Advanced Numerical Methods for Fractional-Order Differential Systems

Fractional-order differential systems have become very popular in the last few years because they can model memory and hereditary effects. However, existing numerical techniques for solving such systems still have certain limitations. We used generalized Caputo-type fractional derivatives for the Abel differential equation in normal form and for the four-dimensional Chen system to avoid such problems. We also used two different methods for their solutions: the adaptive predictor-corrector (P-C) method and the generalized Laplace decomposition method (TρDM). First, the methods are easy to use and give more accurate results than other methods. Second, the methods let you do calculations quickly without having to recalculate fractional sequences for different starting points. Thirdly, the methods are strong enough to accurately identify chaotic attractors and capture the system’s dynamics. Lastly, they are flexible and work well with computers, which makes them useful for many different scientific and engineering systems. A lot of numerical tests and comparisons with the Adams–Bashforth–Moulton (ABM) method show that the proposed methods work well, are reliable, and have real-world benefits.