Application of the Tanh–Coth and HSI methods in deriving exact analytical solutions for the STFKS equation

The stochastic time-fractional Kuramoto–Sivashinsky (STFKS) equation models a wide range of physical phenomena involving spatio-temporal instabilities and noise-driven dynamics. Accurate analytical solutions to this equation are essential for understanding the influence of fractional derivatives and stochasticity in nonlinear systems. This study applies the Tanh–Coth method and He's Semi-Inverse (HSI) method in conjunction with the Truncated M-fractional derivative (TMFD) framework to derive exact solitary wave solutions of the STFKS equation. These approaches are implemented under a traveling wave transformation to reduce the governing partial differential equation to an ordinary differential equation. Exact analytical solutions are obtained for the STFKS equation, and graphical plots are presented to visualize the physical characteristics of the derived wave profiles under varying fractional orders and stochastic conditions. The results confirm that both the Tanh–Coth and HSI methods are effective and reliable for solving the STFKS equation. The graphical analysis reveals the significant impact of fractional parameters and stochastic terms on the solution behavior, demonstrating the practical utility of the proposed methodology in nonlinear stochastic modeling.