Uniqueness and numerical scheme for the Robin coefficient identification of the time-fractional diffusion equation

A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition

“…x ∈ Ω (1.1) * Corresponding author: Prof. J.J.Liu, email: jjliu@seu.edu.cn with time-fractional order derivative, where ν(x) is the outward unit normal direction on ∂Ω, and F (x, t), b(x, t), u 0 (x) are internal source, boundary source and the initial status of the diffusion process, respectively. The Robin boundary condition in (1.1) with impedance coefficient λ(x) > 0 describes the convection between the solute in a body and one in the ambient environment [20], which is physically important. The time-fractional derivative ∂ α 0+ u is the Caputo derivative of order 0 < α < 1 defined by…”

“…For the Robin boundary condition with impedance coefficient depending on (x, t), the existence and continuous dependence of the solution in C(Ω T ) was proved by integral equation method in a bounded domain with Lyapunov boundary [8] under the continuous regularity of (F (x, t), b(x, t), u 0 (x)). In one-dimensional spatial case, Wei and Wang considered the existence and uniqueness of a weak solution with Robin coefficient depending on t [20,22].…”

On the simultaneous reconstruction of boundary Robin coefficient and internal source in a slow diffusion system

“…where ν(x) is the outward normal direction on ∂Ω, and F(x, t), b(x, t), u 0 (x) are internal source, boundary source and initial status of the diffusion process, respectively. The boundary impedance coefficient λ(x) > 0 is the so-called Robin coefficient describing the convection between the solute in a body and one in the ambient environment through the boundary [29].…”

“…As for boundary Robin coefficient inversion for time-fractional diffusion equation, the recovery of time-dependent Robin coefficient was studied from boundary measurement for one-dimensional diffusion equation in [33,34], where the inverse problem was reformulated as a regularized optimization problem for solving a nonlinear Volterra integral equation of the first kind. Recently [29] showed the uniqueness of the inversion of time-dependent Robin coefficient from a nonlocal boundary condition.…”

Hopf's lemma and uniqueness of simultaneously determining source profile and Robin coefficient in a fractional diffusionequation by interior data

“…In pollution tracking and prevention in ground water, lakes and rivers, slow diffusion processes may occur, which can be well modeled by system (1). The spatial component f (x) denotes the spatial distribution of the source and the Robin coefficient γ(x) describes the convection between the solute in a body and one in the ambient environment [39]. Hence, knowing them is indispensable to a good understanding of the slow diffusion processes.…”