In this study, we investigate the space-time fractional simplified modified Camassa-Holm (smCH) equation and the coupled Boussinesq-Burgers (cBB) equation within the M-truncated fractional derivative framework. Using a fractional traveling-wave transform, the nonlinear fractional equations are reduced to ordinary differential forms and investigated analytically through the Sardar sub-equation method. A wide spectrum of closed-form wave solutions is constructed, including singular, periodic, multiple, sharp kink, and kink-type solitons. The mathematical structure of the method enables the generation of solutions systematically, preserving the key properties and analytical consistency of the original fractional model. The impact of the models parameters on wave amplitude, width, and propagation behavior explains through 3D and contour plots which are physically meaningful patterns relevant to nonlinear dispersive media. A comparison with existing literature shows that several of the obtained solutions reduce to known forms for special parameter values, and some other solutions are new. The results contribute to mathematical physics by extending exact-solution space for fractional nonlinear wave equations arising in fluid dynamics, plasma physics, and related applied fields. This study shows that the Sardar sub-equation method is an effective, convenient, and straightforward analytical technique for solving fractional-order nonlinear differential equations for obtaining exact traveling wave solutions.
Sadhu et al. (Thu,) studied this question.