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.