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Mat. Zametki, 2017, Volume 101, Issue 5, Pages 768–778 (Mi mz11468)  

This article is cited in 19 scientific papers (total in 19 papers)

A Regular Differential Operator with Perturbed Boundary Condition

M. A. Sadybekova, N. S. Imanbaevab

a Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science, Republic of Kazakhstan
b South Kazakhstan State Pedagogical institute

Abstract: The operator $\mathcal{L}_{0}$ generated by a linear ordinary differential expression of $n$th order and regular boundary conditions of general form is considered on a closed interval. The characteristic determinant of the spectral problem for the operator $\mathcal{L}_{1}$, where $\mathcal{L}_{1}$ is an operator with the integral perturbation of one of its boundary conditions, is constructed, assuming that the unperturbed operator $\mathcal{L}_{0}$ possesses a system of eigenfunctions and associated functions generating an unconditional basis in $L_{2}(0,1)$. Using the obtained formula, we derive conclusions about the stability or instability of the unconditional basis properties of the system of eigenfunctions and associated functions of the problem under an integral perturbation of the boundary condition. The Samarskii–Ionkin problem with integral perturbation of its boundary condition is used as an example of the application of the formula. \renewcommand{\qed}

Keywords: basis, regular boundary condition, eigenvalue, root function, spectral problem, integral perturbation of the boundary condition, characteristic determinant.

Funding Agency Grant Number
Ministry of Education and Science of the Republic of Kazakhstan 0825/4
This work was supported by the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan under grant 0825/GF4.

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/mzm11468

Full text: PDF file (460 kB)
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English version:
Mathematical Notes, 2017, 101:5, 878–887

Bibliographic databases:

UDC: 517.927
PACS: 02.30.Jr, 02.30.Tb
Received: 15.12.2016
Revised: 20.11.2016

Citation: M. A. Sadybekov, N. S. Imanbaev, “A Regular Differential Operator with Perturbed Boundary Condition”, Mat. Zametki, 101:5 (2017), 768–778; Math. Notes, 101:5 (2017), 878–887

Citation in format AMSBIB
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\paper A Regular Differential Operator with Perturbed Boundary Condition
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\pages 768--778
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    This publication is cited in the following articles:
    1. M. E. Akhymbek, M. A. Sadybekov, “Correct restrictions of first-order functionaldifferential equation”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, UNSP 050014  crossref  isi  scopus
    2. G. Dildabek, M. B. Saprygina, “Volterra property of an problem of the Frankl type for an parabolichyperbolic equation”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, UNSP 050011  crossref  isi  scopus
    3. N. S. Imanbaev, M. A. Sadybekov, “About characteristic determinant of one boundary value problem not having the basis property”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, UNSP 050002  crossref  isi  scopus
    4. T. Sh. Kal'menov, G. Arepova, D. Suragan, “On the symmetry of the boundary conditions of the volume potential”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, UNSP 040014  crossref  isi  scopus
    5. T. Sh. Kal'menov, “Boundary conditions for the Cauchy potential for two-dimensional hyperbolic equations”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, UNSP 040002  crossref  isi  scopus
    6. T. Sh. Kalmenov, G. Besbaev, R. Medetbekova, “Regular boundary value problems for the heat equation with scalar parameters”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, UNSP 040019  crossref  isi  scopus
    7. B. D. Koshanov, G. D. Smatova, “On correct restrictions of bi-Laplace operator and their properties”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, UNSP 050016  crossref  isi  scopus
    8. G. Oralsyn, “Trace formulae for the heat-volume potential of the time-fractional heat equation”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, UNSP 050012  crossref  isi  scopus
    9. B. Sabitbek, “On Hardy and Rellich type inequalities for an Engel-type operator”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, UNSP 050003  crossref  isi  scopus
    10. A. A. Sarsenbi, “On a class of inverse problems for a parabolic equation with involution”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, UNSP 040021  crossref  isi  scopus
    11. A. Sh. Shaldanbayev, M. T. Shomanbayeva, “Solution of singularly perturbed Cauchy problem for ordinary differential equation of second order with constant coefficients by Fourier method”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, UNSP 040017  crossref  isi  scopus
    12. A. B. Imanbayeva, A. Sh. Shaldanbayev, A. A. Kopzhasarova, “Asymptotic decomposition of the solution of a singularly perturbed Cauchy problem for a system of ordinary differential equations with constant coefficients”, Izv. Nats. Akad. Nauk Resp. Kaz. Ser. Fiz.-Mat., 5:315 (2017), 112–126  mathscinet  isi
    13. B. Aibek, A. Aimakhanova, G. Besbaev, M. A. Sadybekov, “About one inverse problem of time fractional evolution with an involution perturbation”, International Conference on Analysis and Applied Mathematics (ICAAM 2018), AIP Conf. Proc., 1997, Amer. Inst. Phys., 2018, UNSP 020012-1  crossref  isi  scopus
    14. N. S. Imanbaev, “Distribution of eigenvalues of a third-order differential operator with strongly regular boundary conditions”, International Conference on Analysis and Applied Mathematics (ICAAM 2018), AIP Conf. Proc., 1997, Amer. Inst. Phys., 2018, UNSP 020027-1  crossref  isi  scopus
    15. M. A. Sadybekov, G. Dildabek, M. B. Ivanova, “One class of inverse problems for reconstructing the process of heat conduction from nonlocal data”, International Conference on Analysis and Applied Mathematics (ICAAM 2018), AIP Conf. Proc., 1997, Amer. Inst. Phys., 2018, UNSP 020069-1  crossref  isi  scopus
    16. A. S. Erdogan, D. Kusmangazinova, I. Orazov, M. A. Sadybekov, “On one problem for restoring the density of sources of the fractional heat conductivity process with respect to initial and final temperatures”, Bull. Karaganda Univ-Math., 91:3 (2018), 31–44  crossref  isi
    17. V. L. Kritskov, M. A. Sadybekov, A. M. Sarsenbi, “Nonlocal spectral problem for a second-order differential equation with an involution”, Bull. Karaganda Univ. Math., 91:3 (2018), 53–60  isi
    18. Kirane M., Sadybekov M.A., Sarsenbi A.A., “On An Inverse Problem of Reconstructing a Subdiffusion Process From Nonlocal Data”, Math. Meth. Appl. Sci., 42:6 (2019), 2043–2052  crossref  isi  scopus
    19. Nurlan S. Imanbaev, “On a problem that does not have basis property of root vectors, associated with a perturbed regular operator of multiple differentiation”, Zhurn. SFU. Ser. Matem. i fiz., 13:5 (2020), 568–573  mathnet  crossref
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