Inverse problems for a nonlocal wave equation with an involution perturbation

Kirane, Mokhtar; Al-Salti, Nasser

doi:10.22436/jnsa.009.03.49

Cited by 53 publications

(47 citation statements)

References 24 publications

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“…The well‐posedness of direct and inverse problems for parabolic equations with involution is considered in previous studies …”

Section: Reduction To a Mathematical Problemmentioning

confidence: 99%

“…The well-posedness of direct and inverse problems for parabolic equations with involution is considered in previous studies. [4][5][6] The solvability of various inverse problems for parabolic equations was studied in papers of Anikonov YuE and Belov YuYa, Bubnov BA, Prilepko AI and Kostin AB, Monakhov VN, Kozhanov AI, Kaliev IA, Sabitov KB, and many others. These citations can be seen in Orazov and Sadybekov.…”

Section: Reduction To a Mathematical Problemmentioning

confidence: 99%

“…The mathematical problem (5) to (8) for a = 0 was considered in Torebek and Tapdigoglu, 28 and for a = = 0 in Kirane et al 29 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.…”

Section: Reduction To a Mathematical Problemmentioning

confidence: 99%

See 2 more Smart Citations

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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“…The well‐posedness of direct and inverse problems for parabolic equations with involution is considered in previous studies …”

Section: Reduction To a Mathematical Problemmentioning

confidence: 99%

Section: Reduction To a Mathematical Problemmentioning

confidence: 99%

See 1 more Smart Citation

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 show that the eigenvalues of the operators L 0,k and L 1,k coincide. We substitute the fundamental system (14), (15) …”

Section: Non Self-adjoint Boundary Value Problemmentioning

confidence: 99%

“…During recent years, the number of publications with the use of an involution operator in various sections of the theory of ordinary differential equations (see [4,10,12,15,16,19]), of partial differential equations (see [3,7,9,14,16,17,20,21]), of linear operators, T-invariant with respect to some group of homeomorphisms (see [8]), differential equations with operator coefficients (see [5][6][7]), PT-symmetric operators (see [1,2]) increased significantly.…”

Section: Introductionmentioning

confidence: 99%

The nonlocal problem for the differential-operator equation of the even order with the involution

Baranetskij¹,

Kalenyuk²,

Kolyasa³

et al. 2018

Carpathian Math. Publ.

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In this paper, the problem with boundary non-self-adjoint conditions for differential-operator equations of the order 2n with involution is studied. Spectral properties of operator of the problem is investigated.By analogy of separation of variables the nonlocal problem for the differential-operator equation of the even order is reduced to a sequence {L k } ∞ k=1 of operators of boundary value problems for ordinary differential equations of even order. It is established that each element L k of this sequence is an isospectral perturbation of the self-adjoint operator L 0,k of the boundary value problem for some linear differential equation of order 2n.We construct a commutative group of transformation operators whose elements reflect the system V(L 0,k ) of the eigenfunctions of the operator L 0,k in the system V(L k ) of the eigenfunctions of the operators L k . The eigenfunctions of the operator L of the boundary value problem for a differential equation with involution are obtained as the result of the action of some specially constructed operator on eigenfunctions of the sequence of operators L 0,k .The conditions under which the system of eigenfunctions of the operator L of the studied problem is a Riesz basis is established.

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Some inverse problems for the nonlocal heat equation with Caputo fractional derivative

Torebek

Tapdıgoglu

2017

Math Methods in App Sciences

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Inverse problems for a nonlocal wave equation with an involution perturbation

Abstract: Two inverse problems for the wave equation with involution are considered. Results on existence and uniqueness of solutions of these problems are presented.

Cited by 53 publications

References 24 publications

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

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

The nonlocal problem for the differential-operator equation of the even order with the involution

Some inverse problems for the nonlocal heat equation with Caputo fractional derivative

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