The inverse problem of finding the time-dependent diffusion coefficient of the heat equation from integral overdetermination data

Kanca, Fatma; Ismailov, Mansur I.

doi:10.1080/17415977.2011.629093

Cited by 49 publications

(26 citation statements)

References 9 publications

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“…Verify that the functions p(t), ω(t) are a solution to the inverse problem (16). Integrating (19), we obtain (18) whose transformation validates equality (17) and, hence,ψ (16) hold.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Parameter Identification and Control in Heat Transfer Processes

Pyatkov¹,

Goncharenko²

2017

Bulletin of the SUSU. MMP

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The article is devoted to the study of some mathematical models describing heat transfer processes. We examine an inverse problem of recovering a control parameter providing a prescribed temperature distribution at a given point of the spatial domain. The parameter is a lower order coecient depending on time in a parabolic equation. This nonlinear problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the xed point theorem. Existence and uniqueness theorems of solutions to this problem are stated and proved. Stability estimates are exposed. The main result is the global (in time) existence of solutions under some natural conditions of the data. The proofs rely on the maximum principle. The main functional spaces used are the Sobolev spaces.

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“…Verify that the functions p(t), ω(t) are a solution to the inverse problem (16). Integrating (19), we obtain (18) whose transformation validates equality (17) and, hence,ψ (16) hold.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Parameter Identification and Control in Heat Transfer Processes

Pyatkov¹,

Goncharenko²

2017

Bulletin of the SUSU. MMP

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“…It is easy to see that the functionũ k,1 (t) = ( k,i ) −1 k,i is a partial solution of the inhomogeneous Equation 13. We use the general solution of the homogeneous Equation 13, which is constructed in Kilbas et al 48, p233 for + > 0.…”

Section: Uniqueness Of the Solution Of The 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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“…Methods of solutions and theoretical aspects for various types of nonlocal BVP for parabolic differential equations are extensively investigated by several researchers (see previous studies [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references therein). Well posedness of nonclassical BVPs for various differential and difference equations were studied in a number of publications (see previous works [1][2][3][4][5][6][7][8] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Stability of Rothe difference scheme for the reverse parabolic problem with integral boundary condition

Ashyralyyev

2020

Math Methods in App Sciences

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In this paper, we study the approximation of reverse parabolic problem with integral boundary condition. The Rothe difference scheme for an approximate solution of reverse problem is discussed. We establish stability and coercive stability estimates for the solution of the Rothe difference scheme. In sequel, we investigate the first order of accuracy difference scheme for approximation of boundary value problem for multidimensional reverse parabolic equation and obtain stability estimates for its solution. Finally, we give numerical results together with an explanation on the realization in one-and two-dimensional test examples.KEYWORDS difference scheme, integral condition, reverse parabolic equation, stability MSC CLASSIFICATION 65N06; 65N12Abbreviations: ADS, accuracy difference scheme; BVP, boundary value problem; DS, difference scheme; SAPD, self-adjoint positive definite.

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The inverse problem of finding the time-dependent diffusion coefficient of the heat equation from integral overdetermination data

Cited by 49 publications

References 9 publications

Parameter Identification and Control in Heat Transfer Processes

Parameter Identification and Control in Heat Transfer Processes

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

Stability of Rothe difference scheme for the reverse parabolic problem with integral boundary condition

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