The Fractional Power Series Method for Solving the Nonlinear Kuramoto-Sivashinsky Equation

Since Kuramoto and Shivashinsky introduced the Kuramoto-Sivashinsky equation in the mid-1970s, it has been the subject of extensive research. This equation is important in modeling phenomena like flame front propagation, plasma instabilities, viscous flow problems, and magnetized plasma’s behaviors. In this work, the nonlinear time-fractional Kuramoto-Sivashinsky problem is approximated using the fractional power series technique. Our goal is to comprehend the numerical solutions of both linear and nonlinear fractional differential equations by solving the Kuramoto-Sivashinsky equation in a simple and efficient way. The existence, uniqueness, and stability of the considered nonlinear equation are investigated. Furthermore, the conditions for convergence and truncation error of the result are discussed. The accuracy of the numerical technique was validated by comparing the approximate results with those from existing literature. The novelty of this paper lies in utilizing the fractional power series approach specifically for the nonlinear fractional Kuramoto-Sivashinsky partial differential equation. The performance of the proposed technique was confirmed using graphical and tabular representations, with numerical simulations run in MATLAB R2016a. The findings show that the approach is effective and simple for studying the behavior of nonlinear models in science and technology.