Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source. (7th September 2021)

• Record Type: Journal Article
• Authors: Jin, Bangti; Zhou, Zhi
• Is Part Of: Inverse problems. Volume 37:Number 10(2021)
• Publication Date: 2021-09-07
• Subjects: inverse potential problem -- subdiffusion -- unknown medium -- order determination -- numerical reconstruction; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ac1f6d
• Languages: English
• ISSNs: 0266-5611

Abstract

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.