Riesz basis property of system of root functions of second-order differential operator with involution

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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“…We note that the case β = 0 was investigated in detail in [10] for the space L 2 and in [11] for the space…”

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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“…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: 94%

“…We use the general solution of the homogeneous Equation , which is constructed in Kilbas et al, p233 for α + β > 0. Combining them, we get

u_{k, i} (t) = \frac{f_{k, i}}{λ_{k, i}} + C_{k, i} E_{α, 1 + \frac{β}{α}, \frac{β}{α}} (- λ_{k, i} t^{α + β}), 0 < t < T, i = 1, 2,

where E α + β ,1,1 − α is the generalized Mittag‐Leffler function (Kritskov and Sarsenbi, p48):

E_{α, 1 + \frac{β}{α}, \frac{β}{α}} (z) = \sum_{k = 0}^{\infty} c_{k} z^{k}; c_{0} = 1, c_{k} = \prod_{j = 0}^{k - 1} \frac{Γ (j (α + β) + β + 1)}{Γ (j (α + β) + α + β + 1)}, k \in N,

and the constants C k , i and f k , i are unknown.…”

Section: Uniqueness Of the Solution Of The Problemmentioning

confidence: 99%

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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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“…In the particular case of the reflection, starting with [5], the computation of Green's functions for differential equations with reflection expanded [6-10, 17, 18]. This has motivated several applications concerning almost periodic solutions [15,16], the obtaining of eigenvalues and explicit solutions of different problems [13,14] or their qualitative properties [2,4].…”

Section: Introductionmentioning

confidence: 99%

On linear differential equations and systems with reflection

Cabada

Tojo

2017

Applied Mathematics and Computation

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In this paper we develop a theory of linear differential systems analogous to the classical one for ODEs, including the obtaining of fundamental matrices, the development of a variation of parameters formula and the expression of the Green's functions. We also derive interesting results in the case of differential equations with reflection and generalize the Hyperbolic Phasor Addition Formula to the case of matrices.

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Riesz basis property of system of root functions of second-order differential operator with involution

Cited by 38 publications

References 22 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 linear differential equations and systems with reflection

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