Numerical method for a Cauchy problem for multi-dimensional Helmholtz equation

Abstract

The main focus of this research is to address the Cauchy problem of the multi-dimensional Helmholtz equation with mixed boundary conditions. This problem is known to be ill-posed according to Hadamard's definition. To tackle this issue, we propose the mollification regularization method based on exponential decay. Using prior rule and posterior rule to create a regular approximation solution and convergence of the solution is also provided. Furthermore, the proposed method is shown to be robust against data disruption noise, making it a reliable approach for solving the Cauchy problem of the multi-dimensional Helmholtz equation with mixed boundary conditions.

Keywords:

• Multi-dimensional Helmholtz equation
• mollification method
• exponential decay kernel
• Cauchy problem
• error estimate

Mathematics Subject Classifications:

• 35-00
• 35C06
• 35C05
• 65M32
• 35R30

Acknowledgments

We would like to express our sincere gratitude to the referees and editor for their diligent review of our paper and for providing valuable recommendations. Their insightful feedback has greatly improved the quality of our research!

Disclosure statement

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

Additional information

Funding

The financial support provided by the National Natural Science Foundation of China (No. 11961054) and the Natural Science Foundation of Ningxia (No. NZ16011).