The fractional Landweber method for identifying the space source term problem for time-space fractional diffusion equati
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The fractional Landweber method for identifying the space source term problem for time-space fractional diffusion equation Fan Yang1 · Qu Pu1 · Xiao-Xiao Li1 Received: 24 July 2019 / Accepted: 17 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper is devoted to solve an inverse problem for identifying the source term of a time-fractional nonhomogeneous diffusion equation with a fractional Laplacian in a non-local boundary. Based on the expression of the solution for the direct problem, the inverse problem for searching the space source term is converted into solving the first kind of Fredholm integral equation. The conditional stability for the inverse source problem is investigated. The fractional Landweber method is used to deal with this inverse problem and the regularized solution is also obtained. Furthermore, the convergence rates for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. Several numerical examples are given to show the proposed method is efficient and stable. Keywords Time-fractional diffusion equation · Fractional Laplacian · Identifying the unknown source · Fractional Landweber regularization · Ill-posed problem Mathematics Subject Classification (2010) 35R25 · 47A52 · 35R30
Fan Yang
[email protected] Qu Pu [email protected] Xiao-Xiao Li [email protected] 1
School of Science, Lanzhou University of Technology, Lanzhou, Gansu, 730050, People’s Republic of China
Numerical Algorithms
1 Introduction Time-fractional diffusion equations have attracted much attention recently because of their successful applications in anomalous diffusion and mechanical fields. They can be used to describe the continuous random walk phenomenon [1], option pricing [2], and superdiffusion and subdiffusion phenomena [3]. Due to the memory property of fractional derivatives, time-fractional diffusion equations have advantages in describing hereditary diffusions. Direct problems for time-fractional diffusion equations had been studied extensively; see [4–8] for examples. In some practical cases, some boundary data or initial data, diffusion coefficients, or source terms may not be given; we need additional measurement data to invert them, and this is the inverse problem of the time-fractional diffusion equation. By reading [9–12], we get most of the inverse problems are ill-posed; we need to use the regularization method to solve them. The inverse problems for time-fractional diffusion equations had been investigated recently. For a backward problem of a time-fractional diffusion equation, a lot of research results had been obtained. In [13], Liu et al. used quasi-reversibility to consider a backward problem for a time-fractional diffusion equation. In [14], Wang et al. utilized Tikhonov regularized method to consider a backward problem for the time-fractional diffusion equation. In [15], Wang et al. made use of the truncation method for a backward about a time-fractional dif
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