Adams-Bashforth Scheme and Kayo-Kengne-Akgül Derivative: Connexion and Chaotic Modelling

Abstract

In this paper, we propose a novel formulation of the Adams–Bashforth scheme for solving fractional differential equations (FDEs) involving the Caputo-type Kayo–Kengne–Akgül derivative. We first recall the formal definitions of this derivative and its associated integral, establishing their existence and boundedness. By verifying the Fundamental Theorem of Calculus for this framework, we demonstrate that the integral is the exact inverse of the Caputo-type operator. We then extend the Adams–Bashforth scheme to FDEs by formulating the corresponding Cauchy problem and applying the inverse relationship. Following temporal discretization, the underlying functions are approximated using two-step Lagrange interpolation. By evaluating the resulting fractional integrals, we derive the explicit Adams–Bashforth iterative scheme. Furthermore, we provide a rigorous analysis of the truncation error, convergence, and stability of the proposed method. Finally, the effectiveness of the proposed numerical framework is demonstrated through its application to a three-dimensional chaotic flow system. © 2026