空间分数阶Schrödinger方程的时间分裂谱方法

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.

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.

近年来，分数阶微积分广泛应用于各个工程物理领域。许多有趣的分数阶应用模型被建立。同时，数值模拟表明，分数阶模型往往比整数阶方程更加符合物理实际。..本项目主要研究分数阶微分方程的定性理论和高效算法。目前，在本项目的支持下，我们获得两项结果：.1：首次建立了时间分数阶非线性ODEs的耗散性和收缩性理论，为时间分数阶ODEs的定性理论和数值稳定性建立了一个理论基础。.2：我们建立了时间分数阶非线性泛函微分方程的耗散性和稳定性理论，为分数阶非线性复杂系统的数值研究提供了统一的理论框架。..这两项成果都发表在国际高水平SCI杂志：.Wang D, Xiao A. Dissipativity and contractivity for fractional-order systems. Nonlinear Dynamics, 2015, 80(1-2): 287-294..Wang D, Xiao A, and Liu H. Dissipativity and stability analysis for fractional functional differential equations. Fractional Calculus and Applied Analysis, 2015, 18(6): 1399-1422...特别是第二篇论文杂志，是分数阶方程方向唯一的国际顶级期刊(TOP，SCI一区)。同时我们的论文得到了审稿人较高的评价：."Importance, significance for FCAA audience and value of contribution: .nice work in using the generalized Halanay-type inequality, interesting and useful work。.The numerical studies are good as well. I have no reservation in recommending this paper for FCAA publication.”