An inverse backward problem for degenerate two-dimensional parabolic equation

• Autorzy: Atifi Khalid , Essoufi El-Hassan , Khouiti Bouchra

Identyfikatory

DOI

10.7494/OpMath.2020.40.4.427

• Języki publikacji: EN

Abstrakty

EN

This paper deals with the determination of an initial condition in the degenerate two-dimensional parabolic equation [formula], where Ω is an open, bounded subset of R2, a [formula] with a ≥0 everywhere, and [formula], with initial and boundary conditions [formula] from final observations. This inverse problem is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. To show the convergence of the descent method, we prove the Lipschitz continuity of the gradient of the Tikhonov functional. Also we present some numerical experiments to show the performance and stability of the proposed approach.

Słowa kluczowe

EN

data assimilation; adjoint method; regularization; heat equation; inverse problem; degenerate equations; optimization

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2020
• Tom: Vol. 40, no. 4
• Strony: 427--449
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Atifi Khalid

• Laboratoire de Mathematiques Informatique et Sciences de Pingenieur (MISI) Universite Hassan 1, Settat 26000, Morocco

autor

Essoufi El-Hassan

• Laboratoire de Mathematiques Informatique et Sciences de Pingenieur (MISI) Universite Hassan 1, Settat 26000, Morocco

autor

Khouiti Bouchra

• Laboratoire de Mathematiques Informatique et Sciences de Pingenieur (MISI) Universite Hassan 1, Settat 26000, Morocco

Bibliografia

• [1] K. Atifi, E.-H. Essoufi, Data assimilation and null controllability of degenerate/singular-parabolic problems, Electron. J. Differential Equations 2017 (2017) 135, 1-17.
• [2] K. Atifi, E.-H. Essoufi, B. Khouiti, Y. Balouki, Identifying initial condition in degenerate parabolic equation with singular potential, Int. J. Differ. Equ. 2017 (2017), Article ID 1467049.
• [3] K. Atifi, B. Khouiti, E.-H. Essoufi, New approach to identify the initial condition in degenerate hyperbolic equation, Inverse Probl. Sci. Eng. 27 (2019), 484-512.
• [4] F. Bourquin, A. Nassiopoulos, Assimilation thermique ID par methode adjointe liberee, Problemes Inverses, Collection Recherche du LCPC, 2006.
• [5] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, 1997.
• [6] M. Ferrara, G. Molica Bisci, Existence results for elliptic problems with Hardy potential, Bull. Sci. Math. 138 (2014), 846-859.
• [7] P. Kuchment, L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math. 19 (2008), 191-224.
• [8] E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, 2nd ed., New York, Cambridge University Press, 2003.
• [9] L.B.L. Santos, L.D. Chiwiacowsky, H.F. Campos-Velho, Genetic algorithm, and varia-tional method to identify initial conditions: worked example in hyperbolic heat transfer, Tend. Mat. Apl. Comput. 2 (2013), 265-276.
• [10] L. Yang, Z.-C. Deng, An inverse backward problem for degenerate parabolic equations Numer. Methods Partial Differential Equations, 33 (2017), 1900-1923.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).