An inverse problem of identifying the source coefficient in a degenerate heat equation

Deng, Zui‐Cha; Qian, Kun; Rao, Xiao-Bo; Yang, Liu; Luo, Guoxi

doi:10.1080/17415977.2014.922079

Cited by 24 publications

(19 citation statements)

References 17 publications

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“…The objective relates to the class of inverse problems, which are related to the reconstruction of the right parts of the differential equations with the partial derivatives [4 -5]. Statement issues and numerical methods for the solution of inverse problems for the reconstruction of the right parts of differential equations with the partial derivatives are investigated in [4][5][6][7][8][9][10].…”

Section: Statement Of the Problem And Solution Methodsmentioning

confidence: 99%

Parallel Numerical Method of an Inverse Problem of Double-Phased Filtration

Алиев

Hamzaev²,

İsmayılova

et al. 2019

AzJHPC

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Was investigated the process of the displacement of oil with water in the horizontally located two-dimensional layer, which is described with the double-phased filtration model of incompressible liquid in non-deformable porous media. Within this model was set inverse problem for the definition flow rate of exploitative wells. Meanwhile, as additional conditions are set downhole pressure in injection wells. For the numerical solution of the problem, firstly was conducted discretization by time. As a result original problem comes down to two independent problems which are solved sequentially in each layer of time: the inverse problem for the definition of pressure distribution and flow rate of exploitative wells and forward problem for the definition of saturation of displacing phase. For the solution of the forward problem was offered particular fission, which gives an opportunity parallelization of received differential problems. In the base of the offered numerical method were carried out experimental results for model tasks. For parallelization of computing, processes was applied Open MP technology.

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Section: Statement Of the Problem And Solution Methodsmentioning

confidence: 99%

Parallel Numerical Method of an Inverse Problem of Double-Phased Filtration

Алиев

Hamzaev²,

İsmayılova

et al. 2019

AzJHPC

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“…If ̸ = 0, the first results in this direction are obtained in [1] for the nondegenerate heat operator (i.e., > 0) with a singular potential. But, the study of numerical reconstruction questions are rarely taken into account; see [13,14].…”

Section: Introductionmentioning

confidence: 99%

An Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracy

Atifi

Boutaayamou

Sidi

et al. 2018

Abstract and Applied Analysis

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The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

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“…Analogous results correspond to the determination of the source term in the following simplified Crocco‐type equation

u_{t} - {true(x^{α} u_{x} true)}_{x} = g, (x, t) \in (0, 1) \times (0, T)

are obtained in (for general Crocco equation, we refer the reader to ). Numerical treatments for identifying the source term with terminal observations can be found in . In , optimization method is proposed to stabilize the following inverse radiative coefficient problem

u_{t} - {true(a (x) u_{x} true)}_{x} + q (x) u = 0, (x, t) \in Q = (0, l) \times (0, T],

where the diffusion coefficient is assumed to vanish at both extremities of the domain.…”

Section: Introductionmentioning

confidence: 99%

“…are obtained in [24] (for general Crocco equation, we refer the reader to [25]). Numerical treatments for identifying the source term with terminal observations can be found in [26]. In [27], optimization method is proposed to stabilize the following inverse radiative coefficient problem…”

Section: Introductionmentioning

confidence: 99%

An inverse backward problem for degenerate parabolic equations

Yang

Deng

2017

Numerical Methods Partial

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This work studies the inverse problem of reconstructing an initial value function in the degenerate parabolic equation using the final measurement data. Problems of this type have important applications in the field of financial engineering. Being different from other inverse backward parabolic problems, the mathematical model in our article may be allowed to degenerate at some part of boundaries, which may lead to the corresponding boundary conditions missing. The conditional stability of the solution is obtained using the logarithmic convexity method. A finite difference scheme is constructed to solve the direct problem and the corresponding stability and convergence are proved. The Landweber iteration algorithm is applied to the inverse problem and some typical numerical experiments are also performed in the paper. The numerical results show that the proposed method is stable and the unknown initial value is recovered very well.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1900–1923, 2017

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An inverse problem of identifying the source coefficient in a degenerate heat equation

Cited by 24 publications

References 17 publications

Parallel Numerical Method of an Inverse Problem of Double-Phased Filtration

Parallel Numerical Method of an Inverse Problem of Double-Phased Filtration

An Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracy

An inverse backward problem for degenerate parabolic equations

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