Research On The Simulation Of The Mobile-immobile Solute Transport Models And Inverse Problems

The research on solute transport mainly focus on the phenomenon and mechanism of all kinds of solutes in soil and groundwater.Among them,the migration and transformation of pollutant solutes in porous media such as soil seriously threatens the sustainable development of the world.The mathematical model is used to quantitatively describe the temporal and spatial distribution and migration of pollutants in porous media,which can provide a theoretical basis for pollution control and remediation.In recent years,fractional calculus has been widely used to describe anomalous diffusion phenomena in complex media,especially in porous media,due to its genetic or memory properties.The fractal mobile-immobile(MIM in short)solute transport model is set forth from the classical MIM model based on the fractional-order dynamics.This paper is dedicated to the solution of two types of solute transport models and the research of related inverse problems.The detailed researching work includes:1.In this paper,an integer-fractal MIM solute transport model is derived based on the longtime dynamic behaviors in the immobile zone in a heterogenous porous media.The unique existence of the solution to the forward problem is obtained.A finite difference scheme is put forward to solve this model and the stability and convergence of this implicit difference scheme are proved.Numerical experiments show that the numerical solution can approximate the exact solution well.The related works is reorganized in Chapter 3.2.Two inverse problems of determining the two source terms and identifying the model parameters are considered.Firstly,the inverse problem of determining the two source terms is considered with the additional Dirichlet-Neumann boundary measurements,and numerical inversions are performed by the boundary functional method.Numerical inversions show the effectiveness and stability of the method.This is the main content of Chapter 4.Next,The inverse problem of identifying the fractional order and the degradation coefficient is considered with the additional data at one interior point and the uniqueness is proved which provides a theoretical basis for the construction of inversion algorithm.The homotopy regularization method is applied to perform numerical inversions with noisy data.From the perspective of optimization,the inverse problem is transformed into a minimal problem of cost function,and the inversion solutions coincide with the exact solutions to demonstrate a numerical stability of the inverse problem.The relevant work is collected in Chapter 5.3.Another kind of model is considered,called fractal MIM model which is derived based on the longtime dynamic behaviors both in the mobile zone and immobile zone.Similarly,the unique existence for the solution of the forward problem is proved.An implicit finite difference scheme has been set forth for solving the coupled system,and stability and convergence of the scheme are proved.Numerical simulations with different parameters are presented to reveal the solute transport behaviors in the fractal case.The relevant work is collected in Chapter 6.4.Finally,in this paper,we investigate an inverse problem of identifying the two fractional orders using the additional observations at one interior point in mobile zone.The uniqueness of the inverse problem is proved in the space of Laplace transform by the maximum principle and numerical inversions with noisy data are presented to demonstrate a numerical stability of the inverse problem.The related work is collected in Chapter 7.