近年来, 分数阶微分方程的研究快速发展, 被广泛应用于物理、化学、生物及金融等多个领域. 分数阶方程在刻画具有“记忆”特征或非局部“长程”效应的物理过程或现象时, 往往比经典整数阶方程更加准确. 但是通常情况下求出其精确解非常困难，使得数值方法成为研究分数阶方程的重要手段.. 本项目研究空间分数阶薛定谔方程的数值方法. 具体研究两个问题：(1) 对于空间分数阶薛定谔方程, 构造时间分裂傅里叶谱方法, 并给出收敛性证明；(2) 研究空间分数阶薛定谔方程基态解的离散规范化梯度流方法, 并给出相应算法和误差分析. 该项目旨在发展分数阶方程数值方法相关理论和为分数阶量子力学方程提供高效算法.

In recent years, the fractional differential equations have made rapid progress, and is widely used in physics, chemistry, biology and financial fields. The fractional equations are more accurate than the classical integer order equations when we use fractional equations in the description of the physical process or phenomenon which has a memory feature or non local long-range effect. But it is often very difficult to obtain the exact solutions of the fractional differential equations, thus the numerical method has become an important tool to study the fractional differential equations.. This project studies the numerical methods for the space fractional Schrödinger equations. Specifically, we consider two problems: (1) For the space fractional Schrödinger equation with periodic boundary conditions, we will construct time splitting Fourier spectral method, and provide rigorous convergence analysis; (2) We study the gradient flow with discrete normalization method for the ground solution of spatial fractional nonlinear Schrödinger equations, and the corresponding numerical algorithms and error analysis. The project aims to develop the theory of numerical methods of fractional equations and to provide efficient algorithms for the fractional order quantum mechanical equations.