| Issue |
ESAIM: M2AN
Volume 53, Number 4, July-August 2019
|
|
|---|---|---|
| Page(s) | 1223 - 1244 | |
| DOI | https://doi.org/10.1051/m2an/2019019 | |
| Published online | 09 July 2019 | |
Variational method for a backward problem for a time-fractional diffusion equation
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P.R. China
* Corresponding author: tingwei@lzu.edu.cn
Received:
18
May
2017
Accepted:
13
March
2019
This paper is devoted to solve a backward problem for a time-fractional diffusion equation by a variational method. The regularity of a weak solution for the direct problem as well as the existence and uniqueness of a weak solution for the adjoint problem are proved. We formulate the backward problem into a variational problem by using the Tikhonov regularization method, and obtain an approximation to the minimizer of the variational problem by using a conjugate gradient method. Four numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed algorithm.
Mathematics Subject Classification: 65M32 / 35R11
Key words: Backward problem / fractional diffusion equation / Tikhonov regularization / variational method / conjugate gradient method
© EDP Sciences, SMAI 2019
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