Numerical Solution of Fractional Differential Systems Using a Hybrid Predictor–Corrector Method

Abstract

This paper extends the Hybrid Kernel Predictor-Corrector (HKPC) method to systems of nonlinear fractional differential equations governed by the hybrid Caputo-Fabrizio–Atangana-Baleanu (HCFAB) operator. A vector-valued Volterra integral formulation is rigorously established, and an extended predictor-corrector scheme achieving second-order temporal accuracy is derived with complete proofs. The method is applied to a fractional SIR epidemic model under the hybrid operator, wherein a memory-dependent basic reproduction num-
ber Rα,θ 0 is introduced and analyzed. Equilibrium analysis, stability conditions, and convergence of the numerical scheme are established through formal theorems. Detailed numerical experiments and supporting visualizations confirm theoretical predictions and demonstrate how the blending parameter θ and fractional order α influence epidemic dynamics and disease thresholds.