Spacfe-time Fractional Advection Dispersion Equation And Inverse Problems

In this dissertation we use the inversion of the fractional order and the diffusion coefficient of the fractional advection dispersion equation as application background.We discuss the homotopy perturbation method in parameter inverse problem for space-time fractional advection dispersion equation and two-term time fractional dispersion equation.Numerical examples are given in different classification focusing on the space-time fractional order,the diffusion coefficient and some factors influencing the algorithm to work.Numerical simulations show that the inversion algorithm can acurrately reflect the fractional diffusion,and the inversion result is effective when there is some disturbance with the additional data.The main contents of this dissertation are given as follows:In chapter 1,background and significance of this dissertation are simply introduced,and the main researching works and structure of this dissertation are discussed.In chapter 2,homotopy method and Sigmoid neural network function is introduced,leading to a homotopy perturbation method based on optimal perturbation.In chapter 3,we investigate the forward problem and inverse problems for the space-time fractional advection-dispersion equation.The forward problem is solved by finite difference scheme.The stability and convergence are proved with several theorems,and numerical simulations are also presented.Executing numerical simulation with homotopy perturbation method in different additional data,inversion of the space and time fractional order is the emphasis as well as the diffusion coefficient depending on space.In chapter 4,we discuss the model of two-term time fractional dispersion equation and the inverse problem of determining the two fractional orders.Finite difference scheme is applied to find the numerical solution.The stability and convergence are proved with similar theorems like in chapter 3.With the terms of Mittag-Leffler function,the exact solution of the forward problem is given.The numerical simulation is also executed with homotopy perturbation method but just in interior point form of additional data,and we focus on the inversion of the two time fractional order.In chapter 5,a summary of the thesis is given,and some related problems which could be considered for the future work are discussed.