Well-posedness of the inverse source problem for parabolic systems

Prilepko, A. I.; Tkachenko, D. S.

doi:10.1007/s10625-004-0012-2

Cited by 11 publications

(4 citation statements)

References 4 publications

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“…Inverse source problems for different type of sources appeared in [11,18,20,31,37]. In [27] there is a complete list of references where identification of various type of sources are listed.…”

Section: Approximate Solution To An Inverse Problem For a Fractional ...mentioning

confidence: 99%

Anomalous Diffusion with Caputo-Fabrizio Time Derivative: an Inverse Problem.

Seminara

Troparevsky

Fabio

et al. 2022

TCAM

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In this work we approximate the source for a non homogeneous fractionaldiffusion equation in 1D, from measurements of the concentration at a finite number ofpoints. We use Caputo-Fabrizio time fractional derivative to model anomalous diffusion.Separating variables, we arrive to a linear system which provides approximate values forthe Fourier coefficients of the unknown source. Numerical examples show the efficiency ofthe method, as well as some of its practical limitations.

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Section: Approximate Solution To An Inverse Problem For a Fractional ...mentioning

confidence: 99%

Anomalous Diffusion with Caputo-Fabrizio Time Derivative: an Inverse Problem.

Seminara

Troparevsky

Fabio

et al. 2022

TCAM

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“…The existence results of an inverse problem to NSE with both the integral as well as final overdetermination data are proved in [34] by using Schauder's fixed point theorem in two and three dimensions. The global well-posedness of the inverse source problem for parabolic systems is examined in [36]. Under the assumption that the initial data u 0 ∈ H and the viscosity constant is sufficiently large, the authors in [14] proved that an inverse problem for 2D NSE with the final overdetermination data is well-posed.…”

Section: Introductionmentioning

confidence: 99%

Existence, uniqueness and stability of an inverse problem for two-dimensional convective Brinkman-Forchheimer equations with the integral overdetermination

Kumar¹,

Mohan²

2022

Preprint

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In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations:in a bounded domain Ω ⊂ R 2 with smooth boundary ∂Ω, where α, β, µ > 0 and r ∈ [1, 3]. The investigated inverse problem consists of reconstructing the vector-valued velocity function u, the pressure field p and the scalar function f . For the divergence free initial data u 0 ∈ L 2 (Ω), we prove the existence of a solution to the inverse problem for two-dimensional CBF equations with the integral overdetermination condition, by showing the existence of a unique fixed point for an equivalent operator equation (using an extension of the contraction mapping theorem). Moreover, we establish the uniqueness and Lipschitz stability results of the solution to the inverse problem for 2D CBF equations with r ∈ [1, 3].

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Section: Introductionmentioning

confidence: 99%

“…Inverse problems with the final overdetermination have been well studied for parabolic equations (see [5,17,21,22,33], etc and the references therein). The works [5,17,21,22,32,33], etc assumed that the initial data is smooth (at least in H 2 (Ω)). The authors in [37] established the solvability results of an inverse problem to the nonlinear NSE with the final overdetermination data in three dimensions using Schauder's fixed point theorem.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness of an inverse problem for two and three dimensional convective Brinkman-Forchheimer equations with the final overdetermination

Kumar¹,

Mohan²

2021

Preprint

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In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations:, 3) with smooth boundary, where α, β, µ > 0 and r ∈ [1, ∞). The CBF equations describe the motion of incompressible fluid flows in a saturated porous medium. The inverse problem under our consideration consists of reconstructing the vector-valued velocity function u, the pressure field p and the vector-valued function f . We have proved the well-posedness result (existence, uniqueness and stablility) for the inverse problem for the 2D and 3D CBF equations with the final overdetermination condition using Schauder's fixed point theorem for arbitrary smooth initial data. The well-posedness results hold for r ≥ 1 in two dimensions and for r ≥ 3 in three dimensions. The global solvability results available in the literature helped us to obtain the uniqueness and stability results for the model with fast growing nonlinearities.

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Well-posedness of the inverse source problem for parabolic systems

Cited by 11 publications

References 4 publications

Anomalous Diffusion with Caputo-Fabrizio Time Derivative: an Inverse Problem.

Anomalous Diffusion with Caputo-Fabrizio Time Derivative: an Inverse Problem.

Existence, uniqueness and stability of an inverse problem for two-dimensional convective Brinkman-Forchheimer equations with the integral overdetermination

Well-posedness of an inverse problem for two and three dimensional convective Brinkman-Forchheimer equations with the final overdetermination

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