Tuan, NH, Lesnic, D orcid.org/0000-0003-3025-2770, Viet, TQ et al. (1 more author) (2019) Regularization of the semilinear sideways heat equation. IMA Journal of Applied Mathematics, 84 (2). pp. 258-291. ISSN 0272-4960
Abstract
A classical physical example of the sideways heat equation is represented by re-entry vehicles in the atmosphere where the temperature at the nozzle of a rocket is so high that any thermocouple attached to it would be destroyed. Instead one could measure both the temperature and heat flux, i.e. Cauchy data, at an interior boundary inward the capsule. In addition, we assume that there exists a heat source which is significantly dependent on space, time and temperature, and hence it cannot be neglected. This gives rise to a non-characteristic Cauchy inverse boundary value problem in the sense that the interior accessible boundary is overspecified, while the exterior hostile boundary is underspecified as nothing is prescribed on it. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the Cauchy data. In order to obtain a stable numerical solution, we propose two regularization methods to solve the semilinear problem in which the heat source is a Lipschitz function of temperature. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in L² uniformly with respect to the space coordinate under some a priori assumptions on the solution. These assumptions place no serious restrictions on the applicability of the results since in practice we always have some control and knowledge about how large the absolute temperature and heat flux are likely to be. Finally, in order to increase the significance of the study, numerical results are presented and discussed illustrating the theoretical findings in terms of accuracy and stability.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This is an author produced version of a paper published in IMA Journal of Applied Mathematics. Uploaded in accordance with the publisher's self-archiving policy. |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 09 Oct 2018 10:24 |
Last Modified: | 24 Oct 2019 00:38 |
Status: | Published |
Publisher: | Oxford University Press |
Identification Number: | 10.1093/imamat/hxy058 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:136856 |