Fractional-Order Numerical Scheme with Symmetric Structure for Fractional Differential Equations with Step-Size Control

This research paper uses two-stage explicit fractional numerical schemes to solve fractional-order initial value problems of ODEs. The proposed methods exhibit structural symmetry in their formulation, contributing to enhanced numerical stability and balanced error behavior across computational steps. The schemes utilize constant and variable step sizes, allowing them to adapt efficiently to solve the considered fractional-order initial value problems. These schemes employ variable step-size control based on error estimation, aiming to minimize computational costs while maintaining good accuracy and stability. We discuss the linear stability of the proposed numerical schemes and observe that a higher-stability region is obtained when the fractional parameter value equals one. We also discuss consistency and convergence analysis of the proposed methods and observe that as the fractional parameter values rise from 0 to 1, the scheme’s convergence rate improves and achieves its maximum at 1. Several numerical test problems are used to demonstrate the efficiency of the proposed methods in solving fractional-order initial value problems with either constant or variable step sizes. The proposed numerical schemes’ results demonstrate better accuracy and convergence behavior than the existing methods used for comparison.