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Differential-algebraic boundary-value problems with the variable rank of leading-coefficient matrix

Abstract

Conditions for the solvability of the linear boundary-value problem for systems of differential-algebraic equations with the variable rank of the leading-coefficient matrix and the corresponding solution construction procedure have been found.

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References

  1. 1.

    S. L. Campbell, Singular Systems of Differential Equations, Pitman, San Francisco, 1980.

    MATH  Google Scholar 

  2. 2.

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

  3. 3.

    A. A. Boichuk and L. M. Shegda, “Degenerate Network Boundary-Value Problems,” Neliniyni Kolyvannya, 10(3), 303–312 (2007).

    Google Scholar 

  4. 4.

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

  5. 5.

    S. M. Chuiko. “On a reduction of the order in a differential-algebraic system,” Ukr. Math. Bull., 15(1), 1–17 (2018); transl. in. J. Math. Sci., 235(1), 2–14 (2018).

  6. 6.

    S. M. Chuiko. “A generalized Green operator for a linear Noetherian differential-algebraic boundary value problem,” Siberian Advances in Mathematics, 30, 177–191 (2020).

    Article  Google Scholar 

  7. 7.

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

    Google Scholar 

  8. 8.

    F. R. Gantmakher, Matrix Theory [in Russian]. Nauka, Moscow, 1988.

    Google Scholar 

  9. 9.

    G. M. Fikhtengolts, Course of Differential and Integral Calculus. Vol. 2 [in Russian]. GIFML, Moscow, 1962.

  10. 10.

    A. A. Boichuk. “Boundary-value problems for systems of difference equations,” Ukr. Math. Journal, 49(6), 930–934 (1997).

    MathSciNet  Article  Google Scholar 

  11. 11.

    V. Ya. Gutlyanskii, O. V. Nesmelova, and V. I. Ryazanov. “On semilinear equations in the complex plane,” Dopovidi Natsionalnoi Akademii Nauk Ukrainy, 7, 9–16 (2019).

  12. 12.

    A. Samoilenko, A. Boichuk, and S. Chuiko. “Hybrid difference differential boundary-value problem,” Miskolc Mathematical Notes, 18(2), 1015–1031 (2017).

    MathSciNet  Article  Google Scholar 

  13. 13.

    A. A. Boichuk and L. M. Shehda. “Conditions for bifurcation of solutions of degenerate boundary-value problems,” Nonlinear Oscillations, 12(2), 149–156 (2009).

    MathSciNet  Article  Google Scholar 

  14. 14.

    S. M. Chuiko. “Green’s operator of a generalized matrix linear differential-algebraic boundary value problem,” Siberian Mathematical Journal, 56(4), 752–760 (2015).

    MathSciNet  Article  Google Scholar 

  15. 15.

    S. M. Chuiko. “A generalized matrix differential-algebraic equation,” Ukr. Math. Bull., 12(1), 11–26 (2015); transl. in J. Math. Sci., 210(1), 9–21 (2015).

  16. 16.

    S. Chuiko. “Weakly nonlinear boundary value problem for a matrix differential equation,” Miskolc Mathematical Notes, 17(1), 139–150 (2016).

    MathSciNet  Article  Google Scholar 

  17. 17.

    S. M. Chuiko. “To the issue of a generalization of the matrix differential-algebraic boundary-value problem,” Ukr. Math. Bull., 14(1), 16–32 (2017); transl. in J. Math. Sci., 227(1), 13–25 (2017).

  18. 18.

    G. W. Stewart. “On the continuity of the generalized inverse,” SZAM J.A. & Math., 17, 33–45 (1969).

  19. 19.

    S. L. Campbell. “On continuity of the Moore-Penrose and Drazin generalized inverses,” Linear algebra and its appl., 53–57 (1977).

  20. 20.

    N. I. Akhiezer, Lectures on Approximation Theory [in Russian]. Nauka, Moscow, 1965.

    Google Scholar 

  21. 21.

    S. M. Chuiko. “On approximate solution of boundary value problems by the least square method,” Nonlinear Oscillations, 11(4), 585–604 (2008).

    MathSciNet  Article  Google Scholar 

  22. 22.

    M. A. Perepelitsa and A. A. Pokutnyi. “Investigation of the solvability of weakly nonlinear differentialalgebraic systems,” Vestnik YuUrGU, Seriya “Matematicheskoe Modelirovanie i Programmirovanie”, 6(4), 55-62 (2013).

  23. 23.

    A. Boichuk and S. Chuiko. “Autonomous weakly nonlinear boundary value problems in critical cases,” Differential Equations, 10, 1353–1358 (1992).

    Google Scholar 

  24. 24.

    S. M. Chuiko and O. V. Nesmelova. “Nonlinear boundary-value problems for degenerate differential-algebraic systems,” Ukr. Math. Bull., 17(3), 313–324 (2020); transl. in J. Math. Sci., 252(4), 463–471 (2021).

  25. 25.

    S. M. Chuiko and I. A. Boichuk, “An autonomous Noetherian boundary value problem in the critical case,” Nonlinear Oscillations, 12(3), 405–416 (2009).

    MathSciNet  Article  Google Scholar 

  26. 26.

    V. Ya. Gutlyanskii, O. V. Nesmelova, and V. I. Ryazanov. “The Dirichlet problem for the Poisson type equations in the plane,” Dopovidi Natsionalnoi Akademii Nauk Ukrainy, 5, 10–16 (2020).

  27. 27.

    I. I. Skrypnik, “Removability of isolated singularities for anisotropic elliptic equations with gradient absorption,” Israel J. Math., 215(1), 163–179 (2016).

    MathSciNet  Article  Google Scholar 

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Correspondence to Sergii M. Chuiko.

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Dedicated to V.Ya. Gutlyanskii on the occasion of his 80th birthday

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 18, No. 3, pp. 303–318, July–September, 2021.

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Chuiko, S.M. Differential-algebraic boundary-value problems with the variable rank of leading-coefficient matrix. J Math Sci 259, 10–22 (2021). https://doi.org/10.1007/s10958-021-05597-8

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Keywords

  • Differential-algebraic equations
  • linear boundary-value problems
  • variable rank of leadingcoefficient matrix