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Applied Mathematics-A Journal of Chinese Universities  2019, Vol. 34 Issue (3): 284-308    DOI: 10.1007/s11766-019-3613-5
    
A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation
CHENG Xiao-liang   YUAN Le-le   LIANG Ke-wei
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation
CHENG Xiao-liang   YUAN Le-le   LIANG Ke-wei
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China
 全文: PDF 
摘要： In this paper, we consider a Cauchy problem of the time fractional diffusion equation
(TFDE) in x ∈ [0,L]. This problem is ubiquitous in science and engineering applications. The illposedness
of the Cauchy problem is explained by its solution in frequency domain. Furthermore,
the problem is formulated into a minimization problem with a modified Tikhonov regularization
method. The gradient of the regularization functional based on an adjoint problem is deduced
and the standard conjugate gradient method is presented for solving the minimization problem.
The error estimates for the regularized solutions are obtained under Hp norm priori bound
assumptions. Finally, numerical examples illustrate the effectiveness of the proposed method.
关键词： Cauchy problem;  time-fractional diffusion equation;  a modified Tikhonov regularization method;  conjugate gradient method;  error estimates    
Abstract: In this paper, we consider a Cauchy problem of the time fractional diffusion equation
(TFDE) in x ∈ [0,L]. This problem is ubiquitous in science and engineering applications. The illposedness
of the Cauchy problem is explained by its solution in frequency domain. Furthermore,
the problem is formulated into a minimization problem with a modified Tikhonov regularization
method. The gradient of the regularization functional based on an adjoint problem is deduced
and the standard conjugate gradient method is presented for solving the minimization problem.
The error estimates for the regularized solutions are obtained under Hp norm priori bound
assumptions. Finally, numerical examples illustrate the effectiveness of the proposed method.
Key words: Cauchy problem    time-fractional diffusion equation    a modified Tikhonov regularization method    conjugate gradient method    error estimates
出版日期: 2019-09-20
CLC:  34A08  
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CHENG Xiao-liang YUAN Le-le LIANG Ke-wei

引用本文:

CHENG Xiao-liang YUAN Le-le LIANG Ke-wei. A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation[J]. Applied Mathematics-A Journal of Chinese Universities, 2019, 34(3): 284-308.

CHENG Xiao-liang YUAN Le-le LIANG Ke-wei. A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation. Applied Mathematics-A Journal of Chinese Universities, 2019, 34(3): 284-308.

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http://www.zjujournals.com/amjcub/CN/10.1007/s11766-019-3613-5    或    http://www.zjujournals.com/amjcub/CN/Y2019/V34/I3/284

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