Fractional calculus has been introduced as an e cient tool for modeling physical phenomena, thanks to its memory and hereditary properties. For example, fractional models have been successfully used to describe anomalous di↵usion processes such as contaminant transport in soil, oil flow in porous media, and groundwater flow. These models capture important features of particle transport such as particles with velocity variations and longrest periods.
Mathematical modeling of physical phenomena requires the identification of pa rameters 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 AdvectionDispersion Equation (FADE), which is used to model solute transport in porous media. Identifying parameters for such an equa tion is important to understand how chemical or biological contaminants are trans ported throughout surface aquifer systems. For instance, an estimate of the di↵eren tiation order in groundwater contaminant transport model can provide information about soil properties, such as the heterogeneity of the medium.
Our main contribution is to propose a novel e cient algorithm based on modulating functions to estimate the coe cients and the di↵erentiation order for space FADE,
which can be extended to general fractional Partial Di↵erential Equation (PDE). We also show 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 twostage 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 nonhomogeneous case is derived and existence and uniqueness of the solution are established. In addition, the uniqueness and stability of the inverse problem is studied. Moreover, the modulating functionsbased method is used to solve the problem and it is compared to a standard Tikhonobased optimization technique.
Date of Award  Nov 12 2015 

Original language  English (US) 

Awarding Institution   Computer, Electrical and Mathematical Sciences and Engineering


Supervisor  Taousmeriem Laleg (Supervisor) 

 Inverse Problem
 modulating functions
 fractional advectiondispersion equation