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：本文描述了一种用于数值求解泛函微分方程与延迟微分方程的切比雪夫谱配置方法框架。该框架通过重心重采样矩阵进行插值，并结合多域方法以处理由非光滑初始数据传播的孤立间断点。针对具有离散型、比例型、连续型及状态依赖型等多种延迟类型的线性与非线性泛函微分方程及延迟微分方程，本文通过多个算例验证了方法的几何收敛性。该框架是基于多项式插值的标准谱配置方法的自然延伸，可便捷地嵌入现有谱离散化系统（如Chebfun/Chebop），从而实现对广泛类型非线性泛函与延迟微分方程的自动化高效求解。