分数阶微分方程的数值计算和动力学行为

项目名称： 分数阶微分方程的数值计算和动力学行为

项目编号： No.10801067

项目类型： 青年科学基金项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 邓伟华

作者单位： 兰州大学

项目金额： 17万元

中文摘要： 分数阶微积分今天已成为一个非常热门的话题，由于它在描述反常扩散等科学问题的成功应用及巨大潜力，近十几年里它得到了迅猛发展，覆盖了自然科学中几乎所有的科学领域。我们针对分数阶微分方程进行研究，主要是研究分数阶微分方程的数值方法，也讨论了分数阶微分方程解析解的光滑性、稳定性、遍历性，以及在物理应用中的动力学模拟。首先，我们研究了空间分数阶和时间分数阶的Fokker-Planck方程的有限元方法，给出了有限元方法求解这类问题的一般框架，严格分析了算法的稳定性、收敛性，并进行了数值模拟说明了算法的有效性；将次扩散机理与Klein-Kramers方程结合就得到了时间分数阶的Klein-Kramers方程，我们讨论了数值求解这类方程的有限差分的隐格式，对算法进行了严格的理论分析，我们还意外的得到了物理约束条件，该约束条件定量的刻画了次扩散的性质、粒子的动能以及流体温度之间的关系；我们还研究了空间分数阶（Levy）Klein-Kramers方程的有限差分格式，推广了离散的极值原理，严格得到了稳定性和收敛性结果；关于分数阶常微分方程的数值计算、理论分析和应用，我们也得到一系列的结果。

中文关键词： 分数阶微分方程；有限元；有限差分；动力学行为

英文摘要： Fractional calculus has become a very hot topic these days. Because of its successful application and huge potential in describing the scientific questions, like anomalous diffusion, fractional calculus developed very fast in the past decade, covering almost all the scienfic fields. Our research focus on fractional differential equations, mainly discussing its numerical methods, also analyzing the smoothness, stability, and ergodicity of its analytical solution, and performing the dynamical simulations of fractional diffusion equations in physical applictions. First, we have studied the finite element method for the space and time fractional Fokker-Planck equation, provided the general framework for solving this kind of problem by finite element method, the numerical stability and convergence are strictly established, and the numerical simulations are also performed to testify the effectiveness of the algorithm. Incorporating subdiffusive mechanisms into the Klein-Kramers formalism leads to the time fractional Klein-Kramers equation, the implicit finite difference methods for solving this equation are provided and detailedly analyzed; a physical constraint is obtained beyond our expectation, quantitatively described the relations among the property of subdiffusion, the kinetic energy of the particles, and the temperature of the fluid. The implicit finite difference methods for space (Levy) fractional Klein-Kramers equation are also discussed, and the discrete maximum principle is generalized to strictly obtain the theoretical results. For the numerical computation, theoretical analysis and application, of fractional ODEs, we also obtain a series of results.

英文关键词： fractional differential equation; finite element method; finite difference method; dynamical behavior