A spectral collocation method for nonlinear fractional initial value problems with nonsmooth solutions

A general class of nonlinear fractional differential equations is considered. Some sufficient conditions for the existence and uniqueness of exact solution are established by using Weissinger's fixed point theorem and the Gronwall-Bellman lemma. A spectral collocation method based on the smoothing technique is presented to solve the problem numerically. Then the rigorous error estimates under the L² and L^∞ norms are derived. The most remarkable feature of the method is its capability to achieve spectral convergence for weakly singular solutions. Finally, numerical results are given to support the theoretical conclusions with smooth and weakly singular solutions.