Theoretical Analysis And Numerical Methods For Stochastic Fractional Partial Differential Equations

Fractional calculus plays a key role in characterizing the power structure of anoma-lous diffusion.Therefore,in recent years,fractional?nonlocal?differential equations have been widely concerned and successfully applied in various fields of science and engineering.With the development of science and technology,it is gradually recognized that stochastic perturbations are unavoidable in physical systems,and sometimes can not be ignored,so we need to add the corresponding stochastic terms to deterministic governing equation-s.Thus the stochastic differential equation,as a new discipline of applied mathematics,has gradually developed into an indispensable branch of mathematics.In order to better characterize the anomalous diffusion phenomena with memory and noise perturbations,the stochastic fractional differential equations appear.At the same time,the study of the theory and computational methods of such equations has also begun to be popular.However,the nonlocal properties of fractional operators and the low regularity and uncer-tainty of noises cause great difficulties for the study of this kind of equations.Therefore,the literature on the asymptotic behavior and effective numerical method of such equa-tions is still very rare,which motivates us to further explore the long time behavior of solutions of the equations and design the effective numerical methods to solve them.This paper is divided into six chapters to elaborate.Chapter 1 summarizes the development of stochastic fractional differential equations,analyzes the current research situation of the equations,and expounds the main contents,research methods and the main innovation points of this paper.In the second chapter,we firstly introduce some necessary preparation knowledge.Secondly,the existence,uniqueness and continuous dependence of the mild solutions of model 1 in this chapter are proved by using the-order fractional resolvent operator the-ory.Then the global existence and asymptotic behavior of the mild solution of model 2are studied by using the resolvent operator theory and the Schauder fixed point theorem.Finally,a global forward attracting set of model 2 in the mean square topology is estab-lished.This is an interesting generalization of the pullback attractor of the stochastic unbounded delay equation with classical derivative in[28].In the third chapter,we use the basis functions in space²to expand the equation into a series form,and obtain the solution of stochastic spatio-temporal fractional equation based on the existing results of fractional ordinary differential equation.Secondly,by discreting the space-time additive noise,the regularized stochastic space-time fractional order wave equation is obtained;and by using the Galerkin finite element method,the regularized stochastic space-time fractional wave equation is discretized in the spatial direction.Finally,the error estimation of the model is established.The convergence order is???^max{12,^-12^-}+?²for?(1,₂³];and?+?²for?(₂³,2],whereis the order of the time fractional derivative,is the order of the space fractional Laplacian,?is the time step,?is the space step,andcan be any sufficiently small positive constant.Numerical experiments verify the convergence orders of the model error and the finite element approximation of the regularized equation.In the fourth chapter,we consider the time-fractional stochastic delay evolution in-clusion with nonlinear multiplicative noise and fractional noise.By using the new result of the non-compactness measure of the stochastic integral term and a fixed point theorem for the multi-valued mapping,we obtain the global existence of mild solutions of the model1 in this chapter.Among them,we solve the problem of calculating the non-compactness measure of the stochastic integral term.Finally,we discuss the asymptotic behavior of mild solutions of the model 2.The research technique of Chapter 5 is similar to Chapter 2.In this chapter,we mainly deal with nonlinear stochastic integral terms with multiplicative white noise and multiplicative fractional noise.By defining and discretizing the space-time white noise and fractional noise,the model errors are introduced,and the regularized nonlinear time tempered fractional stochastic wave equations are obtained;and the convergence orders of the modeling error are well established,which have transition point=₂³,i.e.,the convergence order is-₂¹-for?(1,₂³],and 1 for?(₂³,2],whereis the order of the time fractional derivative andcan be any sufficiently small positive constant.Finally,the Galerkin finite element method is used to derive the numerical scheme for the regularized nonlinear time tempered fractional stochastic wave equation,and the error estimates are deduced in detail.The sixth chapter is the summary of this paper and the prospect of the future work.