延迟和分数阶微分代数系统的迭代方法及应用

Delay differential-algebraic systems and fractional order (delay) differential-algebraic systems are often used for modeling many practical problems in science and engineering, owning time lag, memory, constraint limit, etc. This project focuses on studying the theory of high efficient iterative algorithms for solving integer order and fractional order (delay) differential-algebraic systems as well as the construction, realization and the application of high efficient iterative algorithms. Detailed contents of this project include the following four parts: (1) developing the convergence theory of waveform relaxation method and variational iteration method for solving delay differential-algebraic systems (especially high-index problems) and fractional order (delay) differential- algebraic systems; (2) constructing the efficient waveform relaxation method based on the multiple grid technology, the technologies of splitting and parallel computing for large scale integer order and fractional order (delay) differential-algebraic systems; (3) studying the restrictive variational theory and constructing efficient variational iteration method for the high-index integer order and fractional order delay differential-algebraic systems; (4) seeking for application of the obtained theoretical results and efficient algorithms in numerical simulation of time-domain analysis in large scale integrated circuit. The expected results of this project aim to promote development of numerical methods for solving delay differential- algebraic systems and fractional order (delay) differential-algebraic systems, and apply the results for simulation computation in many fields of scientific engineering, such as electric power system, automatic control and biology.

延迟微分代数系统或分数阶(延迟)微分代数系统常用来描述具有时滞效应、记忆性及约束限制等特征的科学工程问题。本项目重点研究这两类系统的迭代求解方法的理论和相应高效算法的构造、实现及应用。具体内容包括：(1) 探索求解延迟微分代数系统(特别是高指标问题)和分数阶(延迟)微分代数系统的波形松弛法、变分迭代法的收敛理论；(2) 针对大规模整数阶和分数阶(延迟)微分代数系统，以分裂和并行计算为关键技术，引入多重网格快速算法，构造高效波形松弛法；(3) 针对整数阶高指标延迟微分代数系统和分数阶延迟微分代数系统，研究限制变分理论，构造高效变分迭代法；(4) 将所获部分理论结果和高效算法应用于大规模集成电路的时域分析等领域的数值模拟。本项目研究成果旨在促进延迟和分数阶(延迟)微分代数系统数值方法的发展，并应用于电力系统、自动控制和生物学等科学工程领域。

该项目针对奇异摄动微分方程或分数阶（泛函）微分（代数）方程解的适定性、稳定性和散逸性及相应高效迭代方法（波形松弛方法和变法迭代方法）的构造、收敛性分析及应用等方面开展了研究，具体研究内容包括：（1）构造了线性和非线性奇异摄动延迟微分方程的连续和离散波形松弛方法迭代格式, 并证明迭代的收敛性，当小参数趋近于零，该结果可以直接应用到延迟微分代数方程波形松弛法的相应理论；(2) 获得了线性和非线性分数阶延迟微分代数方程的离散波形松弛方法迭代格式, 并证明迭代的收敛性；（3）构造了分数阶延迟积分微分代数方程变分迭代法，获得了收敛性结果；（4）我们证明了一类分数阶非线性延迟积分微分方程的解的存在唯一性，进一步，研究了分数阶泛函微分方程的散逸性与稳定性；（5）在初步应用中，我们研究了基于马尔科夫残差修正的电功率预测及基于动态博弈论的充电桩产量规划模型，在这两个应用工作中，我们均转为为差分方程，借助于我们的分裂思想，从而高效求解了该问题。