Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions

An open problem in the numerical analysis of spectral methods for fractional differential equations is how to maintain the high-order accuracy for non-smooth solutions. The limited regularity of the exact solution to these equations causes a deterioration in the orders of convergence of standard schemes. In this paper, we derive and analyze an exponentially accurate Jacobi spectral-collocation method for the numerical solution of nonlinear terminal value problems involving the Caputo fractional derivative of rational-order theta is an element of(0, 1). The main ingredient of the proposed approach is to regularize the solution by a suitable smoothing transformation, which allows us to adjust a parameter in the solution according to different given data to maximize the convergence rate. We systematically describe the necessary steps in the implementation process. Additionally, a comprehensive numerical analysis including error estimates under the L-infinity- and weighted L-2-norms is derived. The extensive numerical examples that accompany our analysis confirm our theoretical estimates, as well as give additional insights into the convergence behavior of our method for problems with smooth and non-smooth solutions. (C) 2019 Elsevier B.V. All rights reserved.