Quasi Solution of a Nonlinear Inverse Parabolic Problem

In this paper, we study the existence of a quasi solution to nonlinear inverse parabolic problem related to ℵ ( u ) : ≡ u t - ∇ · ( F ( x , ∇ u ) ) where the function F is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function...

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Bibliographic Details
Published inBulletin of the Iranian Mathematical Society Vol. 45; no. 1; pp. 1 - 12
Main Authors Shayegan, Amir Hossein Salehi, Zakeri, Ali, Nikazad, Touraj
Format Journal Article
LanguageEnglish
Published Singapore Springer Singapore 07.02.2019
Subjects
Mathematics
Mathematics and Statistics
Original Paper
Primary 35K55
Quasi solution
Time-dependent problems
Nonlinear inverse parabolic problem
49J20
Secondary 35R30
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Summary:In this paper, we study the existence of a quasi solution to nonlinear inverse parabolic problem related to ℵ ( u ) : ≡ u t - ∇ · ( F ( x , ∇ u ) ) where the function F is unknown. We consider a methodology, involving minimization of a least squares cost functional, to identify the unknown function F . At the first step of the methodology, we give a stability result corresponding to connectivity of F and u which leads to the continuity of the cost functional. We next construct an appropriate class of admissible functions and show that a solution of the minimization problem exists for the continuous cost functional. At the last step, we conclude that the nonlinear inverse parabolic problem has at least one quasi solution in that class of functions.
ISSN:1017-060X
1735-8515
DOI:10.1007/s41980-018-0115-9