Inverse problems of determining parameters 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 related to orders of the fractional derivatives, are often unknown and difficult to be directly measured, which requires one to discuss inverse problems of identifying these physical quantities from some indirectly observed information of solutions. Inverse problems in determining these unknown parameters of the model are not only theoretically interesting, but also necessary for finding solutions to initial-boundary value problems and studying properties of solutions. This chapter surveys works on such inverse problems for fractional diffusion equations.

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

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

“…Theorem 1. 5 gives a positive answer to the problem posed in review article by Z. Li et al [1] (p. 440) in the Conclusions and Open Problems section: Is it possible to identify uniquely the order of fractional derivatives if an additional information about the solution is specified at a fixed time instant as "the observation data"?.…”

“…The inverse problem of determining the order of time fractional derivative in subdiffusion equations has been studied by a number of authors (see a survey paper [1] and references therein, [13]- [23]). It is necessary to note that in all these publications the following relation was taken as an additional condition (1.4) u(x0, t) = h(t), 0 < t < T, at a monitoring point x0 ∈ Ω.…”

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

“…Now we situate the uniqueness results in existing literature. 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.…”