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