A modified fractional Landweber method for a backward problem for the inhomogeneous time-fractional diffusion equation in a cylinder

2020 ◽  
Vol 97 (11) ◽  
pp. 2375-2393
Author(s):  
Shuping Yang ◽  
Xiangtuan Xiong ◽  
Yaozong Han
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Backward Problem ◽  
Time Fractional Diffusion Equation ◽  
Landweber Method

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.

Solving a backward problem for a distributed-order time fractional diffusion equation by a new adjoint technique

2020 ◽  
Vol 28 (4) ◽  
pp. 471-488
Author(s):  
Lele Yuan ◽  
Xiaoliang Cheng ◽  
Kewei Liang
Keyword(s):  
Diffusion Equation ◽  
Convergence Rates ◽  
Series Representation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Conditional Stability ◽  
Initial Condition ◽  
Backward Problem ◽  
Time Fractional Diffusion Equation ◽  
Distributed Order

AbstractThis paper studies a backward problem for a time fractional diffusion equation, with the distributed order Caputo derivative, of determining the initial condition from a noisy final datum. The uniqueness, ill-posedness and a conditional stability for this backward problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization. Based on the series representation of the regularized solution, we give convergence rates under an a-priori and an a-posteriori regularization parameter choice rule. With a new adjoint technique to compute the gradient of the functional, the conjugate gradient method is applied to reconstruct the initial condition. Numerical examples in one- and two-dimensional cases illustrate the effectiveness of the proposed method.

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