The Quasi-Boundary Value Method for Identifying the Initial Value of the Space-Time Fractional Diffusion Equation
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
THE QUASI-BOUNDARY VALUE METHOD FOR IDENTIFYING THE INITIAL VALUE OF THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION∗
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Fan YANG (
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Yan ZHANG (
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Xiao LIU (
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Xiaoxiao LI (
Department of Mathematics, Lanzhou University of Technology, Lanzhou 730050, China E-mail : [email protected]; [email protected]; [email protected]; [email protected] Abstract In this article, we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation. This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem. We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule. Some numerical results in one-dimensional case and two-dimensional case show that our method is efficient and stable. Key words
Space-time fractional diffusion equation; Ill-posed problem; quasi-boundary value method; identifying the initial value
2010 MR Subject Classification
1
35R25; 47A52; 35R30
Introduction
Fractional diffusion equations have attracted much more attentions recently because they were widely used in model anomalous diffusion and mechanical fields. They can be used to describe the physics [1], chemistry and biochemistry [2], signal processing, mechanical engineering and systems identification [3], medicine and finance [4], electrical and fractional dynamics [5]. Owing to the memory property of the fractional derivative, the fractional diffusion equation has an advantage in describing the hereditary diffusion. In some practical applications, for fractional diffusion equations, sometimes the initial value, part of the boundary value, diffusion coefficient or source term is unknown. If we recover some of them by measurement data, we can deduce inverse problems for fractional diffusion equations. There have been some results for the inverse problem of fractional diffusion equations, such as the identification of source term [6–17], the backward diffusion problem [18–21], initial data problem [22–28], the Cauchy problem [29, 33], and the reconstruction of diffusion coefficient [34, 35]. From the above work, we can find that the existing results mainly focus on time-fractional diffusion equations, rather than the more general space-time fractional diffusion equations. As pointed out by B. Jin and ∗ Received September 4, 2019. The project is supported by the National Natural Science Foundation of China (11561045, 11961044) and the Doctor Fund of Lan Zhou University of Technology.
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
W. Rundell’s article [36], the study of space-fractional inverse problem, either theoretical or numerical, is quite scarce. In [37], the authors proved the uniqueness of the inverse sour
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