Finite Element Method For Aaclass Of Time Fractional Order Stochastic Partial Differential Equation

Fractional stochastic partial differential equation is one of the hottest research directions in Mathematics in recent years.Because of its inheritance and memory,fractional calculus operators can describe many anomalous diffusion phenomena with noise perturbation.Therefore,fractional stochastic partial differential equation models have been applied in dispersion of Porous Media,typing theory,neuroscience and other fields.In this paper,we mainly introduce the theoretical analysis and numerical calculation of the mild solution of Caputo fractional stochastic Allen-Cahn equation.This paper is mainly divided into five chapters.In the first chapter,we introduce the research background and historical development of stochastic partial differential equations,and describe the current research status of stochastic fractional differential equations at home and abroad.In the second chapter,some preliminary knowledge and some important theorem lemmas are introduced.Besides,the mild solutions of fractional stochastic Allen-Cahn equation are constructed.In the third chapter,we proof the existence and uniqueness of mild solution to the above-mentioned equation.We take advantage of the Picard's iteration method in the process.Bisides,the regularity of mild solutions is analyzed by fractional calculus and semigroup theory.In the fourth chapter,we introduce the finite element method.Firstly,we make use of Galerkin finite element method to semi-discretize the space,and obtain the regularity of the mild solution of finite element,and the error analyzes are deduced in detail.Then,we deduce the discretization and its error analysis on the basis of Mittag-Leffler special function.The fifth chapter is the summary of this paper and the prospect of future work.