A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation

2019 ◽  
Vol 34 (3) ◽  
pp. 284-308
Author(s):  
Xiao-liang Cheng ◽  
Le-le Yuan ◽  
Ke-wei Liang
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Tikhonov Regularization Method ◽  
Time Fractional Diffusion Equation

Inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach

2019 ◽  
Vol 27 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Xiong Bin Yan ◽  
Ting Wei
Keyword(s):  
Diffusion Equation ◽  
Tikhonov Regularization ◽  
Direct Problem ◽  
Fractional Diffusion ◽  
Adjoint Problem ◽  
Fractional Diffusion Equation ◽  
Gradient Algorithm ◽  
Tikhonov Regularization Method ◽  
Series Expression ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, we consider an inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach; that is, to determine the space-dependent source term from a noisy final data. Based on the series expression of the solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions, and we provide the existence and uniqueness for the adjoint problem. Further, we use the Tikhonov regularization method to solve the inverse source problem and provide a conjugate gradient algorithm to find an approximation to the minimizer of the Tikhonov regularization functional. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.

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