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Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems

Frederick Maes (UGent) and Karel Van Bockstal (UGent)
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Abstract
We study the uniqueness of some inverse source problems arising in thermoelastic models of type-III. We suppose that the source terms can be decomposed as a product of a time dependent and a space dependent function, i.e. g(t)f(x) for the load source and g(t)f(x) for the heat source. In the first inverse source problem, the source f(x) has to be determined from the final in time measurement of the displacement u(x,T) , or from the time-average measurement ∫T0u(x,t)dt . In the second inverse source problem, the source f(x) has to be determined from the time-average measurement of the temperature ∫T0θ(x,t)dt . We show the uniqueness of a solution to these problems under suitable assumptions on the function g(t) . Moreover, we provide some examples showing the necessity of these assumptions. Finally, we conclude the article by studying two combined problems of determining both sources.
Keywords
Applied Mathematics, Thermoelasticity, inverse source problem, uniqueness, HEAT-SOURCE, RECOVERY, EQUATION, MFS

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MLA
Maes, Frederick, and Karel Van Bockstal. “Uniqueness for Inverse Source Problems of Determining a Space-Dependent Source in Thermoelastic Systems.” JOURNAL OF INVERSE AND ILL-POSED PROBLEMS, vol. 30, no. 6, 2022, pp. 845–56, doi:10.1515/jiip-2021-0055.
APA
Maes, F., & Van Bockstal, K. (2022). Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems. JOURNAL OF INVERSE AND ILL-POSED PROBLEMS, 30(6), 845–856. https://doi.org/10.1515/jiip-2021-0055
Chicago author-date
Maes, Frederick, and Karel Van Bockstal. 2022. “Uniqueness for Inverse Source Problems of Determining a Space-Dependent Source in Thermoelastic Systems.” JOURNAL OF INVERSE AND ILL-POSED PROBLEMS 30 (6): 845–56. https://doi.org/10.1515/jiip-2021-0055.
Chicago author-date (all authors)
Maes, Frederick, and Karel Van Bockstal. 2022. “Uniqueness for Inverse Source Problems of Determining a Space-Dependent Source in Thermoelastic Systems.” JOURNAL OF INVERSE AND ILL-POSED PROBLEMS 30 (6): 845–856. doi:10.1515/jiip-2021-0055.
Vancouver
1.
Maes F, Van Bockstal K. Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems. JOURNAL OF INVERSE AND ILL-POSED PROBLEMS. 2022;30(6):845–56.
IEEE
[1]
F. Maes and K. Van Bockstal, “Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems,” JOURNAL OF INVERSE AND ILL-POSED PROBLEMS, vol. 30, no. 6, pp. 845–856, 2022.
@article{8755402,
  abstract     = {{We study the uniqueness of some inverse source problems arising in thermoelastic models of type-III. We suppose that the source terms can be decomposed as a product of a time dependent and a space dependent function, i.e. g(t)f(x) for the load source and g(t)f(x) for the heat source. In the first inverse source problem, the source f(x) has to be determined from the final in time measurement of the displacement u(x,T) , or from the time-average measurement ∫T0u(x,t)dt . In the second inverse source problem, the source f(x) has to be determined from the time-average measurement of the temperature ∫T0θ(x,t)dt . We show the uniqueness of a solution to these problems under suitable assumptions on the function g(t) . Moreover, we provide some examples showing the necessity of these assumptions. Finally, we conclude the article by studying two combined problems of determining both sources.}},
  author       = {{Maes, Frederick and Van Bockstal, Karel}},
  issn         = {{0928-0219}},
  journal      = {{JOURNAL OF INVERSE AND ILL-POSED PROBLEMS}},
  keywords     = {{Applied Mathematics,Thermoelasticity,inverse source problem,uniqueness,HEAT-SOURCE,RECOVERY,EQUATION,MFS}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{845--856}},
  title        = {{Uniqueness for inverse source problems of determining a space-dependent source in thermoelastic systems}},
  url          = {{http://doi.org/10.1515/jiip-2021-0055}},
  volume       = {{30}},
  year         = {{2022}},
}

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