An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems
In this paper, we consider a nonhomogeneous differential operator equation of first order u′t+Aut=ft. The coefficient operator A is linear unbounded and self-adjoint in a Hilbert space. We assume that the operator does not have a fixed sign. We associate to this equation the initial or final conditi...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2021/8425564 |
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| author | Nihed Teniou Salah Djezzar |
| author_facet | Nihed Teniou Salah Djezzar |
| author_sort | Nihed Teniou |
| collection | DOAJ |
| description | In this paper, we consider a nonhomogeneous differential operator equation of first order u′t+Aut=ft. The coefficient operator A is linear unbounded and self-adjoint in a Hilbert space. We assume that the operator does not have a fixed sign. We associate to this equation the initial or final conditions u0=Φ or uT=Φ. We note that the Cauchy problem is severely ill-posed in the sense that the solution if it exists does not depend continuously on the given data. Using a quasi-boundary value method, we obtain an approximate nonlocal problem depending on a small parameter. We show that regularized problem is well-posed and has a strongly solution. Finally, some convergence results are provided. |
| format | Article |
| id | doaj-art-ff35a584dfd24040a64cfab11e9d22dd |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-ff35a584dfd24040a64cfab11e9d22dd2025-02-03T01:04:18ZengWileyJournal of Mathematics2314-46292314-47852021-01-01202110.1155/2021/84255648425564An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy ProblemsNihed Teniou0Salah Djezzar1Laboratory of Differential Equations, Department of Mathematics, University Constantine 1, Constantine 25000, AlgeriaLaboratory of Differential Equations, Department of Mathematics, University Constantine 1, Constantine 25000, AlgeriaIn this paper, we consider a nonhomogeneous differential operator equation of first order u′t+Aut=ft. The coefficient operator A is linear unbounded and self-adjoint in a Hilbert space. We assume that the operator does not have a fixed sign. We associate to this equation the initial or final conditions u0=Φ or uT=Φ. We note that the Cauchy problem is severely ill-posed in the sense that the solution if it exists does not depend continuously on the given data. Using a quasi-boundary value method, we obtain an approximate nonlocal problem depending on a small parameter. We show that regularized problem is well-posed and has a strongly solution. Finally, some convergence results are provided.http://dx.doi.org/10.1155/2021/8425564 |
| spellingShingle | Nihed Teniou Salah Djezzar An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems Journal of Mathematics |
| title | An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems |
| title_full | An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems |
| title_fullStr | An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems |
| title_full_unstemmed | An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems |
| title_short | An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems |
| title_sort | approximate solution for a class of ill posed nonhomogeneous cauchy problems |
| url | http://dx.doi.org/10.1155/2021/8425564 |
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