Advanced Numerical Methods for Generalized Caputo-Type Fractional-Order Differential Systems: Theory, Implementation, and Chaotic Applications
Keywords:
Fractional differential equations, Caputo derivative, adaptive predictor-corrector method, generalized Laplace decomposition, chaos, Abel equation, Chen system, numerical simulation, computational efficiency, memory effectsAbstract
Fractional-order differential systems offer a robust framework for modeling real-world phenomena with memory and hereditary characteristics, yet their numerical solution remains challenging. This study introduces two advanced numerical methods the adaptive predictor-corrector (P-C) and the generalized Laplace decomposition method (TρDM) to efficiently solve equations involving generalized Caputo-type derivatives. The proposed methods are tested on the Abel differential equation and the four-dimensional Chen system, capturing both regular and chaotic dynamics. Numerical results reveal that the new algorithms achieve high accuracy, computational efficiency, and flexibility, outperforming classical approaches such as the Adams-Bashforth-Moulton (ABM) method. The study’s findings demonstrate that these methods are reliable and effective, providing powerful tools for scientists and engineers tackling complex nonlinear fractional-order systems.
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