The nonlinear space-time fractional Boussinesq and the space-time fractional (2+1)-dimensional breaking soliton equations are significant models for analyzing nonlinear dispersive waves, particularly the propagation of waves in shallow water, as well as acoustic, tidal, and tsunami waves. In this article, the new generalized \((G^{\prime}/G)\) -expansion procedure is applied to obtain solitary wave solutions to these equations together with beta derivative ( \(\beta \) -derivative). A traveling wave transformation aligned with the \(\beta \) -derivative is used to transform the fractional-order equations stated above into integer-order equations. The solutions are found in the form of hyperbolic, trigonometric and rational structures. For specific parameter values, the obtained solutions exhibit bright-dark, parabolic, anti-parabolic, bell-shaped, kink, compacton, and flat kink solitons, which are illustrated using three-dimensional and contour plots. A comparison of the attained results with existing research is presented to validate the novelty and significance. The executed procedure is competent and consistent for investigating nonlinear fractional models in mathematical and theoretical physics.
