Skip to main content

A Generalized Green Operator for a Linear Noetherian Differential-Algebraic Boundary Value Problem

Abstract

We find solvability conditions and describe the construction of a generalized Green operator for the Cauchy problem. For a linear Noetherian differential-algebraic boundary value problem, we describe the construction of a generalized Green operator, find existence conditions for equilibrium states, and describe their construction.

This is a preview of subscription content, access via your institution.

REFERENCES

  1. 1

    N. I. Achieser, Lectures on the Theory of Approximation (Nauka, Moscow, 1965) [Theory of Approximation (Dover Publications, Mineola, NY, 2003)].

  2. 2

    N. V. Azbelev, N. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Functional-Differential Equations (Nauka, Moscow, 1991) [Introduction to the Theory of Functional Differential Equations (World Federation Publishers Company, Atlanta, GA, 1995)].

  3. 3

    A. A. Boĭchuk and S. A. Krivosheya, “A critical periodic boundary value problem for a matrix Riccati equation,” Differ. Uravn. 37, 439 (2001) [Differ. Equations 37, 464 (2001)].

    MathSciNet  Article  Google Scholar 

  4. 4

    A. A. Boĭchuk, A. A. Pokutnyĭ, and V. F. Chistyakov, “Application of perturbation theory to the solvability analysis of differential algebraic equations,” Zh. Vychisl. Mat. Mat. Fiz. 53, 958 (2013) [Comput. Math. Math. Phys. 53, 777 (2013)].

    MathSciNet  Article  Google Scholar 

  5. 5

    A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems (De Gruyter, Berlin, 2016).

    Book  Google Scholar 

  6. 6

    O. A. Boĭchuk and L. M. Shehda, “Degenerate Fredholm boundary-value problems,” Neliniĭni Kolyvannya 10, 303 (2007) [Nonlinear Oscil., NY 10, 306 (2007)].

    Article  Google Scholar 

  7. 7

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

    MATH  Google Scholar 

  8. 8

    S. L. Campbell, Singular Systems of Differential Equations (Pitman Advanced Publishing Program, San Francisco–London–Melbourne, 1980).

    MATH  Google Scholar 

  9. 9

    S. L. Campbell, Singular Systems of Differential Equations. II (Pitman Advanced Publishing Program, San Francisco–London–Melbourne, 1982).

    MATH  Google Scholar 

  10. 10

    S. L. Campbell and L. R. Petzold, “Canonical forms and solvable singular systems of differential equations,” SIAM J. Algebraic Discrete Methods 4, 517 (1983).

    MathSciNet  Article  Google Scholar 

  11. 11

    V. F. Chistyakov and A. A. Shcheglova, Selected Chapters of the Theory of Differential Algebraic Equations (Nauka, Novosibirk, 2003) [in Russian].

    MATH  Google Scholar 

  12. 12

    S. M. Chuĭko, “On approximate solution of boundary value problems by the least squares method,” Neliniĭni Kolyvannya 11, 554 (2008) [Nonlinear Oscil., NY 11, 585 (2008)].

    MathSciNet  Article  Google Scholar 

  13. 13

    S. M. Chuĭko, “Linear Noetherian boundary value problem for a linear differential-algebraic system,” Komp. Issled. Model. 5, 769 (2013) [in Russian].

    Google Scholar 

  14. 14

    S. M. Chuĭko, “On the regularization of a linear Fredholm boundary-value problem by a degenerate pulsed action,” Neliniĭni Kolyvannya16, 133 (2013) [J. Math. Sci. 197, 138 (2014)].

    MathSciNet  Article  Google Scholar 

  15. 15

    S. M. Chuĭko, “The Green’s operator of a generalized matrix differential-algebraic boundary value problem,” Sib. Matem. Zh. 56, 942 (2015) [Siberian Math. J. 56, 752 (2015)].

    MathSciNet  Article  Google Scholar 

  16. 16

    S. M. Chuĭko, “On the regularization of a matrix differential-algebraic boundary-value problem,” Ukr. Matem. Visn. 13, 76 (2016) [J. Math. Sci. 220, 591 (2017)].

    MathSciNet  Article  Google Scholar 

  17. 17

    S. M. Chuĭko, “On a reduction of the order in a differential-algebraic system,” Ukr. Matem. Visn. 15, 1 (2018) [J. Math. Sci. 235, 2 (2018)].

    MathSciNet  Article  Google Scholar 

  18. 18

    S. M. Chuĭko, “The method of least squares in the theory of Noetherian differential-algebraic boundary-value problems,” Ukr. Matem. Visn. 15, 475 (2018) [J. Math. Sci. 242, 381 (2019)].

    MathSciNet  Article  Google Scholar 

  19. 19

    S. M. Chuĭko, and D. V. Sysoev, “Periodic matrix boundary-value problems with concentrated delay,” Neliniĭni Kolyvannya 21, 273 (2018) [J. Math. Sci. 243, 326 (2019)].

    MathSciNet  Article  Google Scholar 

  20. 20

    G. V. Demidenko, “Systems of differential equations of higher dimension and delay equations,” Sib. Matem. Zh. 53, 1274 (2012) [Siberian Math. J. 53, 1021 (2012)].

    MathSciNet  Article  Google Scholar 

  21. 21

    E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problems (Springer, Berlin, 2010).

    MATH  Google Scholar 

  22. 22

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

  23. 23

    D. V. Sysoev, “Conditions of the existence of a unique equilibrium position of the Cauchy problem for linear matrix differential-algebraic equations,” Visn. Khar’kov. Univ., Ser. Mat. Prykl. Mat. Mekh. 86, 10 (2017) [in Russian].

    MATH  Google Scholar 

  24. 24

    A. N. Tikhonov and V. Ya. Arsenin, Methods of Solving Ill-Posed Problems (Nauka, Moscow, 1986) [in Russian].

    Google Scholar 

Download references

Funding

The work was supported by the State Foundation for Basic Research (project No. 0118U003390).

Author information

Affiliations

Authors

Corresponding author

Correspondence to S. M. Chuĭko.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chuĭko, S.M. A Generalized Green Operator for a Linear Noetherian Differential-Algebraic Boundary Value Problem. Sib. Adv. Math. 30, 177–191 (2020). https://doi.org/10.3103/S1055134420030037

Download citation

Keywords

  • differential-algebraic boundary value problem
  • Cauchy problem
  • solvability conditions
  • generalized Green operator