William Rundell

Bangti Jin, William Rundell

Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest. In this paper, we employ a formal analytic and numerical way to examine the degree of ill-posedness of several "classical" inverse problems for fractional differential equations involving a Djrbashian-Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm-Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of "fractional" inverse problems.

Bangti Jin, William Rundell

In this paper, we numerically investigate an inverse problem of recovering the potential term in a fractional Sturm-Liouville problem from one spectrum. The qualitative behaviors of the eigenvalues and eigenfunctions are discussed, and numerical reconstructions of the potential with a Newton method from finite spectral data are presented. Surprisingly, it allows very satisfactory reconstructions for both smooth and discontinuous potentials, provided that the order $α\in(1,2)$ of fractional derivative is sufficiently away from 2.

Barbara Kaltenbacher, William Rundell

Reaction-diffusion equations are one of the most common partial differential equations used to model physical phenomenon. They arise as the combination of two physical processes: a driving force $f(u)$ that depends on the state variable $u$ and a diffusive mechanism that spreads this effect over a spatial domain. The canonical form is $u_t - \triangle u = f(u)$. Application areas include chemical processes, heat flow models and population dynamics. The direct or forwards problem for such equations is now very well-developed and understood. However, our interest lies in the inverse problem of recovering the reaction term $f(u)$ not just at the level of determining a few parameters in a known functional form, but recovering the complete functional form itself. To achieve this we set up the usual paradigm for the parabolic equation where $u$ is subject to both given initial and boundary data, then prescribe overposed data consisting of the solution at a later time $T$. For example, in the case of a population model this amounts to census data at a fixed time. Our approach will be two-fold.First we will transform the inverse problem into an equivalent nonlinear mapping from which we seek a fixed point. We will be able to prove important features of this map such as a self-mapping property and give conditions under which it is contractive. Second, we consider the direct map from $f$ through the partial differential operator to the overposed data. We will investigate Newton schemes for this case. In recent decades various anomalous processes have been used to generalize classical Brownian diffusion. Amongst the most popular is one that replaces the usual time derivative by a fractional one of order $α\leq 1$. We will also include this model in our analysis. The final section of the paper shows numerical reconstructions that demonstrate the viability of the suggested approaches.

Barbara Kaltenbacher, William Rundell

This paper considers the inverse problem of recovering both the unknown, spatially-dependent conductivity $a(x)$ and the potential $q(x)$ in a parabolic equation from overposed data consisting of the value of solution profiles taken at a later time $T$. We show both uniqueness results and the convergence of an iteration scheme designed to recover these coefficients. We also allow a more general setting, in particular when the usual time derivative is replaced by one of fractional order and when the potential term is coupled with a known nonlinearity $f$ of the form $q(x)f(u)$.

William Rundell, Zhidong Zhang

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 $\partialΩ$. The case of a parabolic equation was considered in [HettlichRundell:2001]. 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 behaviour of the process. Some previous inverse problems based on fractional diffusion models have shown considerable differences between classical Brownian diffusion and the anomalous case.