Fourier regularization for a final value time-fractional diffusion problem

Abstract

The evolution process of fractional order describes some phenomenon of anomalous diffusion and transport dynamics in complex system. The equation containing time-fractional derivative provides a suitable mathematical model for describing such a process. The backward problem for this system, which means to recover the initial state for some slow diffusion process from its present status, is very hard to solve due to the nonlocal property of fractional derivative and the irreversibility of time. For this ill-posed problem, we construct a regularizing solution using the Fourier transform method. Both the a-priori choice strategy and the a-posteriori choice strategy for the regularizing parameter are given, with the convergence analysis on the regularizing solution. Numerical implementations are presented to show the validity of the proposed scheme.

Keywords:

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

AMS Subject Classifications:

• 35K57
• 35R25
• 65J20

Acknowledgements

We thank Dr. Liyan Wang very much for the numerical implementations.

Notes

1 This work is supported by NSFC [number 91330109], [number 11171064]; NSF of Jiangsu province [number BK2011583], [number BK2011584]; research foundation of Southeast University [number 3207011102].