Reconstruction of Fractional Models from Boundary Measurements

Anomalous diffusion processes in many complex systems such as subsurface flows, human tissues, viscoelastic material, plasma physics, etc., can be modelled by the fractional subdiffusion equation (_0^c)_t^ u(x,t)=_x (p(x) _x u)_ (x,t)+h(t)b(x). Clearly, these processes are characterized by a number of parameters and coefficients such as diffusivity p(x), order of differentiation , boundary conditions at the interface, domain of influence of the source b(x), and the control h(t). In many real world situations, these parameters are not available or accessible for measurements in the interior of the domain of the system. On the other hand, we can only measure a few of these parameters, such as temperature, pressure, and light (tomography) at the boundary. In this project, we propose to develop a new analytical method to identify, from a finite number of boundary measurements the diffusion coefficient and parameters such as the fractional order and boundary conditions. To do so, we will use new tools from spectral estimation to extract spectral data from the measurements at the boundary. Then this data will be used to construct the underlying elliptic operator. The novelty of this new technique is that it enables the reconstruction of the fractional subdiffusion model from just a finite number of measurements at the boundary, and thus avoid the need to access the interior of the system. This is in contrast with the standard methods that require interior data which is called over-determination. Such data would require invasive and sometimes destructive measurements, let alone, it is not always possible to collect. We expect the new technique to be implemented numerically and to lead to fast identification algorithms. As a result, there will be a two-fold impact: The area of inverse problems will witness new non-invasive methods that use data collected on the boundary instead of inside measurements. As we are using only two measurements instead of infinitely many, it follows that control theory engineers can identify parameters in fractional differential equations in finite time. The proposed technique will be useful in many applications. For example, in desalination plants, the porosity of membrane, which is made of porous medium material, varies with time as it gets clogged by salt, sand and debris. Thus by identifying the diffusion coefficient, one can find how to best control the flow at a desired rate.