A fractional-order quasi-reversibility method to a backward problem for the time fractional diffusion equation

2021 ◽  
Vol 394 ◽  
pp. 113552
Author(s):  
Wanxia Shi ◽  
Xiangtuan Xiong ◽  
Xuemin Xue
Keyword(s):  
Diffusion Equation ◽  
Fractional Order ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Backward Problem ◽  
Time Fractional Diffusion Equation

Variational method for a backward problem for a time-fractional diffusion equation

2019 ◽  
Vol 53 (4) ◽  
pp. 1223-1244 ◽  
Author(s):  
Ting Wei ◽  
Jun Xian
Keyword(s):  
Weak Solution ◽  
Variational Method ◽  
Diffusion Equation ◽  
Variational Problem ◽  
Fractional Diffusion ◽  
Adjoint Problem ◽  
Fractional Diffusion Equation ◽  
Tikhonov Regularization Method ◽  
Backward Problem ◽  
Time Fractional Diffusion Equation

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.

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