Logarithmic stable recovery of the source and the initial state of time fractional diffusion equations

Abstract

In this paper we study the inverse problem of identifying a source or an initial state in a timefractional diffusion equation from the knowledge of a single boundary measurement. We derive logarithmic stability estimates for both inversions. These results show that the ill-posedness increases exponentially when the fractional derivative order tends to zero, while it exponentially decreases when the regularity of the source or the initial state becomes larger. The stability estimate concerning the problem of recovering the initial state can be considered as a weak observability inequality in control theory. The analysis is mainly based on Laplace inversion techniques and a precise quantification of the unique continuation property for the resolvent of the time-fractional diffusion operator as a function of the frequency in the complex plane. We also determine a global time regularity for the time-fractional diffusion equation which is of interest itself.

Well-posedness of the stochastic time-fractional diffusion and wave equations and inverse random source problems

“…It would be interesting to investigate what will happen if only the lateral Cauchy data on the partial boundary is used. Moreover, in the very recent paper [22], the authors considered the inverse problem of identifying a source or an initial state in a timefractional diffusion equation from a single boundary measurement on time interval (0, ∞), and the logarithmic stability estimates were established by the use of Laplace inversion techniques and the unique continuation quantification for the resolvent of fractional diffusion operator as a function of the frequency in the complex plane. It will be challenging to investigate the stability of the inverse source problems for the stochastic fractional diffusion-wave equations under the finite time measurements.…”