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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 2314-4629 2314-4785 |