分数阶动力学系统的精细积分算法研究

项目名称： 分数阶动力学系统的精细积分算法研究

项目编号： No.11202146

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 鲍四元

作者单位： 苏州科技学院

项目金额： 24万元

中文摘要： 相对于经典微积分, 分数阶微积分更加复杂, 一是分数阶导数的数学运算复杂, 二是含分数阶导数的系统具有复杂的动力学。这种复杂性既限制了分数阶微积分的广泛应用，又为复杂系统与结构的研究带来了新机遇。钟万勰提出了矩阵指数的精细积分法，使矩阵指数计算达到了计算机精度，随后将精细积分法扩展到两端边值问题的求解。本项目的研究内容：（1）分数阶微分线性系统的精细算法理论研究；（2）分数阶微分非线性系统的精细算法的理论研究；（3）为提高计算效率而对分数阶微分系统的精细算法的改进研究；（4）分数阶时滞微分系统的精细算法的理论研究；（5）含分数阶微分项的耗散系统的模型及精细算法的研究。研究意义：本项目针对具有发展前景的分数阶问题和精细积分的理论和方法，研究适用于分数阶微分系统的高效数值算法，并推广至时滞分数阶微分系统，再基于分数阶变分原理建立耗散系统的模型和算法。本项目将为分数阶问题的数值算法开拓出新方向。

中文关键词： 分数阶微分；精细积分法；修正项；分组精细积分；Mittag-Leffler函数

英文摘要： Comparing with classical calculs，fractional calculs is more complicated. First the mathematical computation of fractional derivative is complicated, second the system including fractional derivative has complex dynamics. The complexity not only limit the general application of fractional calculs, but also bring new opportunity of the research of complex system and structures. Zhong wan-xie presented the precise integration method of exponential matrix, which make the precision of exponential matrix's calculation upto the precise of computer, then the precise integration method's application is extended to the problem of two-end boundary. The project's research content:(1)the theoretical research of precise computational method for fractional differential linear system; (2) the theoretical research of precise computational method for fractional differential nonlinear system; (3) the improved research of precise computational method for fractional differential system to enhence the computational efficiency; (4) the theoretical research of precise computational method for fractional differential time-delay system; (5) the research of the model and the precise computational method for dissipative system including fractional differential item. The research significance: For the fractional calculus problem with broad

英文关键词： fractional differential；precise integration method；correction item；grouping precise integration method；Mittag-Leffler function