Approximate cloaking for the heat equation via transformation optics

Hoai-Minh Nguyen; Tu Nguyen; Hoai-Minh Nguyen; Tu Nguyen

doi:10.3934/mine.2019.4.775

Mathematics in Engineering

2019, Volume 1, Issue 4: 775-788. doi: 10.3934/mine.2019.4.775

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Approximate cloaking for the heat equation via transformation optics

Hoai-Minh Nguyen ^{1
,
,},
Tu Nguyen ²

1.
Department of Mathematics, EPFL SB CAMA, Station 8, CH-1015 Lausanne, Switzerland
2.
Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

Received: 26 November 2018 Accepted: 24 June 2019 Published: 29 August 2019

In this paper, we establish approximate cloaking for the heat equation via transformation optics. We show that the degree of visibility is of the order $\varepsilon$ in three dimensions and $|\ln \varepsilon|^{-1}$ in two dimensions, where $\varepsilon$ is the regularization parameter. To this end, we first transform the problem in time domain into a family of problems in frequency domain by taking the Fourier transform with respect to time, and then derive appropriate estimates in the frequency domain.
- heat equation,
- approximate cloaking,
- frequency analysis
Citation: Hoai-Minh Nguyen, Tu Nguyen. Approximate cloaking for the heat equation via transformation optics[J]. Mathematics in Engineering, 2019, 1(4): 775-788. doi: 10.3934/mine.2019.4.775

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Abstract

In this paper, we establish approximate cloaking for the heat equation via transformation optics. We show that the degree of visibility is of the order $\varepsilon$ in three dimensions and $|\ln \varepsilon|^{-1}$ in two dimensions, where $\varepsilon$ is the regularization parameter. To this end, we first transform the problem in time domain into a family of problems in frequency domain by taking the Fourier transform with respect to time, and then derive appropriate estimates in the frequency domain.

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