Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator
We consider in this paper an abstract parabolic backward Cauchy problem associated with an unbounded linear operator in a Hilbert space 𝐻, where the coefficient operator in the equation is an unbounded self-adjoint positive operator which has a continuous spectrum and the data is given at the final...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2011/913125 |
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| author | Salah Djezzar Nihed Teniou |
| author_facet | Salah Djezzar Nihed Teniou |
| author_sort | Salah Djezzar |
| collection | DOAJ |
| description | We consider in this paper an abstract parabolic backward Cauchy problem associated with an unbounded linear operator in a Hilbert space 𝐻, where the coefficient operator in the equation is an unbounded self-adjoint positive operator which has a continuous spectrum and the data is given at the final time 𝑡=𝑇 and a solution for 0≤𝑡<𝑇 is sought. It is well known that this problem is illposed in the sense that the solution (if it exists) does not depend continuously on the given data. The method of regularization used here consists of perturbing both the equation and the final condition to obtain an approximate nonlocal problem depending on two small parameters. We give some estimates for the solution of the regularized problem, and we also show that the modified problem is stable
and its solution is an approximation of the exact solution of the original problem. Finally, some other convergence results including some explicit convergence rates are also provided. |
| format | Article |
| id | doaj-art-dd08e7a103254cc28bebbc67ff9abdc7 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-dd08e7a103254cc28bebbc67ff9abdc72025-08-20T03:26:30ZengWileyInternational Journal of Differential Equations1687-96431687-96512011-01-01201110.1155/2011/913125913125Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum OperatorSalah Djezzar0Nihed Teniou1Laboratoire Equations Différentielles, Département de Mathématiques, Faculté des Sciences Exactes, Université Mentouri Constantine, Constantine 25000, AlgeriaLaboratoire Equations Différentielles, Département de Mathématiques, Faculté des Sciences Exactes, Université Mentouri Constantine, Constantine 25000, AlgeriaWe consider in this paper an abstract parabolic backward Cauchy problem associated with an unbounded linear operator in a Hilbert space 𝐻, where the coefficient operator in the equation is an unbounded self-adjoint positive operator which has a continuous spectrum and the data is given at the final time 𝑡=𝑇 and a solution for 0≤𝑡<𝑇 is sought. It is well known that this problem is illposed in the sense that the solution (if it exists) does not depend continuously on the given data. The method of regularization used here consists of perturbing both the equation and the final condition to obtain an approximate nonlocal problem depending on two small parameters. We give some estimates for the solution of the regularized problem, and we also show that the modified problem is stable and its solution is an approximation of the exact solution of the original problem. Finally, some other convergence results including some explicit convergence rates are also provided.http://dx.doi.org/10.1155/2011/913125 |
| spellingShingle | Salah Djezzar Nihed Teniou Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator International Journal of Differential Equations |
| title | Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator |
| title_full | Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator |
| title_fullStr | Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator |
| title_full_unstemmed | Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator |
| title_short | Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator |
| title_sort | improved regularization method for backward cauchy problems associated with continuous spectrum operator |
| url | http://dx.doi.org/10.1155/2011/913125 |
| work_keys_str_mv | AT salahdjezzar improvedregularizationmethodforbackwardcauchyproblemsassociatedwithcontinuousspectrumoperator AT nihedteniou improvedregularizationmethodforbackwardcauchyproblemsassociatedwithcontinuousspectrumoperator |