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.

We consider a fractional diffusion equation (FDE) with an undetermined time-dependent diffusion coefficient a(t). Firstly, for the direct problem part, we establish the existence, uniqueness and some regularity properties of the weak solution for this FDE with a fixed Secondly, for the inverse problem part, in order to recover we introduce an operator and show its monotonicity. With this property, we establish the uniqueness of a(t) and create an efficient reconstruction algorithm to recover this coefficient.

In this work the authors consider an inverse source problem in the following stochastic fractional diffusion equationThe interested inverse problem is to reconstruct f (x) and g(x) by the statistics of the final time data u(x, T ). Some direct problem results are proved at first, such as the existence, uniqueness, representation and regularity of the solution. Then the reconstruction scheme for f and g is given. To tackle the ill-posedness, the Tikhonov regularization is adopted. Finally we give a regularized reconstruction algorithm and some numerical results are displayed.

A standard inverse problem is to determine a source which is supported in an unknown domain D from external boundary measurements. Here we consider the case of a time-dependent situation where the source is equal to unity in an unknown subdomain D of a larger given domain Ω. Overposed measurements consist of time traces of the solution or its flux values on a set of discrete points on the boundary ∂Ω. The case of a parabolic equation was considered in [5]. In our situation we extend this to cover the subdiffusion case based on an anomalous diffusion model and leading to a fractional order differential operator. We will show a uniqueness result and examine a reconstruction algorithm. One of the main motives for this work is to examine the dependence of the reconstructions on the parameter α, the exponent of the fractional operator which controls the degree of anomalous behavior of the process. Some previous inverse problems based on fractional diffusion models have shown considerable differences between classical Brownian diffusion and the anomalous case.