Inverse problems of determining coefficients of the fractional partial differential equations

Abstract

When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model, for example, the orders of the fractional derivative or the source term, are often unknown, which requires one to discuss inverse problems to identify these physical quantities from some additional information that can be observed or measured practically. This chapter investigates several kinds of inverse coefficient problems for the fractional diffusion equation.

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

“…Using the elliptic regularity of the operator − + B j • ∇, one can verify the continuity of [0, ∞) ξ −→ N j (ξ) ∈ B(H 3/2 (∂Ω), H 1/2 (∂Ω)) (j = 1, 2). Therefore, making ξ → 0 in condition (35), we obtain…”

“…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.…”

Uniqueness and stability for inverse source problem for fractional diffusion-wave equations

“…For identifying other parameters, such as fractional orders, initial values, coefficients, we can refer to Floridia and Yamamoto [3], Yan and Wei [38], Yan, Zhang and Wei [39]. We also refer to Jin and Rundell [12], Li, Liu and Yamamoto [19], Li and Yamamoto [20] and Liu, Li and Yamamoto [23] for a topical review and a comprehensive list of bibliographies. For numerical treatments on inverse problems for fractional diffusion-wave equations, we refer to Wei and Zhang [35], Xian and Wei [36] and the references therein.…”

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

“…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.…”