Numerical Analysis of Abel Equations and Fractional-Order Chen Systems Using an Adaptive P–C Method

Shahinaz. A.  Elsamani; Khaled Helmi  Khashan

Authors

Shahinaz. A. Elsamani Department of Mathematics, Faculty of Science, Bakht Al-Ruda University, Duaim.
Khaled Helmi Khashan Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Keywords:

The adaptive (P-C) method, Numerical solution, Generalized fractional derivatives, Chaos

Abstract

This study is presented to solve the Abel differential equation in canonical form and the four-dimensional fractional order Chen system under generalized Caputo-type derivatives by applying the adaptive predictor-corrector method, known as the adaptive (P-C) method. Additionally, it conducts a comparative analysis between the proposed technique and the Runge-Kutta Fourth Order (RK4) method. The findings reveal the effectiveness of the proposed approach, generating solutions that closely match the approximate results obtained via the Runge-Kutta Fourth Order (RK4) method. As a result, extending this methodology to a wide array of systems becomes feasible, enhancing the results' precision. Furthermore, this method exhibits utility in accurately identifying instances of attractor chaos through empirical demonstrations. In prospective applications, this method holds promise for numerically solving diverse models relevant to science and engineering.

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