A mollification regularization method with the Dirichlet kernel for two Cauchy problems of three-dimensional Helmholtz equation

Abstract

In this paper, two Cauchy problems of Helmholtz equation in a three-dimensional case are considered. To address these problems, a mollification method with bivariate Dirichlet kernel is proposed. Stable errors estimates are obtained based on appropriate a priori choices of mollification parameters. Convergence estimates show that the regularization solution depends continuously on the data and wavenumber. Numerical examples of our interest show that Dirichlet kernel is more effective than the Gaussian kernel under the same parameter selection rule, and our procedure is stable with respect to perturbations noise in the data.

Keywords:

• Cauchy problem
• three-dimensional Helmholtz equation
• Dirichlet kernel
• regularization solution
• error estimate
• numerical experiments

2010 AMS SUBJECT CLASSIFICATIONS:

• 26D15
• 31A25
• 31B20
• 31B35
• 65N21

Acknowledgments

The authors would like to extend sincere gratitude to professor Xiaoli Feng of Xidian University for her instruction advice and useful suggestions on this paper. All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Disclosure statement

The authors declare that they have no competing interests.

Additional information

Funding

The authors would like to thank the National Science Foundation of China [grant numbers 11161036, 11961054].