几类记忆型微分方程初值问题的数值方法

许多现象和事物的发展过程具有记忆性，常用记忆型微分方程描述。时间分数阶Hamilton系统、分数阶噪声驱动的过阻尼广义Langevin方程、时滞Hamilton系统是三类有重要应用背景的记忆型微分方程，其中，前二类属时间非局部微分方程，具有长时间历史记忆性，而第一、三类具有守恒量及能量耗散性、对称性、稳定性等结构特性。目前，时空非局部微分方程的建模、理论、计算及应用已成为研究热点，但此三类方程数值方法的研究很有限，甚至是空白。. 本项目针对此三类记忆型微分方程初值问题，克服时间非局部性、弱奇异性、时滞、分数阶噪声等复杂因素带来的实质困难，解决在理论分析、算法构造和算法分析中所遇到的一些关键问题，进一步了解方程的结构特性，发展高效数值方法与保结构算法，建立相应算法理论，并拓广到一些相关问题计算；由此深化和拓展记忆型微分方程数值方法领域的研究，为相关应用领域提供帮助。

The development process of many phenomena and things has memory, which can be described by differential equations with memory. Time-fractional Hamilton and Birkhoff systems, fractional noise-driven over-damped generalized Langevin equations and delay Hamilton systems are three kinds of memory differential equations with important application background. Among them, the first two kinds of equations belong to time nonlocal differential equations with the property of long-term historical memory, while the first, third kinds of equations are of some structural characteristics such as the conservation quantity and energy dissipation, symmetry and stability. At present, for time or space nonlocal differential equations, their modeling, theory, computation and application have become the research hotspot, but the research work on numerical methods of the three kinds of equations is limited or even blank. ..For the three types of initial value problems of differential equation with memory, this project will overcome the essence difficulties caused by some complex factors such as non-locality, weak singularity, delay and fractional noise etc., solve some key problems encountered in theoretical analysis, algorithm construction and algorithm analysis, further understand the structural characteristics of these equations, develop efficient numerical methods and the structure-preserving algorithms, establish the corresponding algorithm theory, and outreach these work to some related problems. Our work can deepen and expand the researches into numerical methods for differential equation with memory and provide some helps for related applications.