A regular differential operator with perturbed boundary condition

Sadybekov, Makhmud A.; Иманбаев, Н.С.

doi:10.1134/s0001434617050133

Cited by 28 publications

(10 citation statements)

References 2 publications

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“…cos π m 1 m 2 nx ; (21) x cos(πm 1 nx). (22) System (20) is orthonormal in L 2 (−1, 1) and hence satisfies the Bessel type inequality with constant B = 1. The Bessel property of system (22) follows from the Bessel property of system (20), because the factor x is bounded.…”

Section: Case Of Rational Rmentioning

confidence: 99%

“…Spectral problems arising in connection with differential operators with involution were considered in [14][15][16][17][18] for first-order operators and in [19,20] for second-order operators. Spectral problems for ordinary differential operators with non-strongly regular boundary conditions and their applications for parabolic problems were investigated in [21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

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Nonlocal spectral problem for a second-order differential equation with an involution

Kritskov¹,

Sadybekov²,

Sarsenbi³

2018

Bul. Kar. Univ. - Math.

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Nonlocal spectral problem for a second-order differential equation with an involution For the spectral problem −u (x) + αu (−x) = λu(x), −1 < x < 1, with nonlocal boundary conditions u(−1) = βu(1), u (−1) = u (1), where α ∈ (−1, 1), β 2 = 1, we study the spectral properties. We show that if r = (1 − α)/(1 + α) is irrational, then the system of eigenfunctions is complete and minimal in L2(−1, 1) but is not a basis. In the case of a rational number r, the root subspace of the problem consists of eigenvectors and an infinite number of associated vectors. In this case, we indicated a method for choosing associated functions that provides the system of root functions of the problem is an unconditional basis in L2(−1, 1).

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Section: Case Of Rational Rmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonlocal spectral problem for a second-order differential equation with an involution

Kritskov¹,

Sadybekov²,

Sarsenbi³

2018

Bul. Kar. Univ. - Math.

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“…We solve the problem by the Fourier method. Some new variants for solving nonlocal boundary value problems by the method of separation of variables were used in our papers . In this paper, we shall use a spectral problem for ordinary differential operators with involution.…”

Section: Reduction To a Mathematical Problemmentioning

confidence: 99%

“…Some new variants for solving nonlocal boundary value problems by the method of separation of variables were used in our papers. [31][32][33][34][35] In this paper, we shall use a spectral problem for ordinary differential operators with involution. Such and similar spectral problems are considered in previous studies.…”

Section: Reduction To a Mathematical Problemmentioning

confidence: 99%

On an inverse problem of reconstructing a subdiffusion process from nonlocal data

Kirane

Sadybekov

Sarsenbi

2019

Math Methods in App Sciences

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We consider a problem of modeling the thermal diffusion process in a closed metal wire wrapped around a thin sheet of insulation material. The layer of insulation is assumed to be slightly permeable. Therefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified, and a third term with an involution is added. Modeling of this process leads to the consideration of an inverse problem for a one‐dimensional fractional evolution equation with involution and with periodic boundary conditions with respect to a space variable. This equation interpolates heat equation. Such equations are also called nonlocal subdiffusion equations or nonlocal heat equations. The inverse problem consists in the restoration (simultaneously with the solution) of the unknown right‐hand side of the equation, which depends only on the spatial variable. The conditions for overdefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.

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“…We solve the problem by the Fourier method. Some new variants for solving nonlocal boundary value problems by the method of separation of variables were used in our papers [29][30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%

On one problem for restoring the density of sources of the fractional heat conductivity process with respect to initial and final temperatures

Erdogan¹,

Florida²,

Kusmangazinova³

et al. 2018

Bul. Kar. Univ. - Math.

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On one problem for restoring the density of sources of the fractional heat conductivity process with respect to initial and final temperatures In this paper we consider inverse problems for a fractional heat equation, where the fractional time derivative is taken into account in Riemann-Liouville sense. For the solution of this equation, we have to find an unknown right-hand side depending only on a spatial variable. The problem modeling the process of determining the temperature and density of sources in the process of fractional heat conductivity with respect to given initial and final temperatures is considered. Problems with general boundary conditions with respect to the spatial variable that are not strongly regular are investigated. The existence and uniqueness of classical solution to the problem are proved. The problem is considered independent from a corresponding spectral problem for an operator of multiple differentiation with not strongly regular boundary conditions has the basis property of root functions.

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A regular differential operator with perturbed boundary condition

Cited by 28 publications

References 2 publications

Nonlocal spectral problem for a second-order differential equation with an involution

Nonlocal spectral problem for a second-order differential equation with an involution

On an inverse problem of reconstructing a subdiffusion process from nonlocal data

On one problem for restoring the density of sources of the fractional heat conductivity process with respect to initial and final temperatures

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