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Determination of the time-dependent effective ion collision frequency from an integral observation

  • Kai Cao ORCID logo and Daniel Lesnic EMAIL logo
Published/Copyright: February 1, 2024

Abstract

Identification of physical properties of materials is very important because they are in general unknown. Furthermore, their direct experimental measurement could be costly and inaccurate. In such a situation, a cheap and efficient alternative is to mathematically formulate an inverse, but difficult, problem that can be solved, in general, numerically; the challenge being that the problem is, in general, nonlinear and ill-posed. In this paper, the reconstruction of a lower-order unknown time-dependent coefficient in a Cahn–Hilliard-type fourth-order equation from an additional integral observation, which has application to characterizing the nonlinear saturation of the collisional trapped-ion mode in a tokamak, is investigated. The local existence and uniqueness of the solution to such inverse problem is established by utilizing the Rothe method. Moreover, the continuous dependence of the unknown coefficient upon the measured data is derived. Next, the Tikhonov regularization method is applied to recover the unknown coefficient from noisy measurements. The stability estimate of the minimizer is derived by investigating an auxiliary linear fourth-order inverse source problem. Henceforth, the variational source condition can be verified. Then the convergence rate is obtained under such source condition.

MSC 2020: 35R30

Award Identifier / Grant number: BK20200389

Award Identifier / Grant number: 12101118

Award Identifier / Grant number: 2242023K40011

Funding statement: K. Cao would like to acknowledge support of the Natural Science Foundation of Jiangsu Province of China (No. BK20200389), the National Natural Science Foundation of China (No. 12101118) and the Fundamental Research Funds for the Central Universities (No. 2242023K40011) for this work.

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Received: 2023-03-06
Revised: 2023-10-07
Accepted: 2023-11-19
Published Online: 2024-02-01
Published in Print: 2024-08-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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