A backward problem for the time-fractional diffusion equation

Abstract

We consider a backward problem in time for a time-fractional partial differential equation in one-dimensional case, which describes the diffusion process in porous media related with the continuous time random walk problem. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by the quasi-reversibility with fully theoretical analysis and test its numerical performance. With the help of the memory effect of the fractional derivative, it is found that the property of the initial status of the medium can be recovered in an efficient way. Since our solution is established on the eigenfunction expansion of elliptic operator, the method proposed in this article can be used to higher dimensional case with variable coefficients.

Keywords:

• inverse problem
• diffusion process
• fractional derivative
• regularization
• numerics

AMS Subject Classifications::

• 35K57
• 35R25
• 65J20

Acknowledgements

This work is supported by NSFC (No. 10771033). This article was written during Liu's stay from December of 2008 to February of 2009 at the Graduate School of Mathematical Sciences of University of Tokyo, and the stay was supported by Global COE Program ‘The Research and Training Center for New Development in Mathematics’, and he thanks for the support.