Theory Analysis And Numerical Algorithm Studies On Some Fractional And Stochastic Fractional Nonlinear Schr?dinger Equation

Fractional-order partial differential equations and stochastic fractional-order partial differential equations have been widely used in many fields of engineering and scientific technology.Its theoretical analysis and numerical algorithm research are both core hot issues.This dissertation focuses on theoretical analysis and numerical algorithm of the damped nonlinear space-fractional Schr?dinger equation,theoretical analysis and numerical algorithm of stochastic nonlinear space-fractional Schr?dinger equation driven by Stratonovich type multiplicative noise,theoretical analysis of stochastic nonlinear time-space fractional Schr?dinger equation driven by Gaussian White noise and theoretical analysis and numerical algorithm of stochastic nonlinear time-space fractional Schr?dinger equation driven by It? type multiplicative noise.The first chapter mainly introduces the background and significance of fractional calculus,the background and significance of stochastic fractional partial differential equation,and the research status of stochastic fractional partial differential equation.In Chapter 2,we mainly introduce the basic knowledge of fractional calculus,gaussian noise and stochastic partial differential equation.In Chapter 3,we verify the unique existence of the global smooth solution of the damped nonlinear space-fractional Schr?dinger equation and show it follows a conformal mass conservation law.We propose a conformal mass-preserving linearized scheme.It is rigorously proved that this scheme preserve the discrete conformal mass.Furthermore,we prove that the proposed scheme admits a unique solution and is of second order convergence in space and of first order convergence in time.Some numerical experiments are carried out to validate the theoretical analysis.In Chapter 4,We establish the well-posedness of mild solution for the space fractional stochastic nonlinear Schr?dinger equation driven by Stratonovich type multiplicative noise.Then a fully discrete mass-preserving scheme with weighted shifted Grünwald–Letnikov method in space and splitting method in time are proposed.Finally,we present error estimates based on the regularity of its solution,which give order 2 in spatial direction and order 1 in temporal direction.In Chapter 5,We present the time-spatial regularity of the nonlocal stochastic convolution for Caputo-type time fractional nonlocal stochastic convolution by the generalized Mittag-Leffler functions and Mainardi function,and the stochastic convolution is transformed into the stochastic coefficients to get an equivalent equation.Then we establish the existence and uniqueness of mild solutions for time fractional and space nonlocal stochastic nonlinear Schr?dinger equation driven by Gaussian white noise.In addition,the global mild solution is also shown.In Chapter 6,we establish the existence and uniqueness of the mild solution and the H?lder continuity for the time fractional and space nonlocal stochastic nonlinear Schr?dinger equation driven by It? type multiplicative white noise.A full discrete scheme with spectral Galerkin method in space and exponential method in time is proposed.It is rigorously proved that the numerical scheme admits an unique solution.Finally,we present the optimal error estimates of the proposed scheme based on the regularity analysis.