Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation

Abstract

In this paper, we investigate a Cauchy problem for the semi-linear elliptic equation. This problem is well known to be severely ill-posed and regularization methods are required. We use a modified quasi-boundary value method to deal with it, and a convergence estimate for the regularized solution is obtained under an a priori bound assumption for the exact solution. Finally, some numerical results show that our given method works well.

Keywords:

• ill-posed problem
• Cauchy problem
• semi-linear elliptic equation
• modified quasi-boundary value method
• convergence estimate

2010 AMS Subject Classifications :

• 35J61
• 45D05
• 45Q05
• 65J20
• 65R30

View correction statement:

Corrigendum on ‘Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation’ (Int. J. Comput. Math., 89(12):1689–1703, 2012. 10.1080/00207160.2012.693174)

Acknowledgements

The author thanks the reviewers for their constructive comments and valuable suggestions that improved the quality of our paper. The work described in this paper was supported by the NSF of China (10971089, 11171136) and the YTSRF of Hexi university (QN2011-10).