Inverse Problems For Fractional Advection Dispersion Equation

Fractional advction-dispersion equation (FADE) is a primary model descrbing solute transport with anomalous non-Fickian behaviors in porous medium. In this dissertation, we will mainly deal with some inverse problems of parameters identification for space FADE, including solutions of the forward problem of FADE, and numerical inversion algorithms for determining model parameters and source coefficients for FADE.Firstly, several kinds of definitions of fractional calculus and derivatives are introduced together with their properties, and comparions with that of integer order are presented. Meantime, general fractional advection dispersion equations are given based on Levy stable laws.In chapter 3, the forward problem of FADE with Dirichlet boundary conditions in a finite domain is solved by finite deifference methods, where the fractional derivative of space is discreted by modified shift Grundwald formula. Numerical stability and convergence of the difference scheme are given, and some numerical examples are presented. Finally, impact of the fractional order on the forward problem is discussed, and numerical simulations are carried out showing that the solution's error becomes small when the fractional order tending to 2.In chapter 4, numerical inversions for parameters of FADE are studied with optimal perturbation regularization algorithms. Several key parameters of FADE, such as the order of fractional derivative, the dispersion coefficient and the average pore-water velocity, are determined by ordinary optimal perturbation algorithm, and furthermore, the above parameters and the source term are determined simultaneously by homotopy regularization algorithm. Quite a few numerical simulations are carried out, and impacts of the fractional order, numerical differential step, regularization parameter and initial iterations on the inversion algorithm are also discussed. Numerical results show that the inversion algorithms here are practicable and effective at least for inverse problems of parameters identification of FADE.Finally in chapter 5, a brief review on the dissertation is given, and some related problems which could be considered for future works are discussed.