A comparative study of fractional derivatives to interpret wave structures for the higher order fractional Ramani equation

This study investigates wave solutions to the higher order Ramani equation with beta derivative via the generalized Kudryashov and the extended sinh-Gordon equation methods. The higher order Ramani equation with beta derivative is reduced into an ordinary differential equation (ODE) with the use of a fractional transformation involving the definition of beta derivative. Hereafter, via the generalized Kudryashov method and the extended sinh-Gordon equation method, some novel solutions are constructed of the reduced ODE in terms of trigonometric functions, hyperbolic functions, and their combinations. Then, the considered wave transformation is set back to the solutions of reduced ODE. As a consequence, all explored wave solutions of the fractional Ramani equation are found to be novel in terms of beta derivative and applied methods sense. To demonstrate the fractional effects of the explored wave solutions, the three-dimensional (3D) and their two-dimensional (2D) cross-sectional line plots are presented under the particular selection of any fractional values within \(\beta \in (\mathrm{0,1}\) $\beta \in (0,1$ ). The 3D and 2D cross-sectional line plots of some of the achieved novel solutions confirm the underlying mechanisms of the model. With an increase in fractional parameters, the kink or anti-kink profile takes on complete form, and smoothness rises. Conversely, the singular-periodic wave solutions show an increase in smoothness and periodicity. Furthermore, a thorough comparison of all the investigated solutions to the equation under consideration that integrates beta derivative (BD), conformable derivative (CD), and M-truncated derivative (MTD) is included in this paper. The findings for the model's BD, CD, and MTD investigate how the fractional parameter affects the wave profile's amplitude, using graphs to illustrate this effect by designating exact fractional parameter values. The obtained findings demonstrate the ability of the implemented techniques to identify wave solutions with fractional derivatives for the nonlinear sixth-order Ramani equation, which are practically useful for utilizing optical fiber.