几类离散与分布型变时滞抛物系统的高精度快速算法研究

The problems related to the delay differential system are very attractive project recently in the fields of the differential equations at home and abroad, and the related application is very extensive. The behavior of the system changes with the change of the delay, and even resulting in chaos. Thus, their simulation is very complicated. The current algorithms have lots of drawbacks including in narrow area of stability, low accuracy, long CPU time, failure in long time simulation and so on, which result in the low efficiency. In order to increase the efficiency, the project will analyze the numerical method basing on the area of the stability, combine with the construction of the preconditioner and the technique of compact alternate direction implicit method, establish several classes of methods of the high efficiency and rapid numerical methods, and discuss the stability and convergence of the algorithms. The research of the project will not only enrich and improve the theory of the delay problems, but also provide some new computational methods for the field of neutral network, automatic control, biological engineering and so on.

时滞微分系统的相关问题是当今国内外微分方程领域十分活跃的研究课题，其相关的应用极其广泛。该类系统的性态随着时滞量的变化而变化，甚至出现混沌现象，因而数值计算十分复杂。当前算法存在如下问题，导致其效率不高，主要包括四点：稳定域窄，计算精度低，耗时长，大范围区域的模拟失真等。为了提高算法的效率，本课题拟尝试通过稳定域分析的方法，结合快速预估子构造和紧交替方向技巧，建立几类高阶精度、快速计算的数值方法，并讨论算法的稳定性、收敛性。本项目的研究不仅将进一步丰富和发展时滞问题的相关理论，而且为神经网络、自动控制、计算生物等工程应用领域提供新的计算方法和依据。

时滞偏微分系统的相关问题是当今国内外微分方程领域十分活跃的研究课题，其相关的应用极其广泛。该类时滞系统的性态随着时滞量的变化而变化，甚至出现震荡和混沌现象，因而数值计算十分复杂。本项目旨在探索和拓展时滞微分方程的数值算法，丰富和发展时滞微分方程高精度快速算法的相关理论（如稳定性，收敛性和振荡性等），并推广算法的应用。主要研究内容包括以下：（1）针对时滞微分方程的稳定性，分析了一类时滞抛物方程的渐近稳定区域，改进了算法的计算精度；（2）引入交替方向技巧，提升高维时滞微分方程的计算效率；（3）研究了变系数时滞抛物方程，并对其建立了高阶精度的紧致差分算法。这些算法的研究不仅进一步丰富和发展了时滞偏微分方程的相关理论，而且为神经网络、自动控制、计算生物等工程应用领域提供新的算法理论和依据。