Theoretical Analysis And Finite Element Method For Fractional Cahn-Hilliard-Cook Equation

With the development of science,the corresponding complex systems of various natural phenomena often show some random or uncertainty characteristics,so we need to add the corresponding stochastic terms to deterministic control equations.And we find the fact that the future state relies not only on the present state but also on all previous historical states,more and more people have payed attention to fractional differential equations because they can describe the phenomena with memory and genetic properties.Stochastic fractional differential equation came into being in order to describe the phenomena with noise effect and memory property,which is a generalization of fractional differential equation and stochastic differential equation.In this thesis,we devote to investigate the extended Cahn-Hilliard-Cook equation with Caputo-type fractional derivative on bounded domains,which can be used to describe the important phase separation phenomena with random effect and shape memory property.When Hurst coefficient H=1/2,we construct the formulation of mild solution for the equation by virtue of Mittag-Leffler function,the existence and uniqueness of mild solution are also established by Banach fixed point theorem.With the help of stochastic analysis techniques,fractional calculus and semigroup theory,we prove the Holder regularity of mild solution to the given problem.We discuss the regularity results of the generalized Ornstein-Uhlenbeck process with the Hurst coefficient 1/4<H<1/2 and 1/2<H<1.Then,the existence,uniqueness and regularity properties of mild solution are proved.We propose a Galerkin finite element method to derive the semidiscrete scheme for the given equation when H=1/2.The regularity properties of mild solution to finite element approximation of this equation are presented,and a result concerning the convergence error estimate for corresponding semidiscrete scheme is established.Finally,based on the approximations of Mittag-Leffler function,we also construct the fully discrete scheme and arrive at the strong convergence error estimate in L2-norm.