Numerical Methods For Space Fractional KGS Equations And Fractional Diffusion Equation

Fractional calculus has been playing more and more important roles in some fields such as biology,ecology,mechanics,material science and control system etc.In this thesis,we study the conservative difference schemes and Fourier spectral scheme for the space fractional Klein-Gordon-Schr(?)dinger(KGS)equations,the symplectic difference method of space fractional Schr(?)dinger equation,and the spectral collocation methods for two-sided fractional diffusion equations.In Chapter 2,we give the definitions of some symbols and fractional order operators,and some lemmas used later.In Chapter 3,we give an efficient conservative scheme for the space fractional KGS system with low-degree Yukawa interaction and zero boundary condition,which is based on the fractional central difference scheme,the Crank-Nicolson scheme and the leap-frog scheme.First,we use the fractional central difference scheme to discretize the system in space.Second,we use the Crank-Nicolson and the leap-frog scheme to discretize the system in time.We find that the scheme can be decoupled and linearized,and preserve mass and energy conservation laws.The stability and convergence of the scheme are discussed,and it is shown that the scheme is of the accuracy v(?2+h2).The numerical experiments are given,and verify the correctness of theoretical results and the efficiency of the scheme.In Chapter 4,a conservative finite difference scheme is presented for the high-degree nonlinear space fractional KGS system with high-degree Yukawa in-teraction,where the fractional central difference scheme is used in space,and the Crank-Nicolson scheme to the Schr(?)dinger equation and the leap-frog scheme to the Klein-Gordon equation are applied in time.Our scheme amended the scheme presented in Chapter 3,can be decoupled and preserve mass and energy conserva-tion laws.The existence and uniqueness of the scheme are proven,and it is shown that the scheme is convergent of the accuracy O(?2+h2)in the maximum norm.The numerical experiments are given,and verify the correctness of theoretical re-sults and the efficiency of the scheme.In particular,the effects of the fractional orders and high-degree term coefficient on the behaviors of some solitary wave solutions are investigated numerically,and some interesting phenomena includ-ing quantum subdiffusion and local high oscillation are observed by intuitionistic images.In Chapter 5,we propose the Fourier spectral method to solve space KGS sys-tem with low-degree Yukawa interaction and periodic boundary condition.First,the semi-discrete scheme is given by using Fourier spectral method in space,and the conservativeness and convergence of the semi-discrete scheme are discussed.Second,the fully discrete scheme is obtained based on Crank-Nicolson/leap-frog methods in time.It is shown that the scheme can preserve mass and energy con-servation laws.It is proven that the scheme is of the accuracy O(?2+N-r).Last,based on the numerical experiments,the correctness of theoretical results is veri-fied,and the effects of the fractional orders ?,? on the solitary solution behaviors are investigated.In Chapter 6,the symplectic scheme is presented for solving the space frac-tional Schr(?)dinger equation(SFSE)with one dimension.First,the symplectic con-servation laws are investigated for space semi-discretization systems of the SFSE based on the existing second-order fractional central difference scheme and the ex-isting fourth-order compact scheme.Then,a fourth-order central difference scheme is developed in space,and the resulting semi-discretization system is shown to be a finite dimension Hamiltonian system of ordinary differential equations.Moreover,we get the full discretization scheme for the Hamiltonian system by symplectic midpoint scheme in time.In particular,the space semi-discretization and the full discretization are shown to preserve some properties of the SFSE.At last,numerical experiments are given to verify the efficiency of the scheme.In Chapter 7,we develop spectral collocation methods for a class of two-sided fracional diffusion equations.Since the solutions of these fractional diffusion equations usually exhibit singularities at the end-points,it can not be well ap-proximated by classical polynomial based functions.We first give non-classical in-terpolants based on the associated-Jacobi-Gauss points,and obtain the fractional differentiation matrices by some useful lemmas.Second,we present convergence and stability for the developed spectral collocation method.Finally,several numer-ical examples are considered to demonstrate the validity and applicability of the basis functions to approximate fractional diffusion equations including two-sided,multi-term left-sided,and time-space fractional diffusion equations.