泛函微分方程的高效边值方法及其算法理论

Functional differential equation is one of the important models to describe modern scientific problems and its numerical computation is an effective tool to solve such a model. At present, there have been many numerical methods for the initial problems of functional differential equations, but their computational efficiency and the robustness of long-term computing are still needed to be arised. Using the boundary value methods to solve initial value problems is a type of new technique appearing in the field of scientific computation in recent years. Compared with the classical numerical methods, it has better stability and higher computational efficiency, and is very suitable to the paralell computing. For the initial and boundary problems of functional differential equations, this project plans to construct new numerical methods with high-efficiency and high-precision, investigate their algorithmic theory on linear and nonlinear stability, convergence and computational efficiency. and apply the obtained algorithms to the important problems in physics, biology, economy control science and the related fields.

泛函微分方程是刻画现代科学技术问题的重要模型之一, 其数值计算是求解该类模型的有效手段. 目前, 针对泛函微分方程初值问题已有众多数值方法， 但其计算效率及长时间计算的稳健性仍有待提高. 边值方法求解初值问题是近年来呈现在科学计算领域的一种新技术, 与经典数值方法相比较，其具更佳的稳定性能和更高的计算效率, 且特别适合于并行计算. 本项目拟针对泛函微分方程初、边值问题, 构造新型高效高精度数值算法, 研究其线性与非线性稳定性、收敛性及计算效率等算法理论, 并将其算法应用于物理学、生物学、经济学及控制科学等领域中的重要问题.

针对具实际应用背景的多类泛函微分方程初值问题和初-边值问题，我们分别构制了若干新型高效高精度的数值算法，特别重点研究了边值与块边值方法，获得了系列新颖、实用的泛函微分方程数值算法理论，诸如解的收敛性、稳定性、唯一可解性和长时间动力性等，在国际重要学术期刊上发表SCI收录论文37篇，培养硕士生12名、博士生9名，其中8名硕士生、5名博士生已按时毕业并分别获得硕士和博士学位。此外，我们积极开展了与国内外同行的合作交流，主持和参加了系列相关学术活动。据此，我们已圆满完成了本项目任务。