Numerical Method For Anomalous Diffusion Equations And Stochastic Fractional Partial Differential Equations

The anomalous diffusion equation can well describe the mechanism of anomalous dynamics,including the diffusion of spatial power law distribution and the long-term correlation.Therefore,many workers in different fields establish and study the anomalous diffusion equation.Deterministic equations can present main laws of the development of things.However,stochastic disturbance in the universe is everywhere.Therefore,in order to describe the development law of things comprehensively,noise is introduced to describe stochastic disturbance.Thus theoretical and numerical studies of stochastic differential equation have also become popular.When the equation contains non-local operators and noise,theoretical and numerical studies become more challenging.The complexity of noise also brings difficulties to the research,such as tempered Gaussian noise.Based on these questions,the numerical methods of anomalous diffusion equations,the regularity of solutions and numerical approximation of stochastic fractional partial differential equations are studied in this paper.This paper consists of six chapters.In the first chapterm the research significance and status of anomalous diffusion equations and stochastic fractional partial differential equations are briefly described;the research contents and innovations of this paper are explained in detail.In the second chapterm we introduce some preliminary knowledge,including the defini-tion of fractional Laplace operator,fractional Gaussian process,several simulation meth-ods of fractional Gaussian process,and selecting the most suitable simulation method by comparison.In the third chapter,we study the numerical schemes for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion.The main challenges of the numerical schemes come from the singu-larity in the time direction.When 0<H<1/2,a change of variables(?)(t2H)=2Ht2H-1(?)t avoids the singularity of numerical computation at t=0,which naturally results in nonuniform time discretization and greatly improves the computational efficiency.For H>1/2,the time span dependent numerical scheme and nonuniform time discretization are introduced to ensure the effectiveness of the calculation and the computational ef-ficiency.The stability and convergence of the numerical schemes are demonstrated by using Fourier method.By numerically solving the corresponding Fokker-Planck equation,we obtain the mean squared displacement of stochastic processes,which conforms to the characteristics of the tempered fractional Brownian motion.In the fourth chapter,we discuss the first exit and Dirichlet problems of the non-isotropic tempered Levy process Xt.The upper bounds of all moments of the first exit position |X?D| and the first exit time ?D are explicitly obtained.It is found that the prob-ability density function of |X?D| or ?D exponentially decays with the increase of |X?D| or?D,and E[?D]?|E[X?D]|,E[?D]?E[|X?D-E[X?D]|2].Since ?m?/2,? is the infinites-imal generator of the anisotropic tempered stable process,we obtain the Feynman-Kac representation of the Dirichlet problem with the operator ?m?/2,?.Furthermore,averaging the generated trajectories of the stochastic process leads to the solution of the Dirichlet problem,which is also verified by numerical experiments.In the fifth chapter,we discusses the fractional diffusion equation forced by a tem-pered fractional Gaussian noise.The fractional diffusion equation governs the probability density function of the subordinated killed Brownian motion.The tempered fractional Gaussian noise plays the role of fluctuating external source with the property of localiza-tion.We first establish the regularity of the infinite dimensional stochastic integration of the tempered fractional Brownian motion and then build the regularity of the mild solution of the fractional stochastic diffusion equation.The spectral Galerkin method is used for space approximation;after that the system is transformed into an equivalent for-m having better regularity than the original one in time.Then we use the semi-implicit Euler scheme to discretize the time derivative.In terms of the temporal-spatial error splitting technique,we obtain the error estimates of the fully discrete scheme in the sense of mean-squared L2-norm.Extensive numerical experiments confirm the theoretical esti-mates.In the sixth chapter,the infinitesimal generator(fractional Laplacian)of a process obtained by subordinating a killed Brownian motion catches the power-law attenuation of wave propagation.This paper studies the numerical schemes for the stochastic wave equation with fractional Laplacian as the space operator,the noise term of which is an infinite dimensional Brownian motion or fractional Brownian motion.Firstly,we establish the regularity of the mild solution of the stochastic fractional wave equation.Then a spectral Galerkin method is used for the approximation in space,and the space convergence rate is improved by postprocessing the infinite dimensional Gaussian noise.In the temporal direction,when the time derivative of the mild solution is bounded in the sense of mean-squared Lp-norm,we propose a modified stochastic trigonometric method,getting a higher strong convergence rate than the existing results,i.e.,the time convergence rate is bigger than 1.Particularly,for time discretization,the provided method can achieve an order of 2 at the expenses of requiring some extra regularity to the mild solution.The theoretical error estimates are confirmed by numerical experiments.In the end of this paper,we summarize this paper and look forward to future work.