An improved fractional predictor-corrector method for nonlinear fractional differential equations with initial singularity

The solution and source term of nonlinear fractional differential equations (NFDEs) with initial values generally have the initial singularity. As is known that numerical methods for NFDEs usually occur the phenomenon of order reduction due to the existence of initial singularity. In this paper, an improved fractional predictor-corrector (PC) method is developed for NFDEs based on the technique of variable transformation. This improved fractional PC method can achieve the optimal convergence order, i.e., the 1+alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+\alpha $$\end{document} order convergence rate for fractional order alpha is an element of(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document}, of the classical fractional PC method under the high smoothness requirement on the solution and source term. Furthermore, the detailed error analysis also exhibits the relationship between the convergence rate of the improved fractional PC method and the regularities of the solution and source term. Finally, the theoretical error estimate is verified through numerical experiments.