分数阶微分方程的高精度数值方法和反常动力学行为

项目名称： 分数阶微分方程的高精度数值方法和反常动力学行为

项目编号： No.11271173

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 邓伟华

作者单位： 兰州大学

项目金额： 60万元

中文摘要： 经典牛顿力学和统计物理学在描述一些自然现象时受到挑战，如湍流速度场的不规则起伏、复杂系统中的反常扩散、黏弹性材料的记忆性等。与此同时分数阶微分算子越来越多的被证明是描述中间过程和临界现象的重要而有效的工具。本项目拟开展分数阶微分方程的高精度数值方法研究，并以分数阶微分方程为模型研究反常动力学。具体地说：（1）利用谱方法是整体方法（应用整个区域上的信息）及雅可比多项式的权与分数阶积分算子核的一致性并使用算子分裂技术设计计算格式，实现计算分数阶问题与经典问题在计算量上相当；（2）发挥间断有限元法在做hp逼近的灵活性及方便处理复杂边界条件的优势，设计二维或三维不规则几何区域上分数阶微分方程的间断有限元法，给出分数阶算子数值流选取的一般策略；（3）构造新颖的稳定的高精度有限差分格式（主要思想见正文）；（4）设计分数阶时间导数的高精度离散格式；(5)应用动力系统的理论和数值模拟的手段研究反常动力学。

中文关键词： 分数阶微分方程；高精度数值方法；反常动力学行为；；

英文摘要： The classical Newtonian mechanics and statistic physics are challenged in describing some natural phenomena, such as the irregular fluctuation of the turbulence velocity field, the anomalous diffusion of complex system, the memory of the viscoelastic materials. At the same time, fractional differential operators are more and more proved to be an important and effective tool in characterizing the intermediate processes and critical phenomena. This project aims to develop the high accurate numerical methods for fractional differential equations, and study the anomalous dynamics by taking the fractional differential equations as concrete models. Specifically: (1) Based on the global properties (using the information of the whole domain) of the spectral methods, the consistency of the weights of the Jacobi polynomials and the kernel of fractional integral operators, using the operator splitting techniques to design the numerical schemes, and finally realize taking almost the same computational costs for computing the fractional problems and the classical ones; (2) Using the flexibility of DG methods in doing HP approximations and its convenience in dealing with the complex boundary conditions, we design the DG methods to solve 2D or 3D fractional differential equations in irregular domain, and try to provide the gen

英文关键词： Fractional differential equation；High order accurate numerical method；Anomalous dynamical behavior；；