Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source

Abstract

This paper is concerned with an inverse problem of recovering a potential term and fractional order in a one-dimensional subdiffusion problem, which involves a Djrbashian-Caputo fractional derivative of order α ∈ (0, 1) in time, from the lateral Cauchy data. In the model, we do not assume a full knowledge of the initial data and the source term, since they might be unavailable in some practical applications. We prove the unique recovery of the spatially-dependent potential coefficient and the order α of the derivation simultaneously from the measured trace data at one end point, when the model is equipped with a boundary excitation with a compact support away from t = 0. One of the initial data and the source can also be uniquely determined, provided that the other is known. The analysis employs a representation of the solution and the time analyticity of the associated function. Further, we discuss a two-stage procedure, directly inspired by the analysis, for the numerical identification of the order and potential coefficient, and illustrate the feasibility of the recovery with several numerical experiments.