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

Abstract

We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary valueWe establish the uniqueness for an inverse problem of determining an order α of fractional derivatives by data u(x 0 , t) for 0 < t < T at one point x 0 in a spatial domain Ω. The uniqueness holds even under assumption that Ω and A are unknown, provided that the initial value does not change signs and is not identically zero.The proof is based on the eigenfunction expansions of finitely dimensional approximating solutions, a decay estimate and the asymptotic expansions of the Mittag-Leffler functions for large time.

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

“…This work is also a natural continuation of the authors' recent work [9], where the unique recovery of one single fractional order was proved for the model (1.4) within an unknown medium (e.g. diffusion coefficient, potential coefficient) and scatterer from lateral flux data at one single point (see also [36] for recent work in the case of nonself-adjoint elliptic operators.) Note that the analysis in the work [9] relies heavily on the analyticity of the solution at large time, large-time asymptotic of the two-parameter Mittag-Leffler function E α,β (z) and the strong maximum principle and Hopf's lemma for elliptic problems.…”

“…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].…”

Uniqueness in determining fractional orders of derivatives and initial values

“…The inverse problem of determining parameters including α have been studied in Alimov and Ashurov [3,4], Ashurov and Fayziev [5], Ashurov and Umarov [6], Ashurov and Zunnunov [7], Cheng et al [8], Hatano et al [13], Janno [15], Janno and Kinash [16], Jin and Kian [17], Krasnoschok et al [19], Li et al [21], Li and Yamamoto [24], Sun et al [37], Tatar and Ulusoy [39], Yamamoto [41,42], Yu et al [43], and so on. See Li et al [23] as a survey.…”