(untitled) (Contemporary Mathematics, 2025)
In this research, the Aboodh residual power series method is applied to investigate the behavior of the fractional order of the Kuramoto-Sivashinsky (KS) equation. The KS equation is a nonlinear partial differential equation (PDE) that can model the chaotic behavior of different physical processes. The presence of fractional derivatives leads to memory effects and non-local interactions, making the analysis non-trivial and fundamental for understanding various processes. The Aboodh residual power series method (ARPSM) is a highly effective analytical approach for analyzing more complicated problems and deriving highly accurate approximations. Using a combination of analytical derivation and computational studies, we discuss the features of the fractional KS equation and explain how the fractional derivatives affect its behavior. Altogether, the results indicate that the ARPSM is beneficial in obtaining necessary information about the complexity of this equation for further analysis and contributions to nonlinear dynamics field and fractional calculus. This method does not require complex calculations and is characterized by ease and flexibility in analyzing more complicated problems. In addition, you do not need highly efficient computers; a personal computer with normal capabilities can be used to perform all calculations for the most complicated problems.