Numerical Study of the Fractional Van der Pol Equation Using the Modified Atangana–Baleanu–Caputo Derivative
Year: 2026
Type: article
Abstract: This study investigated the nonlinear fractional Van der Pol equation using the modified Atangana–Baleanu–Caputo (mABC) fractional derivative operator. The Van der Pol oscillator is a fundamental model in nonlinear dynamics, with applications spanning electrical circuits, cardiac rhythms, neural activity, mechanical systems, fluid dynamics, control engineering, robotics, and economic cycles. Unlike the classical ABC fractional derivative, the modified operator offers enhanced capability for modeling systems with long-term memory and hereditary properties. The main objective of this study is to test the applicability of the modified operator for solving this nonlinear oscillator problem, and to analyze how the fractional order and the damping term influence the system dynamics. A predictor–corrector numerical scheme based on Lagrange interpolation was developed and implemented for the coupled fractional system. The key findings demonstrate that the fractional order [Formula: see text] significantly influences the damping characteristics and oscillatory behavior. The proposed numerical scheme achieves good agreement with the reference methods, including the seventh-order Runge–Kutta (RK7) and three-step Adams–Bashforth (AB3) schemes, with observed quantitative differences attributable to the distinct kernel structures of the fractional operators. The numerical simulations performed in MATLAB R2017b yielded graphical and tabular results. The novelty of this study lies in the application of a mABC derivative to the Van der Pol system.
Source: Mathematics Open
Institutions: Wollo University, Samara University, Arsi University
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7162684039