Opuscula Math. 40, no. 4 (2020), 427-449
https://doi.org/10.7494/OpMath.2020.40.4.427

 
Opuscula Mathematica

An inverse backward problem for degenerate two-dimensional parabolic equation

Khalid Atifi
El-Hassan Essoufi
Bouchra Khouiti

Abstract. This paper deals with the determination of an initial condition in the degenerate two-dimensional parabolic equation \[\partial_{t}u-\mathrm{div}\left(a(x,y)I_2\nabla u\right)=f,\quad (x,y)\in\Omega,\; t\in(0,T),\] where \(\Omega\) is an open, bounded subset of \(\mathbb{R}^2\), \(a \in C^1(\bar{\Omega})\) with \(a\geqslant 0\) everywhere, and \(f\in L^{2}(\Omega \times (0,T))\), with initial and boundary conditions \[u(x,y,0)=u_0(x,y), \quad u\mid_{\partial\Omega}=0,\] 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.

Keywords: data assimilation, adjoint method, regularization, heat equation, inverse problem, degenerate equations, optimization.

Mathematics Subject Classification: 15A29, 47A52, 34A38, 93C20, 60J70, 35K05, 35K65.

Full text (pdf)

  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 1D par méthode adjointe libérée, Problèmes 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 variational 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.
  • Khalid Atifi (corresponding author)
  • ORCID iD https://orcid.org/0000-0001-6425-0015
  • Laboratoire de Mathématiques, Informatique et Sciences de l'ingénieur (MISI), Université Hassan 1, Settat 26000, Morocco
  • Communicated by Massimiliano Ferrara.
  • Received: 2019-02-25.
  • Revised: 2020-03-31.
  • Accepted: 2020-04-03.
  • Published online: 2020-07-09.
Opuscula Mathematica - cover

Cite this article as:
Khalid Atifi, El-Hassan Essoufi, Bouchra Khouiti, An inverse backward problem for degenerate two-dimensional parabolic equation, Opuscula Math. 40, no. 4 (2020), 427-449, https://doi.org/10.7494/OpMath.2020.40.4.427

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.