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Quasi Solution of a Nonlinear Inverse Parabolic Problem

  • Amir Hossein Salehi Shayegan
  • Ali Zakeri
  • Touraj Nikazad
Original Paper
  • 21 Downloads

Abstract

In this paper, we study the existence of a quasi solution to nonlinear inverse parabolic problem related to \( \aleph (u):\equiv u_{t}-\nabla \cdot (F(x,\nabla u)) \) 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.

Keywords

Quasi solution Nonlinear inverse parabolic problem Time-dependent problems 

Mathematics Subject Classification

Primary 35K55 Secondary 35R30 49J20 

Notes

Acknowledgements

We wish to thank two anonymous referees for constructive criticism and helpful suggestions which improved our paper.

References

  1. 1.
    Di Nardo, R.: Nonlinear elliptic and parabolic equations with measure data. PhD thesis, Università degli Studi di Napoli Federico II (2009)Google Scholar
  2. 2.
    Faragó, I., Karátson, J.: The gradient-finite element method for elliptic problems. Comput. Math. Appl. 42(8–9), 1043–1053 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Faragó, I., Karátson, J.: Numerical solution of nonlinear elliptic problems via preconditioning operators: Theory and applications, vol. 11. Nova Science Publishers, Inc., New York (2002)Google Scholar
  4. 4.
    Hasanov, A.: Inverse coefficient problems for monotone potential operators. Inverse Probl. 13(5), 1265–1278 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hasanov, A., Liu, Z.H.: An inverse coefficient problem for a nonlinear parabolic variational inequality. Appl. Math. Lett. 21(6), 563–570 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ladyzhenskaia, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and quasi-linear equations of parabolic type, vol. 23. American Mathematical Society, Providence (1988)Google Scholar
  7. 7.
    Ou, Y.H., Hasanov, A., Liu, Z.H.: Inverse coefficient problems for nonlinear parabolic differential equations. Acta Math. Sin. (Engl. Ser.) 24(10), 1617–1624 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Rektorys, K.: The method of discretization in time and partial differential equations. D. Reidel Publishing Company, Dordrecht (1982)zbMATHGoogle Scholar
  9. 9.
    Wilansky, A.: Topology for analysis. Ginn, Boston (1970)zbMATHGoogle Scholar
  10. 10.
    Zakeri, A., Salehi Shayegan, A.H.: Gradient WEB-spline finite element method for solving two-dimensional quasilinear elliptic problems. Appl. Math. Model 38(2), 775–783 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  • Amir Hossein Salehi Shayegan
    • 1
  • Ali Zakeri
    • 1
  • Touraj Nikazad
    • 2
  1. 1.Faculty of MathematicsK. N. Toosi University of TechnologyTehranIran
  2. 2.School of MathematicsIran University of Science and TechnologyTehranIran

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