An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions

This paper aims to provide a rigorous analysis of exponential convergence of an adaptive spectral collocation method for a general nonlinear system of rational-order fractional initial value problems. The key idea of the proposed method is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions to a class of fractional initial value problems, leading to spectrally accurate approximation. Numerical examples are presented, which verify the theoretical predictions and demonstrate that the new formulation of the spectral method leads to better performance compared to other known numerical approaches with a relatively smaller number of degree of freedoms. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.