非线性时滞微分方程的谱配置方法研究

时滞微分方程是一种应用广泛的数学模型，近几十年来有关它的数值方法研究倍受关注，但传统的方法多采用差分方法。而谱方法作为求解微分方程的一种重要数值方法，它的主要优点是高精度，已被应用于科学和工程计算的众多领域。本项目旨在研究非线性时滞微分方程的谱配置方法。我们首先考虑非线性时滞微分方程初值问题的Legendre配置方法，建立相应的误差估计理论，并提出简单、快速、稳定的高精度算法；然后考虑非线性时滞抛物型和双曲型方程初边值问题，空间和时间方向均采用Legendre配置方法进行离散，进而设计相应的时空高精度算法。本项目的研究成果将发展和丰富时滞微分方程的数值解法，并对解决更加复杂的时滞偏微分方程的数值模拟具有一定的启发意义。

As a widely used mathematical model, delay differential equation has attracted much attention over the recent decades, and the traditional method for solving the delay differential equation numerically is the difference method. Spectral method as an important numerical method for solving differential equations, its main advantage is the high accuracy. It has been increasingly widely used in many fields of science and engineering computing. This project focuses on the spectral collocation method for nonlinear delay differential equations. We first consider the Legendre collocation method for initial value problem for nonlinear delay differential equations, estabilish the error estimation theory, and propose a simple, fast and stable algorithm with high accuracy. We then consider the initial-boundary value problem for nonlinear delay parabolic and hyperbolic equations, use the Legendre collocation method for discretization in both the spatial and temporal directions, and design the corresponding space-time high accuracy algorithm. The research results of this project will further develop and enrich the numerical methods for delay differential equations, and it will shed light on the numercial simulation for the more complex delay partial differential equations.