Identification of a constant coefficient in an elliptic equation

Lyubanova, Anna Sh.

doi:10.1080/00036810802189654

Cited by 22 publications

(11 citation statements)

References 12 publications

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“…The inverse problem considered in this paper is solved under boundary conditions one of that is an integral condition on the surface of the measuring electrode. The problems with similar conditions are also considered by Lyubanova [9] for elliptic equations of filtration earlier.…”

Section: Introductionmentioning

confidence: 85%

A parameter identification problem for spontaneous potential logging in heterogeneous formation

Pan

Liu

2013

Journal of Inverse and Ill-Posed Problems

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In this paper, we want to estimate the true resistivity of permeable bed using the spontaneous potential on the measuring electrode in heterogeneous formation, where the resistivity of the objective layer varies continuously as a function of position. The problem is a generalization of Peng's work for the traditional three-parameter invasion model, where the resistivity of formation is piecewise constant. The mathematical model of the problem is described by a variational problem of elliptic type, where the formation resistivity appears as a nonlinear coefficient. We study an extended problem and then apply its results to prove the existence of solutions to the parameter identification problem arising from SP logging. Finally, a sufficient condition is given to ensure the uniqueness of solutions.

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Section: Introductionmentioning

confidence: 85%

A parameter identification problem for spontaneous potential logging in heterogeneous formation

Pan

Liu

2013

Journal of Inverse and Ill-Posed Problems

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“…We take into account of limitations (12) and (13) in the following (10). Then, recalling that both x and x are admissible data, by Proposition 1, the limitation in Lemma 3.3 and the mean value theorem, we get the estimate…”

Section: Injectivitymentioning

confidence: 99%

“…Finally, it is also noteworthy mentioning that, to the best of our knowledge, there are only a few papers (see [2,4,8,9,11,12,19]) where a constant rather than a function is identified. The main novelty in our formulation of the inverse problem consists in imposing the final conditions of energy type, which, although natural by the viewpoint of measurements, leads to nonlinear (quadratic) conditions, which appear to be new in literature.…”

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confidence: 99%

Recovering a large number of diffusion constants in a parabolic equation from energy measurements

Mola¹

2018

Inverse Problems &Amp; Imaging

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Let (H, •, •) be a separable Hilbert space and A i : D(A i) → H (i = 1, • • • , n) be a family of nonnegative and self-adjoint operators mutually commuting. We study the inverse problem consisting in the identification of a function u : [0, T ] → H and n constants α 1 , • • • , αn > 0 (diffusion coefficients) that fulfill the initial-value problem u (t) + α 1 A 1 u(t) + • • • + αnAnu(t) = 0, t ∈ (0, T), u(0) = x, and the additional conditions A 1 u(T), u(T) = ϕ 1 , • • • , Anu(T), u(T) = ϕn, where ϕ i are given positive constants. Under suitable assumptions on the operators A i and on the initial data x ∈ H, we shall prove that the solution of such a problem is unique and depends continuously on the data. We apply the abstract result to the identification of diffusion constants in a heat equation and of the Lamé parameters in a elasticity problem on a plate.

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“…Here M is an elliptic linear differential operator of the second order in the space variables. We establish the existence, uniqueness and stability of the strong solution of the inverse problems for (1.1) and the associated stationary equation with an unknown coefficient k under the Dirichlet boundary condition and the additional integral boundary data akin to the conditions of overdetermination considered in [1][2][3][4]. An exact statement of the problems will be given below.…”

Section: Introductionmentioning

confidence: 99%

“…An exact statement of the problems will be given below. Following the idea of [1][2][3][4] based on the method of [5] we prove the existence of the solution by reducing the inverse problem to an operator equation of the second type for the unknown coefficient. We show that the operator of this equation is a contraction on a set constructed with the use of the comparison theorems for elliptic and pseudoparabolic equations.…”

Section: Introductionmentioning

confidence: 99%

Inverse problems for the stationary and pseudoparabolic equations of diffusion

Lyubanova

Velisevich

2018

Applicable Analysis

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The identification of an unknown coefficient in the lower term of pseudoparabolic differential equation of diffusion (u + ηM u)t + M u + ku = f and elliptic second order differential equation M u + ku = f with the Dirichlet boundary condition is considered. The identification of k is based on an integral boundary data. The local existence and uniqueness of generalized strong solutions for the inverse problems are proved. The stability estimates are exposed.

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Identification of a constant coefficient in an elliptic equation

Cited by 22 publications

References 12 publications

A parameter identification problem for spontaneous potential logging in heterogeneous formation

A parameter identification problem for spontaneous potential logging in heterogeneous formation

Recovering a large number of diffusion constants in a parabolic equation from energy measurements

Inverse problems for the stationary and pseudoparabolic equations of diffusion

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