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

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 ∈ 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.