# PhD Defense | Methods and Algorithms for Solving Inverse Problems for Fractional Advection Dispersion Equations

Start Date: October 29, 2015
End Date: October 29, 2015

​​By Abeer Aldoghaither PhD Candidate of Professor Taous-Meriem Laleg (KAUST)

Fractional calculus has been introduced as an efficient tool for modeling physical phenomena, thanks to its memory and hereditary properties. For example, fractional models have been successfully used to describe anomalous diffusion processes such as contaminants transport in soil, oil flow in porous media, and groundwater flow. These models capture important features of particles transport such as particles with velocity variations and long rest periods. Mathematical modeling of physical phenomena requires the identification of parameters and variables from available measurements. This is referred to as an inverse problem. In this work, we are interested in studying theoretically and numerically inverse problems for space Fractional Advection Dispersion Equation (FADE), which is used to model so- lute transport in porous media. Identifying parameters for such an equation is important to understand how chemical or biological contaminants are trans- ported throughout surface aquifer system. For instance, an estimate of the differentiation order in groundwater contaminant transport model can provide information about soil properties, such as the heterogeneity of the media. Our main contribution is to propose an efficient novel algorithm based on modulating functions to estimate the coefficients and the differentiation order for space FADE, which can be extended to general fractional Partial Differential Equation (PDE). We also showed how the method can be applied to the source inverse problem. This work is divided into two parts: In part I, the proposed method is described and studied through an extensive numerical analysis. The local convergence of the proposed two-stage algorithm is proven for 1D space FADE. The properties of this method are studied along with its limitations. Then, the algorithm is generalized to the 2D FADE. In part II, we analyze direct and inverse source problems for a space FADE. The problem consists of recovering the source term using final observations. An analytic solution for the non-homogeneous case is derived and existence and uniqueness of the solution are established. In addition, the uniqueness and stability of the inverse problem are studied. Moreover, the modulating functions based method is used to solve the problem and it is compared to a standard Tikhonov based optimization technique.
Biography: Abeer Aldoghaither received her Bachelor’s degree in Mathematics from King Abdulaziz University, Jeddah, Saudi Arabia, and her Master’s degree in Mathematics from George Washington University, Washington, DC, USA. Her research interests are inverse problems for fractional pde’s. Throughout her PhD, Abeer published quality papers and participated in several international conferences.