A spectral collocation method for functional and delay differential equations

Abstract

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 nonsmooth initial data. Geometric convergence in the number of degrees of freedom 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 and can be readily incorporated into existing spectral discretizations, such as in Chebfun/Chebop, allowing the automated and efficient solution of a wide class of nonlinear FDEs and DDEs.