Mathematical Sciences: Studies in Numerical Solution of Functional Differential Equations

本研究的目的是研究常微分方程和泛函微分方程、Volterra积分和积分微分方程的数值方法的性质。本文提出了一种基于变步长变阶Adams公式的中立型时滞微分方程和Volterra积分微分方程数值解的高效可靠的通用算法。本文将研究各种数值方法对Volterra积分方程和非卷积核积分微分方程的递归关系的渐近性质。此外，还将研究大型常微分方程系统的时间点松弛和波形松弛龙格-库塔方法的收敛性、阶性和稳定性。这些方法适用于并行计算环境。最后，将研究微分方程的变步长两步龙格-库塔方法。这些方法的主要应用之一是为连续龙格-库塔方法提供有效的误差估计。这项研究是在数值分析领域，它有广泛的应用于各种物理和工程应用。

The purpose of the research is to study the properties of numerical methods for ordinary and functional differential equations and Volterra integral and integrodifferential equations. An efficient and reliable general purpose algorithm for the numerical solution of neutral delay-differential equations and Volterra integro-differential equations based on variable stepsize variable order Adams formulas will be developed. The asymptotic behavior of recurrence relations resulting from application of various numerical methods to Volterra integral and integro-differential equations with nonconvolution kernels will be studied. Moreover, convergence, order and stability properties of time-point relaxations and waveform relaxation Runge-Kutta methods for large systems of ordinary differential equations will be examined. These methods are appropriate in a parallel computing environment. Finally, variable stepsize two-step Runge-Kutta methods for differential equations will be investigated. One of the main applications of these methods is to provide efficient error estimates for continuous Runge-Kutta methods. This research is in the area of numerical analysis, which has applications to a wide variety of physical and engineering applications.