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A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems

Boundary Value Problems volume 2006, Article number: 37524 (2006) Cite this article

Abstract

We study a final value problem for first-order abstract differential equation with positive self-adjoint unbounded operator coefficient. This problem is ill-posed. Perturbing the final condition, we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed and that their solutions converge if and only if the original problem has a classical solution. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.

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References

  1. 1.

    Ababna M: Regularization by nonlocal conditions of the problem of the control of the initial condition for evolution operator-differential equations. Vestnik Belorusskogo Gosudarstvennogo Universiteta. Seriya 1. Fizika, Matematika, Informatika 1998, (2):60-63, 81.

    Google Scholar 

  2. 2.

    Ames KA, Payne LE: Asymptotic behavior for two regularizations of the Cauchy problem for the backward heat equation. Mathematical Models & Methods in Applied Sciences 1998,8(1):187-202. 10.1142/S0218202598000093

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Ames KA, Payne LE, Schaefer PW: Energy and pointwise bounds in some non-standard parabolic problems. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2004,134(1):1-9. 10.1017/S0308210500003036

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Clark GW, Oppenheimer SF: Quasireversibility methods for non-well-posed problems. Electronic Journal of Differential Equations 1994,1994(8):1-9.

    MathSciNet  Google Scholar 

  5. 5.

    Denche M, Bessila K: A modified quasi-boundary value method for ill-posed problems. Journal of Mathematical Analysis and Applications 2005,301(2):419-426. 10.1016/j.jmaa.2004.08.001

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Ivanov VK, Mel'nikova IV, Filinkov AI: Operator-Differential Equations and Ill-Posed Problems. Fizmatlit "Nauka", Moscow; 1995:175.

    Google Scholar 

  7. 7.

    Lattès R, Lions J-L: Méthode de Quasi-Réversibilité et Applications, Travaux et Recherches Mathématiques, no. 15. Dunod, Paris; 1967:xii+368.

    Google Scholar 

  8. 8.

    Lavrentiev MM: Some Improperly Posed Problems of Mathematical Physics, Springer Tracts in Natural Philosophy. Volume 11. Springer, Berlin; 1967.

    Google Scholar 

  9. 9.

    Mel'nikova IV: Regularization of ill-posed differential problems. Sibirskiĭ Matematicheskiĭ Zhurnal 1992,33(2):125-134, 221. translated in Siberian Math. J. 33 (1992), no. 2, 289–298

    MathSciNet  Google Scholar 

  10. 10.

    Miller K: Stabilized quasi-reversibility and other nearly-best-possible methods for non-well-posed problems. In Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Edinburgh, 1972), Lecture Notes in Mathematics. Volume 316. Springer, Berlin; 1973:161-176. 10.1007/BFb0069627

    Google Scholar 

  11. 11.

    Payne LE: Some general remarks on improperly posed problems for partial differential equations. In Symposium on Non-Well-Posed Problems and Logarithmic Convexity (Heriot-Watt Univ., Edinburgh, 1972), Lecture Notes in Mathematics. Volume 316. Springer, Berlin; 1973:1-30. 10.1007/BFb0069621

    Google Scholar 

  12. 12.

    Showalter RE: The final value problem for evolution equations. Journal of Mathematical Analysis and Applications 1974,47(3):563-572. 10.1016/0022-247X(74)90008-0

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Showalter RE: Cauchy problem for hyperparabolic partial differential equations. In Trends in the Theory and Practice of Nonlinear Analysis (Arlington, Tex, 1984), North-Holland Math. Stud.. Volume 110. North-Holland, Amsterdam; 1985:421-425.

    Google Scholar 

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Author information

Affiliations

  1. Laboratoire Equations Differentielles, Département de Mathématiques, Faculté des Sciences, Université Mentouri Constantine, Constantine, 25000, Algeria

    M Denche & S Djezzar

Authors
  1. M Denche
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  2. S Djezzar
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Correspondence to M Denche.

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Denche, M., Djezzar, S. A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems. Bound Value Probl 2006, 37524 (2006). https://doi.org/10.1155/BVP/2006/37524

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Convergence Rate