Modeling,Discretization,Simulation,Analysis And Application Of Anomalous And Ultra-Slow Diffusion

With the development of research,people have discovered anomalous and ultra-slow diffusion phenomena in quantum physics,economics,random analysis of option market and various biological systems.This thesis concentrates on the topic of "Exploring the physical mechanism of anomalous and ultra-slow diffusion phenomena".As we all known that,as a non-local operator,the fractional partial differential equation has a very important application in describing anomalous diffusion,such as material transport in heterogeneous porous media.The integral forms in the definitions of the Hadamard and Caputo-Hadamard fractional derivatives contain the logarithmic kernel.Therefore,fractional partial differential equations containing these two types of derivatives have a natural advantage in describing ultra-slow diffusion behavior such as the logarithmic change of mean square error of particles with diffusion time in complex media and tissues.Therefore,the first and second parts of this thesis focuses on the numerical analysis and parameter inversion of Caputo-Hadamard time-fractional partial differential equations.The diffusion of water through biological tissues is not completely random and can be hindered and restricted by various factors.Diffusion-weighted magnetic resonance imaging is a sub-microscale sensitivity tool for characterizing relevant material and tissue structural features based on measurements of endogenous water diffusion displacement.It can quickly identify tissue deficiency or injuries and has high tissue contrast and repeatability,so it can not only make qualitative diagnosis of diseases,but also contribute to the quantitative evaluation of subsequent treatment.Therefore,the third part of this thesis mainly focus on the research of the correlation model of signal attenuation in diffusion-weighted magnetic resonance imaging.The full text consists of the following six chapters.In first chapter,we briefly introduce several common definitions of fractional derivatives and give the main content and structure of this paper.Next we briefly introduce the principle of diffusion-weighted magnetic resonance imaging and mention the importance of modeling in the field of magnetic resonance imaging.Then we briefly introduce anomalous diffusion and finally give the main content and structure of this paper.In chapter 2,we prove the existence and uniqueness of the solution to the initial value problem of Caputo-Hadamard fractional ordinary differential equation.The existence,uniqueness and regularity theorems are briefly introduced according to the existing articles.Then,we give the fully discrete finite element scheme of the equation,the corresponding truncation error analysis and the optimal error estimation of the finite element numerical scheme.The numerical results verify the good agreement with the theoretical analysis.In chapter 3,we concentrate the parameter inversion problem of CaputoHadamard time fractional partial differential equation.A Levenberg-Maxquardt finite element method is developed.Numerical examples also verify the effectiveness of the proposed method.In chapter 4,we prove the existence and uniqueness of the solution for a variably distributed-order Caputo-Hadamard time-fractional diffusion equation and analyze the wellposedness of the solution.Accordingly,we give a numerical scheme of the finite element method for the model problem.Then we give the analysis of the truncation errors.In chapter 5,we investigate the correlation models that characterizes signal attenuation in diffusion-weighted magnetic resonance imaging.We mainly investigate the performance and difference of each model in distinguishing between normal and abnormal diffusion sets contained in experimental data of different directions.We reduce the fitting error by Optimization of algorithm or fine-tuning existing model structural components based on physical mechanisms.Then through physical analysis and biological characteristics analysis,some specific parameters of the existing model are characterized as functions of b value,and a variety of new variable parameter models are proposed.Then we introduce machine learning method to inversely evolve the unknown parameter expression in the parameter model according to the existing experimental data.In chapter 6,we give the summary of this dissertation and the future research work prospects.