Fractional diffusion: recovering the distributed fractional derivative from overposed data

Abstract

Abstract. There has been considerable recent study in "subdiffusion" models that replace the standard parabolic equation model by a one with a fractional derivative in the time variable. There are many ways to look at this newer approach and one such is to realize that the order of the fractional derivative is related to the time scales of the underlying diffusion process. This raises the question of what order α of derivative should be taken and if a single value actually suffices. This has led to models that combine a finite number of these derivatives each with a different fractional exponent α k and different weighting value c k to better model a greater possible range of time scales. Ultimately, one wants to look at a situation that combines derivatives in a continuous way -the so-called distributional model with parameter µ(α).However all of this begs the question of how one determines this "order" of differentiation. Recovering a single fractional value has been an active part of the process from the beginning of fractional diffusion modeling and if this is the only unknown then the markers left by the fractional order derivative are relatively straightforward to determine. In the case of a finite combination of derivatives this becomes much more complex due to the more limited analytic tools available for such equations, but recent progress in this direction has been made, [16,14]. This paper considers the full distributional model where the order is viewed as a function µ(α) on the interval (0, 1]. We show existence, uniqueness and regularity for an initial-boundary value problem including an important representation theorem in the case of a single spatial variable. This is then used in the inverse problem of recovering the distributional coefficient µ(α) from a time trace of the solution and a uniqueness result is proven.

Initial-boundary value problem for distributed order time-fractional diffusion equations

“…which entails t → t It remains to prove that u is expressed by (24) with F = 0. This can be done by noticing for all s ∈ γ(ε, θ) that the function t → Y (s)e st u 0 is twice continuously differentiable in R + and for all t ∈ (0, +∞) that s → s k Y (s)e st u 0 ∈ L 1 (γ(ε, θ)) with k = 1, 2.…”

“…One of them uses time-fractional diffusion equations of distributed order (DO), see e.g. [13,20,24], but we also mention that a diffusion model with variable fractional order time-derivative was proposed in [27] to depict ultraslow diffusion processes. In this paper, we are concerned with the DO fractional diffusion model.…”

An inverse random source problem in a stochastic fractional diffusion equation

“…The article [32] considered an inverse source problem in an FDE, which was close to this work. The authors in [31,42] analyzed the distributed differential equations, in which the assumption (∆x) 2 ∝ t α was extended to a more general case (∆x) 2 ∝ F (t), and studied some inverse problems in such equations. For an extensive review of the field we refer to [25] and references therein.…”

Inverse problems of determining parameters of the fractional partial differential equations

“…Theorem 5.2 (Rundell and Zhang [27]). In (2), we assume that Ω = (0, 1), a(x) = 0 for 0 < x < 1 and u(1, t) = 0, u(0, · ) ∈ L ∞ (0, ∞), ≡ 0.…”