Misuse and correction of the multistage Adomian decomposition method for fractional ordinary differential equations.

Abstract

The multistage Adomian decomposition method (MADM) is a widely used algorithm for the numerical solution of fractional ordinary differential equations (FODEs). However, this study reveals that the traditional MADM yields distorted results for fractional-order problems due to a fundamental conceptual flaw: the improper handling of the nonlocal memory effect. By analyzing the derivation of the Adomian decomposition method we identify that the traditional multistage implementation neglects the accumulation of historical memory. A comparative analysis using a fractional FitzHugh-Nagumo neuron model demonstrates that the traditional MADM produces physically inconsistent results compared to the established predictor-corrector method. To address this, we propose a revised MADM that correctly incorporates the memory term via numerical integration. Comprehensive performance analysis reveals that the revised MADM is rigorously convergent and stable, and achieves superior absolute accuracy in the high-precision regime. Therefore, this research holds theoretical and practical value by preventing the continued use of a flawed method and by proposing a correct and efficient alternative, thereby opening new avenues for the numerical solution of FODEs.