TY - JOUR T1 - Ill-Posedness of Inverse Diffusion Problems by Jacobi's Theta Transform AU - Ben Belgacem , Faker AU - Kateb , M. Djalil AU - Robin , Vincent JO - Journal of Mathematical Study VL - 2 SP - 115 EP - 130 PY - 2018 DA - 2018/06 SN - 51 DO - http://doi.org/10.4208/jms.v51n2.18.01 UR - https://global-sci.org/intro/article_detail/jms/12458.html KW - Integral operators, regular kernels, Jacobi transform, separated variables approximation. AB -
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.