An Application of Fractional Calculus to Anomalous Diffusion and Imaging in Inhomogeneous Media

In recent years several studies have shown that field transport phenomena in media with non-homogeneous properties are characterized by unconventional behaviors. These processes, usually denoted as anomalous transport phenomena, are accurately described by fractional order mathematical models, whereas the classical integer order models fail to capture their properties. There are several examples of anomalous diffusion throughout the different fields of physics, such as wave propagation and diffusion processes in viscoelastic and heterogeneous media (e.g. soil, porous materials, etc.) fluid flow in porous media, non classical heat transfer. In particular, the diffusion processes in heterogeneous materials have shown to develop anomalous features characterized by non-local behavior due to the onset of long-range interactions. While integer order transport models are not able to explain these effects, fractional order models have shown to be able to capture these phenomena. The aim of this thesis is to investigate the occurrence of anomalous transport mechanisms associated to wave-like fields propagating in highly scattering media and to diffusive fields propagating in inhomogeneous media. Anomalous diffusion models are applicable to complex and inhomogeneous environments where classical diffusion theory ceases to be valid. Anomalous diffusion shows a nonlinear time dependence for the mean-squared displacement, and predicts stretched exponential decay for the temporal evolution of the system response. These unique characteristics of anomalous diffusion enable to probe complex media, with an approach that is not permitted by classical diffusion imaging. The behavior of the initial wave-like field turning into a diffused one will be governed by a classical or anomalous diffusive mechanism depending on the density of the medium. In this work this conversion phenomenon will be studied via a combination of stochastic molecular and fractional continuum models in order to capture the non-local nature of the underlying physics. The numerical analysis presented in this work shows a very distinctive feature of this anomalous mechanism consisting in the transition from Gaussian to heavy-tailed distributions. The unique properties shown by fractional-order models in describing anomalous diffusion phenomena in complex physical systems also suggest their application to inverse problems for tomographic imaging in inhomogeneous and scattering media. In this thesis we consider an application to thermal tomography approach where the heat diffusion in an inhomogeneous medium is described by a fractional differential model. The integration of fractional models in classical tomographic methods, may lead to significant improvements in the sensitivity accuracy and resolution of the reconstruction procedure. These characteristics may be particularly useful in those applications requiring higher level of localized information on the internal structure of the medium investigated.