Adams-Bashforth Scheme and Kayo-Kengne-Akgül Derivative: Connexion and Chaotic Modelling
Universal Journal of Mathematics and Applications, vol.9, no.1, pp.53-63, 2026 (Scopus)
- Publication Type: Article / Article
- Volume: 9 Issue: 1
- Publication Date: 2026
- Doi Number: 10.32323/ujma.1879909
- Journal Name: Universal Journal of Mathematics and Applications
- Journal Indexes: Scopus, Applied Science & Technology Source, Central & Eastern European Academic Source (CEEAS), Directory of Open Access Journals
- Page Numbers: pp.53-63
- Keywords: Advanced mathematical literacy, Complex systems modelling for sustainability, Computational tools for disaster resilience, Multi-disciplinary data analysis, Non-linear dynamics for global goals, Scientific research and innovation
- Open Archive Collection: AVESIS Open Access Collection
- Karadeniz Technical University Affiliated: Yes
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.