Simultaneous determination of different class of parameters for a diffusion equation from a single measurement

Abstract

This article is devoted to the simultaneous resolution of several inverse problems, among the most important formulation of inverse problems for partial differential equations, stated for some class of diffusion equations from a single boundary measurement. Namely, we consider the simultaneous unique determination of several class of coefficients, some internal sources (a source term and an initial condition) and the fractional order in a diffusion equation from a single boundary measurement. Our problem can be formulated as the simultaneous determination of information about a diffusion process (velocity field, density of the medium) and of the source of diffusion. We consider this problems in the context of a classical diffusion process described by a convection-diffusion equation as well as an anomalous diffusion phenomena described by a time fractional diffusion equation.

Inverse source problem with a posteriori boundary measurement for fractional diffusion equations

“…Most of the above‐mentioned results are stated with measurement throughout the full interval of time $$$ \left(0,T\right) $$$. We are only aware of the three results [24–26] considering measurement during an interval of time of the form $$$ \left(T-\epsilon, T\right) $$$ with $$$ \epsilon \in \left(0,T\right) $$$ arbitrary small. In all these three works, the authors use the memory effect of time‐fractional diffusion equations exhibited by Kinash and Janno [24, Theorem 1] which cannot be applied in the context of the inverse problems (IP1)–(IP3).…”

“…Let us observe that the results of Theorems 1.1,1.2, and 1.3 are all stated with a posteriori boundary measurement restricted to an arbitrary small interval of time of the form $$$ \left(T-\epsilon, T\right) $$$ where $$$ T $$$ denotes the final time. We are only aware of the two articles [25, 26] studying this class of inverse source problems with such data. While the results of Kian et al [26] are stated with internal data, in the results of Kian [25], the measurements are given by $$$ {\partial}_{\nu_a}u\left(x,t\right) $$$ and $$$ {\partial}_t^{\alpha }{\partial}_{\nu_a}u\left(x,t\right),\left(x,t\right)\in \Gamma \times \left(T-\epsilon, T\right) $$$ with $$$ \Gamma $$$ an arbitrary open subset of $$$ \mathrm{\partial \Omega } $$$ and with $$$ \epsilon \in \left(0,T\right) $$$ arbitrary small.…”

“…We are only aware of the two articles [25, 26] studying this class of inverse source problems with such data. While the results of Kian et al [26] are stated with internal data, in the results of Kian [25], the measurements are given by $$$ {\partial}_{\nu_a}u\left(x,t\right) $$$ and $$$ {\partial}_t^{\alpha }{\partial}_{\nu_a}u\left(x,t\right),\left(x,t\right)\in \Gamma \times \left(T-\epsilon, T\right) $$$ with $$$ \Gamma $$$ an arbitrary open subset of $$$ \mathrm{\partial \Omega } $$$ and with $$$ \epsilon \in \left(0,T\right) $$$ arbitrary small. In both of these results, the authors apply the memory effect exhibited in Kinash and Janno [24, Theorem 1] in order to determine from these class of data restricted to in interval of time of the form $$$ \left(T-\epsilon, T\right) $$$ similar type of data on the full interval $$$ \left(0,T\right) $$$.…”

“…Let us observe that the results of Theorems 1.1,1.2, and 1.3 are all stated with a posteriori boundary measurement restricted to an arbitrary small interval of time of the form (T − 𝜖, T) where T denotes the final time. We are only aware of the two articles [25,26] studying this class of inverse source problems with such data. While the results of Kian et al [26] are stated with internal data, in the results of Kian [25], the measurements are given by 𝜕 𝜈 a u(x, t) and 𝜕 𝛼 t 𝜕 𝜈 a u(x, t), (x, t) ∈ Γ×(T − 𝜖, T) with Γ an arbitrary open subset of 𝜕Ω and with 𝜖 ∈ (0, T) arbitrary small.…”

The enclosure method for the detection of variable order in fractional diffusion equations

“…We refer to [31] for a survey about this topic (see also [25] for an overview of inverse problems for fractional diffusion equations). Without being exhaustive we can mention the works of [2,7,11,21,22,26,34,30,32,33,41] devoted to the determination of single or multiple constant fractional orders, sometimes together with other parameters (coefficients or internal sources), from several class of observational data. We mention also the recent works [23,24] where the determination of constant fractional order have been studied in the context of an unknown medium (unknown source, coefficients, domain...).…”