Fraction diffusion: Recovering the distributed fractional derivative from overposed data

Abstract

There has been considerable recent study in "sub-diffusion" 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 ? 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, [9, 8]. 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.