分数阶动力学系统的精细积分算法研究

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 development prospect and the precise integration's theory and method, the applying project will discuss the numerical method for fractional differential system with high efficiency, then extended the method to fractional differential time-delay system, and establish the model and computational method for dissipative system based on fractional varational principal.The applying project will exploit a new way for the numerical computation of fractional differential system.

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。

本项目阐述了分数阶精细积分法的建立及其在工程和具体实际问题中的一些应用。结合工程实际常采用Caputo型分数阶导数定义的分数阶常微分方程，研究了分数阶微分系统动力学的数值计算理论和应用，包括：.①研究由钟万勰教授创立的指数函数精细积分理论的MittagLeffer函数推广，建立分数阶微分问题的精细积分理论，并与已有的分数阶变分迭代法进行比较；.②研究了线性分数阶微分问题的积分理论和方法，将线性分数阶微分问题的的积分理论和方法拓展到非线性分数阶系统，以完善精细积分的积分理论；.③.在计算分数阶微分问题时提出了分组的方法，适合于并行计算，从而提高计算效率。. 这些工作丰富和发展了分数阶微分系统动力学的精细积分数值求解理论，提供了研究和解决分数阶微分系统动力学问题的一个新途径。. 应用前景：复杂介质（如土壤、淤泥、裂隙岩体和生物组织等介于理想固体和液体之间的复杂状态物质）的力学行为比普通固体、液体和气体要复杂得多，具有记忆和路径依赖特性，难以用基于梯度变化的经典物理力学模型来描述，这些问题适合用分数阶微分建立相应的模型并计算。