兰州大学机构库 >数学与统计学院
(回火的)分数阶扩散方程的差分数值算法
其他题名Finite difference method of (tempered) fractional diffusion equations
于妍妍
导师伍渝江 ; 邓伟华
学位类别博士
2016-05-29
关键词有限差分法 回火的分数阶导数 生成函数 拟紧格式 稳定性
中文摘要连续时间随机行走(CTRW)模型, 包含随机等待时间和随机跳跃步长两个基本要素. 基于相应的 CTRW 模型, 可以分别导出粒子的概率密度函数(PDF)满足的时间分数阶, 空间分数阶和时间-空间分数阶扩散方程. 当CTRW模型的随机跳跃步长是回火的形式时, 其对应的PDF满足回火的空间分数阶扩散方程. 本文的具体研究内容如下:一、 对标量的非线性欠扩散方程组, 本文构造了半隐式的数值格式. 作为一个具体的模型, 本文分析了欠扩散捕食模型数值解的性质. 用构造的数值格式计算了欠扩散捕食模型, 并从理论上证明了所给格式可以很好地保持解析解的正性和有界性. 二、 本文提供了高阶拟紧离散Riemann–Liouville 回火的分数阶导数的基本策略. 对空间分数阶扩散方程, 本文用一个高阶紧算法离散, 并用Fourier分析法讨论了格式的稳定性. 此外, 数值地求解了回火的分数阶扩散方程, 数值实验结果可以很好地吻合理论结果.三、 寻求离散回火的分数阶导数的高阶拟紧数值算法. 由于回火的分数阶导数中回火因子的存在, 导致方程离散格式的稳定性分析方法与经典的分数阶扩散方程有很大差异. 本文引入了一些分析技术来证明稳定性: 结合矩阵的生成函数和Weyl定理, 严格证明了格式的稳定性. 本文还从多个方面验证了格式的有效性及误差阶.
英文摘要The continuous time random walk (CTRW) model compose waiting times and jump lengths. Based on the corresponding CTRW models, the time, space, or time-space fractional diffusion equations are derived to govern the probability density function (PDF) of the particles. For the CTRW with the distribution of the tempered jump length, the corresponding PDF of the particles satisfies the tempered space fractional diffusion equation. The main research contents of the study are as follows: Firstly, we design semi-implicit schemes for the scalar time fractional reaction-diffusion equations. As a concrete model, the subdiffusive predator-prey systems are discussed in detail. We use the provided numerical schemes to solve the subdiffusive predator-prey system, and theoretically prove and numerically verify that the numerical schemes preserve the positivity and boundedness. Secondly, we provide the basic strategy of deriving the high order quasi-compact discretizations for Riemann-Liouville fractional derivative and tempered space fractional derivative. The fourth order quasi-compact discretization for space fractional derivative is applied to solve space fractional diffusion equation and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore, the tempered space fractional diffusion equation is effectively solved by its counterpart of the fourth order quasi-compact scheme; and the convergence orders are verified numerically. Thirdly, we focus on providing the quasi-compact schemes for the tempered fractional diffusion equations. Not only its derivation but the proof of its numerical stability and convergence are different from the ones of the fractional diffusion equations. The detailed theoretical results are presented, and some techniques are introduced in the analysis: by using the generation function of the matrix and Weyl's theorem, the stability and convergence of the derived schemes are strictly proved. Extensive numerical simulations are performed to show the effectiveness of the schemes, and the third order convergence is confirmed.
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授予单位兰州大学
授予地点兰州
语种中文
文献类型学位论文
条目标识符http://ir.lzu.edu.cn/handle/262010/225750
Collection数学与统计学院
Recommended Citation:
GB/T 7714
于妍妍. (回火的)分数阶扩散方程的差分数值算法[D]. 兰州. 兰州大学,2016.
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