Source inversion of heat conduction from a finite number of observation data

Abstract

This article deals with a backward heat conduction type problem. Namely, the Gaussian convolution is here analysed in a new way so that inverse source formulae to the heat conduction problem are obtained from a finite number of observation data at time and space points. In view of obtaining this main goal, different reproducing kernel Hilbert spaces, iteration schemes and Tikhonov regularization procedures are used and combined in an unified way.

Keywords:

• Gaussian convolution
• Weierstrass transform
• heat conduction
• reproducing kernel
• Paley–Wiener space
• Tikhonov regularization

AMS Subject Classifications:

• 44A15
• 46E22
• 35K05
• 30C40

Acknowledgements

This work was supported in part by Research Unit Mathematics and Applications, University of Aveiro, Portugal, through FCT – Portuguese Foundation for Science and Technology. Q. Chen (School of Informatics, Guangdong University of Foreign Studies) was also supported by NSF of Guangdong Province (Project No. 8151042001000005), the National Research Foundation for the Doctoral Program of Higher Education of China (SRFDP) under grant 20070512001 and the starting grant of GDUFS. S. Saitoh was also supported in part by the Grant-in-Aid for the Scientific Research (C)(2) (No. 21540111) from the Japan Society for the Promotion of Science. The authors are grateful to Prof Tsutomu Matuura for pointing out some numerical errors in certain integrals from an earlier version of the present work.