Simultaneous inversion of the potential term and the fractional orders in a multi-term time-fractional diffusion equation

Abstract

In the present paper, we devote our effort to a nonlinear inverse problem for simultaneously recovering the potential function and the fractional orders in a multi-term time-fractional diffusion equation from the noisy boundary Cauchy data in the one-dimensional case. The uniqueness for the inverse problem is derived based on the analytic continuation, the Laplace transformation and the Gel’fand–Levitan theory. Finally, the Levenberg–Marquardt regularization method with a regularization parameter chosen by a sigmoid-type function is applied for finding a stable approximate solution. Three numerical examples are provided to show the effectiveness of the proposed method.

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

“…Note that to recover the orders αj and weights rj, one classical approach is to apply the regularization method, e.g. Tikhonov regularization [27], which involves a misfit on the measured data (and proper regularization), with the forward map defined implicitly by problem (1.1) (as done recently in [12,17] for recovering one single order). Unfortunately, this approach does not apply in the setting of this work, since the medium (and thus the forward map) is unknown.…”

“…diffusion or potential coefficients, given certain observational data. The only works on recovering multiple orders are [15–17]. Li & Yamamoto [16] proved the unique recovery of multiple orders in two cases: (i) the uniqueness in simultaneously identifying falsefalse{false(αi,rifalse)falsefalse}i=1N when d=1 and u0=δfalse(x−x∗false) (the Dirac delta function concentrated at x∗∈Ω) by measured data at one endpoint; and (ii) the uniqueness in determining falsefalse{false(αi,rifalse)falsefalse}i=1N when d≥1 and u0∈L2false(Ωfalse) by interior measurement.…”

“…Sun et al . [17] proved the unique recovery of the orders and the potential q (for a one-dimensional problem) using the Gel’fand–Levitan theory for Sturm–Liouville problems. All these existing works assume a fully known forward model.…”

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

“…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]. Li and Yamamoto [20] proved the unique recovery of multiple orders in two cases: (i) the uniqueness in simultaneously identifying {(α i , r i )} N i=1 when d = 1 and u 0 = δ(x − x * ) (the Dirac delta function concentrated at x * ∈ Ω) by measured data at one endpoint; (ii) the uniqueness in determining {(α i , r i )} N i=1 when d ≥ 1 and u 0 ∈ L 2 (Ω) by interior measurement.…”

Uniqueness in determination of the fractional order in the TFDE using one measurement

“…The inverse problems of identifying parameters including fractional orders have been studied during the last decade. We refer to Alimor and Ashurov [3], Ashurov and Umarov [4], Ashurov and Zunnunov [5], Cheng et al [9], Hatano et al [18], Janno [19], Janno and Kinash [20], Jin and Kian [23], Li et al [26], Li et al [30], Li and Yamamoto [28], Sun et al [40], Tatar and Ulusoy [41], Yamamoto [45,46], Yu et al [47], and so on.…”