Numerical solutions of 3D Cauchy problems of elliptic operators in cylindrical domain using local weak equations and radial basis functions

ABSTRACT

This paper is concerned with the numerical solutions of 3D Cauchy problems of elliptic differential operators in the cylindrical domain. We assume that the measurements are only available on the outer boundary while the interior boundary is inaccessible and the solution should be obtained from the measurements from the outer layer. The proposed discretization approach uses the local weak equations and radial basis functions. Since the Cauchy problem is known to be ill-posed, the Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system of equations. Numerical results of a different kind of test problems reveal that the method is very effective.

KEYWORDS:

• Cauchy problem
• elliptic partial differential equations
• MLPG
• radial basis functions
• Ill-conditioning
• regularization techniques

2010 AMS SUBJECT CLASSIFICATIONS:

• 65N20
• 65N21
• 65N99
• 65D30

Acknowledgments

The author would like to thank Dr Ming Li (Taiyuan University of Technology, China) and Dr Leevan Ling (Hong Kong Baptist University, Hong Kong) for their helpful discussions and introducing me the subject of inverse problems.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Financial supports from the research council of Persian Gulf University is greatly acknowledged.