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.
Similar content being viewed by others
References
S. L. Campbell, Singular Systems of Differential Equations, Pitman, San Francisco, 1980.
Yu. E. Boyarintsev and V. F. Chistyakov, Algebraic-Differential Systems. Methods of Solution and Studies [in Russian]. Nauka, Novosibirsk, 1998.
A. A. Boichuk and L. M. Shegda, “Degenerate Network Boundary-Value Problems,” Neliniyni Kolyvannya, 10(3), 303–312 (2007).
A. M. Samoilenko, M. I. Shkil’, and V. P. Yakovets’, Linear Systems of Differential Equations with Degeneration [in Ukrainian]. Vyshcha Shkola, Kyiv, 2000.
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).
S. M. Chuiko. “A generalized Green operator for a linear Noetherian differential-algebraic boundary value problem,” Siberian Advances in Mathematics, 30, 177–191 (2020).
V. F. Chistyakov, Algebraic-Differential Operators with Finite-Dimensional Kernel [in Russian]. Nauka, Novosibirsk, 1996.
F. R. Gantmakher, Matrix Theory [in Russian]. Nauka, Moscow, 1988.
G. M. Fikhtengolts, Course of Differential and Integral Calculus. Vol. 2 [in Russian]. GIFML, Moscow, 1962.
A. A. Boichuk. “Boundary-value problems for systems of difference equations,” Ukr. Math. Journal, 49(6), 930–934 (1997).
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).
A. Samoilenko, A. Boichuk, and S. Chuiko. “Hybrid difference differential boundary-value problem,” Miskolc Mathematical Notes, 18(2), 1015–1031 (2017).
A. A. Boichuk and L. M. Shehda. “Conditions for bifurcation of solutions of degenerate boundary-value problems,” Nonlinear Oscillations, 12(2), 149–156 (2009).
S. M. Chuiko. “Green’s operator of a generalized matrix linear differential-algebraic boundary value problem,” Siberian Mathematical Journal, 56(4), 752–760 (2015).
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).
S. Chuiko. “Weakly nonlinear boundary value problem for a matrix differential equation,” Miskolc Mathematical Notes, 17(1), 139–150 (2016).
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).
G. W. Stewart. “On the continuity of the generalized inverse,” SZAM J.A. & Math., 17, 33–45 (1969).
S. L. Campbell. “On continuity of the Moore-Penrose and Drazin generalized inverses,” Linear algebra and its appl., 53–57 (1977).
N. I. Akhiezer, Lectures on Approximation Theory [in Russian]. Nauka, Moscow, 1965.
S. M. Chuiko. “On approximate solution of boundary value problems by the least square method,” Nonlinear Oscillations, 11(4), 585–604 (2008).
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).
A. Boichuk and S. Chuiko. “Autonomous weakly nonlinear boundary value problems in critical cases,” Differential Equations, 10, 1353–1358 (1992).
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).
S. M. Chuiko and I. A. Boichuk, “An autonomous Noetherian boundary value problem in the critical case,” Nonlinear Oscillations, 12(3), 405–416 (2009).
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).
I. I. Skrypnik, “Removability of isolated singularities for anisotropic elliptic equations with gradient absorption,” Israel J. Math., 215(1), 163–179 (2016).
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-021-05597-8