Higher-order predictor-corrector methods with an enhanced predictor for fractional differential equations

Abstract

In this paper, we consider a predictor-corrector method for solving fractional differential equations (FDEs). We construct a 4th-order Adams-Moulton formula suitable for FDEs and develop a higher-order predictor-corrector technique based on this 4th-order formula within the traditional framework. In addition to the traditional predictor structure, which uses all lower-order schemes as predictors for FDEs, we propose an enhanced predictor that employs only one higher-order Adams-Bashforth type predictor. This new approach aims to complete a higher-order predictor-corrector technique by constructing the Adams-Bashforth formula for FDEs. The convergence orders of the proposed schemes are theoretically proved through error analysis, and numerically demonstrated through several experiments. The numerical results show that the higher-order proposed scheme achieves a higher convergence order compared to existing techniques. Furthermore, the enhanced predictor technique enhances the overall convergence order compared to the traditional predictor technique.