Determination of a part of boundary for the Cauchy problem for the Laplace equation

Abstract

In this article, we use the method of lines combined with a quasi-reversibility method to reconstruct a corrosion boundary for the Laplace equation. This problem is ill-posed, that is, the corrosion boundary does not depend continuously on the given Cauchy data. To restore the stability of approximate boundary on the noises in Cauchy data, we use a modified fourth-order partial differential equation to replace the original Laplace equation. Through the suitable choice of a regularization parameter, one can obtain a stable numerical approximation to the unknown boundary. Numerical examples show that the proposed method is feasible and stable.

Keywords:

• Cauchy problem
• Laplace equation
• quasi-reversibility method
• the method of lines
• boundary identification

Acknowledgements

The work described in this article was supported by the NSF of China (10971089, 10671085) and the program of NCET.