This research addresses the modified F-expansion, the newly established extended modified F-expansion, and the unified method for computing the exact solution of the time-fractional regularized long-wave (Tf-RLW) model with Conformable fractional derivative. The Tf-RLW model is widely applicable to solitary wave propagation in shallow water waves, plasma waves, and ion-acoustic waves. The three analytical techniques are employed to simplify this nonlinear model and justify the obtained waves in relation to the existing waves of this model. Fractional derivatives are non-integer-order operators widely used to model complex phenomena in science and engineering, with notable applications in ocean and coastal engineering for tsunami-wave mitigation. This research presented the bright-dark bell waves, periodic rogue waves, periodic waves, singular kink waves, and kinky-periodic bell waves as solutions to the governing model. 3-D, density, and 2-D sketches are given to analyze the intense behavior of the attained waves. Stability investigation of the obtained solutions is also included. All the solutions obtained in this research are substantiated using Maple 18.
Hossain et al. (2026) studied this question.