An optimal method for fractional heat conduction problem backward in time

Abstract

In this article, we consider a fractional order backward heat conduction problem in two-dimensional space which is associated with a deblurring problem. The problem is seriously ill-posed. We propose an optimal regularization method to solve the problem in the presence of noisy data, and obtain the optimal stability error estimation. A deblurring and denoising experiment shows that the optimal method is comparable with the Tikhonov method.

Keywords:

• ill-posed problems
• image deblurring
• regularization
• error estimate

AMS Subject Classifications::

• 35K37
• 35R25

Acknowledgements

The authors thank the reviewers for their very careful reading and for pointing out several mistakes as well as for their useful comments and suggestions. The first author would like to thank Prof. Y.C. Hon and Dr Leevan Ling for their hospitality whilst visiting City University of Hong Kong. The work described in this article was partially supported by a grant from the Research Council of the Hong Kong Special Administrative Region, China (Project No. CityU 101310), the National Natural Science Foundation of China (Nos 11001223 and 11061030), the Research Fund for the Doctoral Program of Higher Education of China (No. 20106203120001) and the Doctoral Foundation of Northwest Normal University, China (No. 5002-577).