Numerical Analyses For Anomalous Dynamic Models Driven By Internal And External Noise

Mathematically,scholars often use noise to describe the uncertain factors of various complex systems.In recent years,anomalous dynamic models driven by internal and external noise have played an important role in the study of anomalous diffusion.However,due to the singularity and non-locality of fractional operators and the low regularity of noise,it is usually difficult to give analytical solutions of such equations.Therefore,establishing numerical schemes of relevant equations and giving accurate numerical analyses are of great significance to explore the mechanism of anomalous dynamics,but they are also very challenging.In this paper,we will give numerical algorithms for several kinds of fractional and stochastic fractional diffusion equations,and establish the complete numerical analyses for them.The details are as follows:In Chapter 1,we introduce the background of the problems studied in this paper,review and summarize the development and research status of numerical solutions for fractional and stochastic fractional diffusion equations;then we briefly describe the main research works and innovations of this paper.In Chapter 2,we build the numerical algorithm of the time-space fractional Fokker–Planck equations with two internal states.Because of the coupling of the solutions and the involved two different fractional Laplacian operators in the system,it is difficult to make full use of the regularity of initial value in the regularity and convergence analyses.Here we first use the elliptic regularity of fractional Laplacian and interpolation theory to establish the accurate estimates of the resolvent operators in different operator norms,and then the optimal regularity estimates of the solutions are given under different initial value regularity assumptions.Also,we establish the fully discrete scheme and its convergence analysis.In Chapter 3,we develop the numerical scheme for the time fractional diffusion equation driven by fractional Gaussian noise with Hurst index H∈(1/2,1).We use the properties of resolvent operator to build the regularity theory of solution,and then the error estimate of the fully discrete scheme is provided by means of Laplace transform and other tools.In this work,we improve the existing numerical analysis methods of stochastic fractional partial differential equations,so that they no longer rely on the explicit expression of the solution,and can reflect the relationship between the noise regularity and the convergence order.In Chapter 4,the numerical algorithm and theoretical analysis are provided for the time-space fractional diffusion equation driven by fractional Gaussian noise with Hurst index H∈(1/2,1).Due to the singularity and non-locality of integral fractional Laplacian,it is difficult to give the properties of their eigenfunctions,which makes a great challenge for the regularity and error analysis.First,thanks to the built resolvent estimate of integral fractional Laplacian by variational method,the optimal regularity estimate of the solution is obtained,and then we provide the convergence of finite difference-finite element scheme by introducing the fractional Ritz projection operator.In Chapter 5,we establish the unified regularity and convergence analysis of the time-space fractional diffusion equation driven by fractional Gaussian noise with Hurst index H∈(0,1).Because It^o isometry of fractional Brownian motion is quite different when H∈(0,1/2), = 1/2 and H∈(1/2,1),there is still no research work to give a unified numerical analysis for H∈(0,1).Here,we establish a unified characterization of It^o isometry with Hurst index H∈(0,1)for fractional Brownian motion by using the equivalence of different fractional Sobolev spaces,and provide the regularity of solution and the effect of noise on the convergence based on this characterization.In Chapter 6,the numerical algorithm of fractional diffusion equation driven by fractional Brownian sheet noise is discussed.We first establish a new characterization of the It^o isometry of fractional Brownian sheet motion to overcome the difficulties of numerical analysis caused by the time-space coupled fractional Brownian sheet noise,thus the regularity of the solution is built.Moreover,we propose a new analysis framework to establish the convergence of the regularized solution and numerical solution by using the approximation theory and the regularity of solution operator.In Chapter 7,we summarize the research work of this paper and look forward to the future work.