Inverse problems of determining sources of the fractional partial differential equations

Abstract

In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order α ∈ (0, 1). Our survey covers the following types of inverse problems:• determination of time-dependent functions in interior source terms • determination of space-dependent functions in interior source terms • determination of time-dependent functions appearing in boundary conditionsKeywords Fractional diffusion equations · Inverse source problems · Uniqueness · Stability MR(2010) Subject Classification 35R11 · 26A33 · 35R30 · 65M32

The uniqueness of inverse problems for a fractional equation with a single measurement

“…Combining (36) with [40,Theorem 1.1] and using the density of H 3/2 (∂Ω) in H 1/2 (∂Ω), we deduce that B 1 = B 2 . Now choosing ξ = 1 and applying [40, Proposition 2.1] (see also [30,45] for equivalent results stated for magnetic Schrödinger operators), we deduce that ρ 1 = ρ 2 .…”

“…In a recent paper [21], the recovery of a Riemannian manifold without boundary was proved from a single internal measurement of the solution of a fractional diffusion equation with a suitable internal source. Finally, we refer to the review articles [34][35][36] as summaries on the recent progress of inverse problems for time-fractional evolution equations. In the one-dimensional case, we mention also the work of [17], where the recovery of a conductivity coefficient appearing in a parabolic equation from a single measurement at one point was considered.…”

Inverse problems for a half-order time-fractional diffusion equation in arbitrary dimension by Carleman estimates

“…Here we used ψ(x, t) ≤ ψ(x, t 0 ), (x, t) ∈ Q δ . Combining the above inequality with (15), we obtain…”

“…Owing to the practical background, inverse problem for fractional diffusion equations has been a hot topic and there are many theoretical and numerical researches on it. For instance, we refer to the survey papers [15,16] and the references therein.…”

An inverse potential problem for subdiffusion: stability and reconstruction*

“…Note that due to the drastic difference in solution operators, i.e., the fractional case involves Mittag-Leffler functions, the extension is nontrivial. We refer interested readers to [25,26,36] and references therein for related inverse source problems, which are often employed to analyze the generic well-posedness for the inverse potential problem.…”