@Article{JMS-51-115,
author = {Ben Belgacem , FakerKateb , M. Djalil and Robin , Vincent},
title = {Ill-Posedness of Inverse Diffusion Problems by Jacobi's Theta Transform},
journal = {Journal of Mathematical Study},
year = {2018},
volume = {51},
number = {2},
pages = {115--130},
abstract = {<p style="text-align: justify;">The subject is the ill-posedness degree of some inverse problems for the
transient heat conduction. We focus on three of them: the completion of missing
boundary data, the identification of the trajectory of a pointwise source and the recovery
of the initial state. In all of these problems, the observations provide over-specified
boundary data, commonly called Cauchy boundary conditions. Notice that the third
problem is central for the controllability by a boundary control of the temperature.
Presumably, they are all severely ill-posed, a relevant indicator on their instabilities,
as formalized by G. Wahba. We revisit these issues under a new light and with different
mathematical tools to provide detailed and complete proofs for these results.
Jacobi Theta functions, complemented with the Jacobi Imaginary Transform, turn out
to be a powerful tool to realize our objectives. In particular, based on the Laptev work
[Matematicheskie Zametki 16, 741-750 (1974)], we provide new information about
the observation of the initial data problem. It is actually exponentially ill-posed.</p>},
issn = {2617-8702},
doi = {https://doi.org/10.4208/jms.v51n2.18.01},
url = {http://global-sci.org/intro/article_detail/jms/12458.html}
}