Simultaneous determination of time-dependent coefficients and heat source

Hussein, M. S.; Lesnic, D.

doi:10.1080/15502287.2016.1231241

Cited by 15 publications

(15 citation statements)

References 15 publications

(14 reference statements)

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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. Proceed with stability estimates.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 80%

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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Section: Proofs Of the Main Resultsmentioning

confidence: 80%

Parameter Identification and Control in Heat Transfer Processes

Pyatkov¹,

Goncharenko²

2017

Bulletin of the SUSU. MMP

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“…To give more physical meaning to the inverse problem, we have in mind a process in which a finite slab is undertaking radioactive decay such that its diffusivity, convection and reaction coefficients are unknown but they depend on time [1,Chap.13], [16]. We finally mention that extensions to cases when the time-dependent heat source is also unknown or when some unknown coefficients may depend on space as well have recently been considered elsewhere, [7,8]. The initial condition is u(x, 0) = ϕ(x), 0 ≤ x ≤ h(0) =: h 0 ,…”

Section: Mathematical Formulationmentioning

confidence: 99%

Multiple time-dependent coefficient identification thermal problems with a free boundary

Hussein

Lesnic

Ivanchov

et al. 2016

Applied Numerical Mathematics

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Multiple time-dependent coefficient identification thermal problems with an unknown free boundary are investigated. The difficulty in solving such inverse and ill-posed free boundary problems is amplified by the fact that several quantities of physical interest (conduction, convection/advection and reaction coefficients) have to be simultaneously identified. The additional measurements which render a unique solution are given by the heat moments of various orders together with a Stefan boundary condition on the unknown moving boundary. Existence and uniqueness theorems are provided. The nonlinear and ill-posed problems are numerically discretised using the finite-difference method and the resulting system of equations is solved numerically using the MATLAB toolbox routine lsqnonlin applied to minimizing the nonlinear Tikhonov regularization functional subject to simple physical bounds on the variables. Numerically obtained results from some typical test examples are presented and discussed in order to illustrate the efficiency of the computational methodology adopted.

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“…, a multiple time‐dependent coefficient inverse problem with unknown free boundary is investigated by using additional observation such as

\begin{matrix} true \\ right \int_{0}^{s false(t false)} u (x, t) d t = m (t) or s^{'} (t) + u_{x} (s false(t false), t) = m (t), t \in [0, T] . \end{matrix}

Since s is unknown, these kinds of measurements are difficult to obtain. For other kinds of inverse problems related to free boundary, we refer to and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Since is unknown, these kinds of measurements are difficult to obtain. For other kinds of inverse problems related to free boundary, we refer to [26][27][28][29][30][31][32][33][34] and the references therein. In Ref.…”

Section: Introductionmentioning

confidence: 99%

A coefficient identification problem for a mathematical model related to ductal carcinoma in situ

Chen

et al. 2019

Stud Appl Math

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We consider a coefficient identification problem for a mathematical model with free boundary related to ductal carcinoma in situ (DCIS). This inverse problem aims to determine the nutrient consumption rate from additional measurement data at a boundary point. We first obtain a global‐in‐time uniqueness of our inverse problem. Then based on the optimization method, we present a regularization algorithm to recover the nutrient consumption rate. Finally, our numerical experiment shows the effectiveness of the proposed numerical method.

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Simultaneous determination of time-dependent coefficients and heat source

Cited by 15 publications

References 15 publications

Parameter Identification and Control in Heat Transfer Processes

Parameter Identification and Control in Heat Transfer Processes

Multiple time-dependent coefficient identification thermal problems with a free boundary

A coefficient identification problem for a mathematical model related to ductal carcinoma in situ

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