sredy, 12, No. 6, 135–149 (1981).

    Google Scholar 

  18. M. V. Bulatov and V. F. Chistyakov, “A method for the numerical solution of linear singular systems of ordinary differential equations of index higher than one,” in: Numerical Methods of Analysis and Their Applications [in Russian], SÉI SD, Academy of Sciences of the USSR, Irkutsk (1987), pp. 100–105.

  19. V. F. Chistyakov, “On singular systems of ordinary differential equations and their integral analogs,” in: Lyapunov Functions and Their Applications [in Russian], Nauka, Novosibirsk (1987), pp. 231–239.

  20. V. F. Chistyakov, On the Relationship between the Properties of Degenerate Systems and the Problems of Variational Calculus [in Russian], Preprint No. 5, Irkutsk Computing Center, Siberian Division of the Academy of Sciences of the USSR, Irkutsk (1989).

  21. L. M. Silverman and R. S. Bucy, “Generalizations of theorem of Dolezal,” Math. System Theory, 4, 334–339 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  22. A. F. Sidorov, V. P. Shapeev, and N. N. Yanenko, Method of Differential Relations and Its Applications to the Gas Dynamics [in Russian], Nauka, Novosibirsk (1984).

    MATH  Google Scholar 

  23. V. I. Arnold, Additional Chapters of the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  24. V. F. Chistyakov, “On the relationship between the structure of a pencil of matrices and the existence of solutions of an implicit system of ordinary differential equations,” in: Methods of Optimization and Investigation of Operations [in Russian], Institute of Systems of Energetics, Siberian Division of the Academy of Sciences of USSR, Irkutsk (1984), pp. 194–202.

  25. S. L. Campbell, “Non-BDF methods for the solution of linear time varying implicit differential equations,” in: Proc. of the American Control Conf. (San Diego, California, June 5–6, 1984), 3, pp. 1315–1318.

  26. M. V. Bulatov and V. F. Chistyakov, The Properties of Differential-Algebraic Systems and Their Integral Analogs, Preprint, Memorial University of Newfoundland, September (1997).

  27. V. F. Chistyakov, “On the solvability of systems of integral Volterra equations of the fourth kind. I,” Differents. Uravn., 38, No. 5, 698–707 (2002).

    Google Scholar 

  28. S. K. Godunov, A. G. Antonov, O. P. Kirilyuk, and V. I. Kostin, Guaranteed Accuracy of the Solutions of Systems of Linear Equations in Euclidean Spaces [in Russian], Nauka, Novosibirsk (1988).

    MATH  Google Scholar 

  29. O. V. Bormotova and V. F. Chistyakov, “On the methods of numerical solution and investigation of systems not of the Cauchy–Kovalevskaya type,” Zh. Vychisl. Mat. Mat. Fiz., 44, No. 8, 1380–1387 (2004).

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of the Dynamics of Systems and Control Theory, Siberian Division of the Russian Academy of Sciences, Irkutsk, Russia

    V. F. Chistyakov & E. V. Chistyakova

Authors
  1. V. F. Chistyakov
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  2. E. V. Chistyakova
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Correspondence to V. F. Chistyakov.

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Translated from Neliniini Kolyvannya, Vol. 20, No. 2, pp. 274–288, April–June, 2017.

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Chistyakov, V.F., Chistyakova, E.V. Evaluation of the Index and Singular Points of Linear Differential-Algebraic Equations of Higher Order. J Math Sci 231, 827–845 (2018). https://doi.org/10.1007/s10958-018-3852-7

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  • DOI: https://doi.org/10.1007/s10958-018-3852-7

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