分数阶偏微分方程的高阶数值格式与理论分析-应用数学专业论文.docxVIP

I I 摘 要 分数阶扩散方程源于基本随机游走模型, 近年来它在数学, 物理, 工程以 及应用领域上获得了广泛应用, 引起了研究人员的极大兴趣. 许多复杂无序 系统包含分形介质上反常扩散现象可以用分数阶扩散方程去精确描述. 随着 实际应用的进一步深化, 分数阶扩散方程的解析解研究方法与数值解研究方 法得到了迅速的发展. 本文研究了时间分数阶扩散方程, 加热下广义二级流体的分数阶Stokes第 一问题的高阶数值格式, 数值解的存在唯一性, 格式的稳定性, 收敛性. 本文 的结构安排如下： 第一章简要阐述了分数阶微积分的由来与发展现状, Gru¨nwald-Letnikov, Riemann-Liouville, Caputo分数阶导数的定义及其性质. 并对分数阶偏微分 方程的数值方法做了详细介绍. 第二章讨论了时间分数阶扩散方程的高阶数值格式, 给出了一种比较简 洁的离散方法, 得出了高阶隐格式. 并对格式的可解性, 稳定性作了严格论证, 并改进了格式的时间误差阶. 第三章考虑带有初边值条件的分数阶Stokes第一问题, 利用分数阶导数的 定义对方程作积分离散. 引进了四阶紧致差分算子逼近空间二阶导数, 得到了 一种新的空间四阶时间一阶的隐式格式. 采用了Fourier方法证明了格式的无 条件稳定性, 利用矩阵法证明了格式的收敛性. 还对格式的时间误差阶作了 改进. 第四章通过数值仿真验证了所作的理论分析的正确性. 将本文所给的数 值结果与文献[33], [43]所给出的数值结果作比较, 表明文中所给的数值格式具 有高效性. 关键词: 分数阶微积分; 分数阶扩散方程; Stokes第一问题; 稳定性; 收敛 性. ABSTRACT Fractional di?usion equations (FDEs) which are derived from the basic random walk models are widely used in mathematics, physics, engineering and application in the recent year, thus it has attracted a considerable interest. Many complex disordered systems including phenomena of anomalous di?u- sion through fractal media can be well described as FDEs. Along with the further practical application, analytic and numerical methods for FDEs are developed rapidly. In this paper, we investigate the higher-order numerical schemes for frac- tional order Stokes’ ?rst problem for a heated generalized second grade ?uid and the di?usion equation with derivative of fractional order in time, respec- tively. The solvability, stability, convergence and improvement for the schemes are also discussed. The paper is consist of four parts as follows. In chapter one, we give a brief description on the origin of fractional calcu- lus and current situation. The classic de?nitions together with their properties of fractional derivatives, namely Gru¨nwald-Letnikov, Riemann-Liouville, Ca- puto derivatives are listed. In addition, the overview on numerical methods for fractional di?erential equations is introduced. In chapter two, we use a simple discrete method on the di?usion equ