Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation

Abstract

An inverse problem for determining the order of the Caputo time-fractional derivative in a subdiffusion equation with an arbitrary positive self-adjoint operator A with discrete spectrum is considered. By the Fourier method it is proved that the value of {\|Au(t)\|}, where {u(t)} is the solution of the forward problem, at a fixed time instance recovers uniquely the order of derivative. A list of examples is discussed, including linear systems of fractional differential equations, differential models with involution, fractional Sturm–Liouville operators, and many others.

Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations

“…As for inverse problems of determining orders and other parameters, we refer to Alimov and Ashurov [3], Ashurov and Umarov [4], Cheng, Nakagawa, Yamamoto and Yamazaki [5], Hatano, Nakagawa, Wang and Yamamoto [9], Janno [10], Janno and Kinash [11], Jin…”

Recovering multiple fractional orders in time-fractional diffusion in an unknown medium

“…The recovery of fractional orders probably has been extensively studied; see [7] for a survey. However, most existing studies focus on recovering one single order in the model (1.4) [8–14], sometimes together with other parameters, e.g. diffusion or potential coefficients, given certain observational data.…”

Recovering Multiple Fractional Orders in Time-Fractional Diffusion in an Unknown Medium

“…The recovery of fractional orders probably is one of the most important inverse problems in the literature, and has been extensively studied; see [19] for a recent survey on many important results the topic. However, most existing studies focus on the case of recovering one single order in the model (1.4) [1,3,9,6,22,21,36], sometimes together with other parameters, e.g., diffusion or potential coefficients, given certain observational data. The only works that we are aware of on recovering multiple fractional orders are [16,20,32].…”