A framework for Chebyshev spectral collocation methods for the numerical solution of functional and delay differential equations (FDEs and DDEs) is described. The framework combines interpolation via the barycentric resampling matrix with a multidomain approach used to resolve isolated discontinuities propagated by non-smooth initial data. Geometric convergence is demonstrated for several examples of linear and nonlinear FDEs and DDEs with various delay types, including discrete, proportional, continuous, and state-dependent delay. The framework is a natural extension of standard spectral collocation methods based on polynomial interpolants and can be readily incorporated into existing spectral discretisations, such as in Chebfun/Chebop, allowing the automated and efficient solution of a wide class of nonlinear functional and delay differential equations.

翻译：描述了一种用于数值求解泛函微分方程（FDEs）和延迟微分方程（DDEs）的切比雪夫谱配置法框架。该框架结合了基于重心重采样矩阵的插值方法与多域方法，用以解析由非光滑初始数据传播的孤立间断点。针对包含离散、比例、连续及状态相关延迟等多种延迟类型的线性和非线性泛函微分方程与延迟微分方程，本文通过多个算例证明了其几何收敛性。该框架是基于多项式插值的标准谱配置法的自然延伸，可便捷地融入现有谱离散化工具（如Chebfun/Chebop），从而实现对广泛非线性泛函微分方程与延迟微分方程的自动化高效求解。