A kind of operator regularization method for Cauchy problem of the Helmholtz equation in a multi-dimensional case

Abstract

In this paper, a Cauchy problem of Helmholtz equation in a multi-dimensional case is investigated. This problem is severely ill-posed and small perturbations to measurement data can result in large changes in the solution. A kind of operator regularization method is proposed. The stable error estimates are obtained in the $L^2-$norm and $H^r-$norm under the conditions that m is even, md>x, mk>1, and suitable choice of regular parameters. Error estimates show that the regularized solution depends continuously on the perturbation noisy data and wave number. Our method makes up the limitation of small waves. Three numerical experiments show that our proposed method is effective and stable, especially in the case of a large wave number.

Keywords:

• Cauchy problem
• Helmholtz equation
• multi-dimensional
• operator regularization method
• error estimate

2010 Mathematics Subject Classifications:

• 26D15
• 31A25
• 31B30
• 31B35

Acknowledgements

The authors would like to thank the referees for the valuable comments and suggestions that improve the quality of our paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The project is supported by the National Science Foundation of China [grant number 11961054], the National Science Foundation of Ningxia Province [grant number 2020AAC03253] and the Foundation of North Minzu University [grant number 2020KYQD15].