隐式中立型Volterra泛函积分微分方程的数值方法研究

Functional differential Equations with delay arguments are powerful tools for the simulation of dynamical systems influenced by delay phenomenon arising in scientific and engineering fields. Among the rest, the neutral functional integral-differential equations with distributed delay items have been widely used in modelling delay dynamical systems, because they can reflect the actual time delay phenomenon much better. So far for the numerical analysis of neutral functional integral-differential equations, there are still a number of open problems. This project focuses on one of these open problems: the numerical treatment for implicit .neutral Volterra functional integral-differential equations, which presented in the energetics,materials science, viscoelastic mechanics, physics, and other scientific and engineering problems. This kind of Volterra functional integral-differential equations have significant differences with the common neutral functional integral.-differential equations. The neutral iterms are not explicit but implicit, and this kind of equations may contain singular systems, for which the theoretical analysis and numerical simulation are much more complicated, and the study of their numerical methods is still in its infancy. In view of this, our project is devoted to the numerical analysis for such a kind of implicit neutral Volterra functional integro-differential equations. We plan to design new numerical algorithms with high precision and stability for the implicit neutral Volterra functional integral-differential equations, by extending the existing excellent numerical methods for ordinary differential equations and partial differential equations, and using various techniques for interpolation and iteration. The convergence, solvability, stability, dissipativity and other important properties of the numerical methods will be investigated to provide the theoretic basis. We will also explore the techniques for efficient implementation, and the applications of the derived numerical methods in the models of thermal conduction, aeronautical Materials, the semiconductor device, etc. This project will expand the numerical research for such a kind of VFIDEs, and enrich the numerical methods and their theoretical results for FIDEs. Furthermore, the achievements can provide new numerical simulation algorithms and analytic tools for related scientific applications.

时滞泛函微分方程是模拟自然和工程中受时滞因素影响的动力学系统的有力工具，其中含分布型时滞项的中立型泛函积分微分方程及其数值方法研究，是该领域的前沿开放性课题。本项目关注在热力学、材料学、黏弹性力学以及物理学等重要科学工程领域中具共性特征的隐式中立型Volterra泛函积分微分方程，由于中立项的隐式性以及包含了奇异系统，其理论分析和数值模拟都更为复杂，现有数值研究成果匮乏。鉴于此,本项目拟针对此类隐式中立型Volterra泛函积分微分方程，拓展现有常及偏泛函微分方程的优秀的数值方法，利用各种插值与迭代技巧，构制新型高效的数值算法，研究数值方法的收敛性、可解性、稳定性、耗散性等重要性质，并探索方法的高效实现技巧及在热传导、航空材料学、半导体装置等实际数学模型中的应用。本项目将填补该类方程的数值研究空白，丰富时滞泛函微分方程的数值方法及理论成果，为相关实际科学问题提供新的数值仿真算法和分析工具。

本项目关注在热力学、材料学、黏弹性力学等重要科学工程领域中具共性特征的隐式中立型 Volterra 泛函积分微分方程(FIDEs)，由于此类方程的特殊复杂性，现有数值研究成果匮乏。鉴于此，本项目研究了几类隐式中立型Volterra 泛函积分微分方程的数值方法及其性质。分别针对几类具体的方程，包括含固定长度区间上分布型延迟的FIDEs、含无界区间上分布型延迟的FIEDs和含奇异核的Volterra FIDEs，设计了高效的数值求解方法，主要包括扩展的Runge-Kutta方法、扩展的单支方法、扩展的线性多步法和扩展的数值积分法。在适当的理论框架下，得到了保障数值方法的收敛性、可解性、稳定性和耗散性的一些理论条件。同时研究了时滞偏微分方程和随机时滞微分方程等特殊模型的数值方法及其相关理论，并探讨了数值方法的有效实现策略。利用数值仿真实验，我们验证了数值方法的高效性及所得理论结果的正确性。本项目填补了该类方程的数值研究空白，丰富了时滞泛函微分方程的数值方法及理论成果，并为相关实际科学问题提供了新的数值仿真算法和分析工具。