- Research Article
- Open Access
A modified quasi-boundary value method for a class of abstract parabolic ill-posed problems
- M Denche1Email author and
- S Djezzar1
https://doi.org/10.1155/BVP/2006/37524
© Denche and Djezzar 2006
- Received: 14 October 2004
- Accepted: 9 August 2005
- Published: 13 February 2006
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.
Keywords
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Equation
- Convergence Rate
Authors’ Affiliations
References
- 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
- 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/S0218202598000093MathSciNetView ArticleMATHGoogle Scholar
- 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/S0308210500003036MathSciNetView ArticleMATHGoogle Scholar
- Clark GW, Oppenheimer SF: Quasireversibility methods for non-well-posed problems. Electronic Journal of Differential Equations 1994,1994(8):1-9.MathSciNetGoogle Scholar
- 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.2004.08.001MathSciNetView ArticleMATHGoogle Scholar
- Ivanov VK, Mel'nikova IV, Filinkov AI: Operator-Differential Equations and Ill-Posed Problems. Fizmatlit "Nauka", Moscow; 1995:175.MATHGoogle Scholar
- Lattès R, Lions J-L: Méthode de Quasi-Réversibilité et Applications, Travaux et Recherches Mathématiques, no. 15. Dunod, Paris; 1967:xii+368.Google Scholar
- Lavrentiev MM: Some Improperly Posed Problems of Mathematical Physics, Springer Tracts in Natural Philosophy. Volume 11. Springer, Berlin; 1967.View ArticleGoogle Scholar
- 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, 289–298MathSciNetGoogle Scholar
- 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/BFb0069627View ArticleGoogle Scholar
- 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/BFb0069621View ArticleGoogle Scholar
- 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)90008-0MathSciNetView ArticleMATHGoogle Scholar
- 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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This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
