The nonlinear fractional differential equations (FDEs) are produced by mathematical modelling of some nonlinear physical systems. The study of such nonlinear physical models through wave solutions analysis corresponding to their FDEs, has a dynamic role in applied sciences. In this paper, we are going to explore the conformable time-fractional KdV equations using the exp _a function method. The way to reach explicit exact wave solutions is to transform the fractional order PDE into a nonlinear ODE of discrete order through travelling wave transforms. The subsequent equation has been explored by utilizing the exp _a function approach. Consequently, some new explicit exact wave solutions of the said equations are effectively formulated and graphically conveyed with the help of numerical simulation.

This article concerns new analytical wave solutions of the Kuralay-II equations (K-IIAE and K-IIBE) with exploration of a new definition of the derivative. This model is used in various fields, like nonlinear optics, ferromagnetic materials and optical fibers. For this purpose, the ${exp}_a$ function, the extended sinh-Gordon equation expansion scheme, and the generalized Kudryashov schemes were utilized. The resulting solutions are dark, bright, dark-bright, periodic, singular and other kinds of solitons. These results are obtained and also verified by the Mathematica tool. Some of the solutions are explained with 2-D, 3-D and contour plots using the Mathematica tool. The solutions obtained succede the present solutions in the literature. For the first time, the effect of the fractional derivative on the solutions is also shown graphically for this model. The analytical wave solutions are highly desirable as they offer insights into the underlying physics or mathematics of a system and provide a framework for further analysis. The results obtained can also be fruitful for the development of models in the future. The schemes used in this research are effective, easy to apply, and reliably handle other fractional non-linear partial differential equations.