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This paper is concerned with an inverse problem of recovering a potentialterm and fractional order in a one-dimensional subdiffusion problem, whichinvolves a Djrbashian-Caputo fractional derivative of order $\alpha\in(0,1)$ intime, from the lateral Cauchy data. In the model, we do not assume a fullknowledge of the initial data and the source term, since they might beunavailable in some practical applications. We prove the unique recovery of thespatially-dependent potential coefficient and the order $\alpha$ of thederivation simultaneously from the measured trace data at one end point, whenthe model is equipped with a boundary excitation with a compact support awayfrom $t=0$. One of the initial data and the source can also be uniquelydetermined, provided that the other is known. The analysis employs arepresentation of the solution and the time analyticity of the associatedfunction. Further, we discuss a two-stage procedure, directly inspired by theanalysis, for the numerical identification of the order and potentialcoefficient, and illustrate the feasibility of the recovery with severalnumerical experiments.

Recovering the Potential and Order in One-Dimensional Time-Fractional Diffusion with Unknown Initial Condition and Source