ABSTRACT
In this chapter, the stochastic models with Ito noise have been studied including stochastic partial differential equations and stochastic wave equations. The models discussed in this Chapter involve in the fractional Laplacian in the spectral sense, white noise and fractional Gaussian noise. In Chapter 3.1, we first review some basic properties of stochastic processes which are used to estimate regularity and approximation error. We prove the existence and uniqueness of the mild solution to SPDEs forced by a tempered fractional Gaussian noise in Chapter 3.2. Moreover, the Holder continuity of the mild solution is discussed. Then, the spectral Galerkin method is used for space approximation; after that the system is transformed into an equivalent form having better regularity than the original one in time. And we use the semi-implicit Euler scheme to discretize the time derivative. In Chapter 3.3, we establish the regularity of the mild solution of the stochastic fractional wave equations involving space-fractional operators and time-fractional derivatives. We design numerical methods for stochastic fractional wave equations. And the higher order approximation for stochastic space fractional wave equation is presented.
