Regularized Solution of the Cauchy Problem for the Biharmonic Equation

In this paper, the Cauchy problem associated with the biharmonic equation is investigated. We prove that in principle, the problem is severely ill-posed in the sense of Hadamard. Therefore, we propose a quasi-boundary value-type regularization method for stabilizing the ill-posed problem. Very sharp...

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Bibliographic Details
Published in:Bulletin of the Malaysian Mathematical Sciences Society 2020-01, Vol.43 (1), p.757-782
Main Authors: Luan, Tran Nhat, Khieu, Tran Thi, Khanh, Tra Quoc
Format: Article
Language:English
Subjects:
Applications of Mathematics
Biharmonic equations
Cauchy problems
Ill posed problems
Mathematics
Mathematics and Statistics
Regularization
Regularization methods
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Summary:In this paper, the Cauchy problem associated with the biharmonic equation is investigated. We prove that in principle, the problem is severely ill-posed in the sense of Hadamard. Therefore, we propose a quasi-boundary value-type regularization method for stabilizing the ill-posed problem. Very sharp convergence estimates are established based on some a priori information on the exact solution. Finally, several numerical examples with random data are provided to show the effectiveness of the proposed method.
ISSN:0126-6705
2180-4206
DOI:10.1007/s40840-018-00711-7