A Comparative Analysis of Five Numerical Methods for Hybrid Caputo - Fabrizio -- Atangana - Baleanu Equations

Abstract

We present a systematic comparison of five numerical methods for fractional differential equations under the hybrid Caputo-Fabrizio--Atangana-Baleanu (HCFAB) operator: Hybrid Kernel Predictor-Corrector (HKPC), Adams--Bashforth--Moulton (ABM), Fractional Euler (FE), Variational Iteration Method (VIM), and Adomian Decomposition Method (ADM). The methods are evaluated on benchmark problems using maximum absolute error and convergence order. HKPC achieves second-order accuracy comparable to ABM with superior error profiles for hybrid memory. VIM and ADM provide semi-analytical solutions for short horizons but diverge beyond $t > T^*$, while FE is cheapest per step but requires $\mathcal{O}(h^{-1})$ more steps for equivalent accuracy. A decision framework for method selection is derived based on problem horizon, Lipschitz constant, and accuracy requirements.