L1 scheme for solving an inverse problem subject to a fractional diffusion equation

2020/06/08 by Li, Binjie, Xie, Xiaoping, Yan, Yubin
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2006.04291

Abstract

This paper considers the temporal discretization of an inverse problem subject to a time fractional diffusion equation. Firstly, the convergence of the L1 scheme is established with an arbitrary sectorial operator of spectral angle < π/2 , that is the resolvent set of this operator contains \z∈\mathbb C∖\0\: |Arg z|< θ\ for some π/2 < θ< π. The relationship between the time fractional order α∈ (0, 1) and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as α approaches 1 . Then an inverse problem of a fractional diffusion equation is analyzed, and the convergence analysis of a temporal discretization of this inverse problem is given. Finally, numerical results are provided to confirm the theoretical results.

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