Boundary-Value Problems for Systems of Linear Difference-Algebraic Equations

Чуйко, С. М.; Chuiko, E. V.; Kalinichenko, Ya. V.

doi:10.1007/s10958-021-05307-4

Cited by 4 publications

(6 citation statements)

References 5 publications

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“…The posed problem ( 1) is a generalization of the problem solved by Boichuk in [9] and also an analog of the Noetherian boundary-value problems for systems of ordinary differential equations in the case of parametric resonance [7,10,11]. Under the condition [12]…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…Hence, the scheme proposed in [9] cannot be used for the construction of the general solution to the Cauchy problem z.0/ D c 2 R n for system (5). At the same time, the Green operator of the Cauchy problem for system (5) can be determined as follows [12]:…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…L c 0 / is the pseudoinverse Moore-Penrose matrix [8]. For the construction of approximate solutions of the operator system (12) in the case where…”

mentioning

confidence: 99%

“…x.k; "/W x.k; �/ 2 COE0; " 0 ç given by the operator system (12) and associated with a continuous eigenfunction h."/W h.0/ WD h ⇤ 0 2 R q :…”

mentioning

confidence: 99%

“…For the construction of the operator system (12) for " 2 ⇥ 0; " ⇤ ⇤ ; it is possible to apply the iterative scheme…”

mentioning

confidence: 99%

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Nonlinear Difference-Algebraic Boundary-Value Problem in the Case of Parametric Resonance

2022

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We determine necessary and sufficient conditions for the existence of solutions of a nonlinear boundaryvalue problem for a system of difference-algebraic equations in the case of parametric resonance. We construct a convergent iterative algorithm for finding approximations to the solutions of nonlinear boundaryvalue problem for the system of difference-algebraic equations in the case of parametric resonance. As an example of application of the constructed iterative scheme, we obtain approximations to the solutions of a two-point boundary-value problem for a system of Mathieu-type difference-algebraic equations with parametric perturbation. To check the accuracy of the obtained approximations to the solutions of the two-point boundary-value problem for the system of Mathieu-type difference-algebraic equations with parametric perturbation, we use the discrepancies in the original equation.

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Section: Statement Of the Problemmentioning

confidence: 99%

Section: Statement Of the Problemmentioning

confidence: 99%

“…L c 0 / is the pseudoinverse Moore-Penrose matrix [8]. For the construction of approximate solutions of the operator system (12) in the case where…”

mentioning

confidence: 99%

“…x.k; "/W x.k; �/ 2 COE0; " 0 ç given by the operator system (12) and associated with a continuous eigenfunction h."/W h.0/ WD h ⇤ 0 2 R q :…”

mentioning

confidence: 99%

“…For the construction of the operator system (12) for " 2 ⇥ 0; " ⇤ ⇤ ; it is possible to apply the iterative scheme…”

mentioning

confidence: 99%

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Nonlinear Difference-Algebraic Boundary-Value Problem in the Case of Parametric Resonance

2022

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Conditions of Solvability of the Problem Inverse to the Cauchy Problem for a Difference-Algebraic Equation

2023

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Linear Noetherian boundary value problem for a matrix difference-algebraic Lyapunov equation

2020

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The study of differential-algebraic boundary value problems was initiated in the works of K. Weierstrass, N. N. Luzin and F. R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the work of S. Campbell, Yu. E. Boyarintsev, V. F. Chistyakov, A. M. Samoilenko, M. O. Perestyuk, V. P. Yakovets, O. A. Boichuk, A. Ilchmann and T. Reis. The study of the differential-algebraic boundary value problems is associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, theory of control, theory of motion stability. At the same time, the study of differential algebraic boundary value problems is closely related to the study of boundary value problems for difference equations, initiated in A. A. Markov, S. N. Bernstein, Ya. S. Besikovich, A. O. Gelfond, S. L. Sobolev, V. S. Ryaben’kii, V. B. Demidovich, A. Halanay, G. I. Marchuk, A. A. Samarskii, Yu. A. Mitropolsky, D. I. Martynyuk, G. M. Vayniko, A. M. Samoilenko, O. A. Boichuk and O. M. Standzhitsky. Study of nonlinear singularly perturbed boundary value problems for difference equations in partial differences is devoted to the work of V. P. Anosov, L. S. Frank, P. E. Sobolevskii, A. L. Skubachevskii and A. Asheraliev. Consequently, the actual problem is the transfer of the results obtained in the articles by S. Campbell, A. M. Samoilenko and O. A. Boichuk on linear boundary value problems for difference-algebraic equations, in particular finding the necessary and sufficient conditions for the existence of the desired solutions, and also the construction of the Green’s operator of the Cauchy problem and the generalized Green operator of a linear boundary value problem for a difference-algebraic equation. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A. M. Samoilenko and O. A. Boichuk on the linear boundary value problems for the differential-algebraic boundary value problem for a matrix Lyapunov equation, in particular, finding the necessary and sufficient conditions of the existence of the desired solutions of the linear differential-algebraic boundary value problem for a matrix Lyapunov equation. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian differential-algebraic boundary value problem for a matrix Lyapunov equation. The proposed scheme of the research of the linear differential-algebraic boundary value problem for a matrix Lyapunov equation in the critical case in this article can be transferred to the seminonlinear differential-algebraic boundary value problem for a matrix Lyapunov equation.

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Boundary-Value Problems for Systems of Linear Difference-Algebraic Equations

Cited by 4 publications

References 5 publications

Nonlinear Difference-Algebraic Boundary-Value Problem in the Case of Parametric Resonance

Nonlinear Difference-Algebraic Boundary-Value Problem in the Case of Parametric Resonance

Conditions of Solvability of the Problem Inverse to the Cauchy Problem for a Difference-Algebraic Equation

Linear Noetherian boundary value problem for a matrix difference-algebraic Lyapunov equation

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