Efficient hybrid ann-accelerated two-stage implicit schemes for fractional differential equations

Abstract

This paper introduces a hybrid two-stage implicit scheme for efficiently solving fractional differential equations, with particular emphasis on fractional initial value problems formulated using the Caputo derivative. Classical numerical approaches to fractional differential equations often encounter challenges related to stability, convergence rate, and memory efficiency. To overcome these limitations, we propose a new discretization framework that directly embeds nonlinear source terms into the time-stepping process, thereby enhancing both stability and accuracy. Our method embeds nonlinear source terms directly into the time-stepping process, enhancing stability and accuracy. Nonlinear systems are efficiently solved using a parallel iterative algorithm with adaptive convergence control, yielding up to 35–50% faster convergence compared with conventional solvers. A rigorous theoretical analysis establishes the scheme’s convergence, stability, and consistency, extending earlier proofs to a broader class of fractional systems. Extensive numerical experiments on benchmark fractional problems confirm that the hybrid approach achieves markedly lower local and global errors, broader stability regions, and substantial reductions in computational time and memory usage compared with existing implicit methods. The results demonstrate that the proposed framework offers a robust, accurate, and scalable solution for nonlinear fractional simulations.