A generalized integral transform and Adomian decomposition scheme for solving the nonlinear fractional Kuramoto–Sivashinsky equation

In this work, we approximate the NTFK-S equation, a model that describes a number of chemical and physical phenomena, including hydrodynamics, reaction-diffusion systems, and long waves on the boundary of viscous fluids. We use a hybrid approach that combines the Adomian decomposition method with the generalized integral transform to get an approximate analytical solution. We solve multiple numerical examples, including three different cases, to show the effectiveness and applicability of both methodologies while confirming the precision and dependability of the suggested strategies. The outcomes demonstrate the usefulness and practicality of the integral transform and the Adomian decomposition approach in analyzing the behavior of nonlinear models. Convergence and error analysis are verified. Numerical simulations confirm the method’s effectiveness. This study advances fractional calculus in nonlinear applications and uses 2D/3D plots to illustrate the impact of the fractional parameter, plotted using MATLAB R2016a, enabling clear visualization of the obtained results.