Exploring Fluid Behavior With Space-Time Fractional-Coupled Boussinesq Equation

The propagation of shallow-water waves is described by the space-time fractional-coupled Boussinesq equation, which is used in the study of fluid flow in dynamic systems. Coastal engineers can efficiently use the nonlinear water wave model to harbor and coastal designs by having a solid understanding of its solutions. In our study, we analyze the Boussinesq equation using the conformable derivative, a modern approach that accounts for fractional effects in wave dynamics. To solve this equation, we employ two advanced methods: the modified extended tanh function method and the ( 1 φ(ξ), φ′(ξ) φ(ξ) ) method. These techniques provide different soliton solutions, including dark-singular, dark, singular, periodic-singular, and bright solitons having unique propagation characteristics. We also perform a bifurcation analysis of the equation, which gives knowledge into the stability and behavior of the solutions by changing the conditions. To visualize the dynamics of the solutions, we give a graphical representations via 3d, 2d and density plot, which allow for a comprehensive examination of their behavior. We also compare the solutions at different fractional order values to highlight the impact of the order on the wave dynamics. The results indicate that the applied methods are not only effective but also give an efficient and robust mathematical insight for studying the governed equation. The symbolic computation software enhances the reliability and speed of the solutions to underscore the potential of these methods in advancing the study of nonlinear water wave phenomena.