A re-scaling spectral collocation method for the nonlinear fractional pantograph delay differential equations with non-smooth solutions

Abstract

The convergence analysis of the spectral methods of the fractional differential equations is generally carried out on the assumption that the underlying solution is sufficiently smooth. Due to the limited smoothing property of the solution operator, these methods fail to achieve spectral accuracy. This work aims to study a spectral collocation approach for the fractional nonlinear pantograph delay differential equations with nonsmooth solutions. The fractional-order derivative is considered in the Caputo sense. An auxiliary transformation is adapted to match the singularity in the corresponding solution and to maximize the convergence rate of the proposed scheme. Therefore, the solution of the resulting equation will possess better regularity and then the numerical method can achieve the spectral accuracy, which serves as an improvement compared with the existing results in the literature. The spectral convergence rate for the proposed approach is discussed in the weighted L2-norm and the L∞-norm. Finally, numerical results are given to confirm our theoretical analysis.