A fast time-stepping method based on the hp-version spectral collocation method for the nonlinear fractional delay differential equation

In this paper, we propose a fast time-stepping method based on the hp-version spectral collocation method for the nonlinear fractional delay differential equation. To reduce the interaction between the subintervals caused by the delay term, a mesh of bidirectional partition is designed, such that each subinterval after delay falls exactly into a single preceding subinterval. Moreover, with the aid of the sum-of-exponentials approximation for the kernel function (t-s)& alpha;-1, a fast time-stepping method is developed to deal with the weakly singular integral, which overcomes the weak singularity of the integral kernel, and greatly reduces the cost for the computation of the history integral. In addition, the hp-convergence of the suggested method is fully analyzed and characterized, which implies that the interplay between h and p can significantly enhance the numerical accuracy. Numerical experiments confirm the theoretical expectations.& COPY; 2023 Elsevier B.V. All rights reserved.