Two Classes Of Conservative Numerical Methods For Fractional Schr?dinger Equations

As we all know,the research of fractional differential equations covers many fields,such as physics,biology and engineering.Schr?dinger equation also has important applications in the fields of Bose-Einstein condensation,plasma,nonlinear optics and fluid dynamics.The solution of the fractional Schr?dinger equation has important geometric structures such as energy conservation,mass conservation and multi-symplectic structure.Therefore,these properties should be maintained as much as possible in the construction of numerical methods.However,the research results of the conservative methods for the fractional order Schr?dinger equation are much less than the ones of integer order equation.Therefore,how to extend the energy conservation algorithm of the integer order Schr?dinger equation to the fractional order Schr?dinger equation is a subject worthy of further study.This paper studies this issue,and the specific research contents are as follows:In Chapter 1,this paper briefly introduces the research process of fractional differential equation,the research status of fractional Schr?dinger equation numerical method and some preparatory knowledge to be used in this paper.In Chapter 2,for solving a class of fractional Schr?dinger equations with wave operators,Crank-Nicolson Fourier Galerkin method and Crank-Nicolson Fourier method are used to discrete it respectively.In addition,the ability of the constructed numerical method to maintain the conservation of mass and energy of the original system is studied in detail.Meanwhile,the convergence of the constructed conservative numerical method is analyzed in detail,and the theoretical results of this chapter are verified by numerical experiments.In Chapter 3,by separating the real and imaginary parts of a class of fractional Schr?dinger equation,it is rewritten into infinite dimensional Hamiltonian partial differential equations.Fourier spectral method is employed to discrete the spatial variables of the original equation,and the semi-discrete form was written as equivalent Hamilton system with finite dimensions.The semi-discrete form were discretized by Crank-Nicolson method and average vector field method.Then,the full discrete numerical method is obtained.Moreover,the constructed numerical method can maintain the conservation of mass and energy of the original system,and the convergence of the method is analyzed in detail.Finally,Numerical results verify the validity of the theoretical results.