In this article, we investigate the nonlinear model describing the various physical and chemical phenomena named the Kuramoto–Sivashinsky equation. We implemented the natural decomposition method, a novel technique, mixed with the Caputo–Fabrizio (CF) and Atangana–Baleanu deriavatives in Caputo manner (ABC) fractional derivatives for obtaining the approximate analytical solution of the fractional Kuramoto–Sivashinsky equation (FKS). The proposed method gives a series form solution which converges quickly towards the exact solution. To show the accuracy of the proposed method, we examine three different cases. We presented proposed method results by means of graphs and tables to ensure proposed method validity. Further, the behavior of the achieved results for the fractional order is also presented. The results we obtain by implementing the proposed method shows that our technique is extremely efficient and simple to investigate the behaviour of nonlinear models found in science and technology.

This work combines a ZZ transformation with the Adomian decomposition method to solve the fractional-order Fokker-Planck equations. The fractional derivative is represented in the Atangana-Baleanu derivative. It is looked at with graphs that show that the accurate and estimated results are close to each other, indicating that the method works. Fractional-order solutions are the most in line with the dynamics of the targeted problems, and they provide an endless number of options for an optimal mathematical model solution for a particular physical phenomenon. This analytical approach produces a series type result that quickly converges to actual answers. The acquired outcomes suggest that the novel analytical solution method is simple to use and very successful at assessing complicated equations that occur in related research and engineering fields.