A comparative study about the propagation of water waves with fractional operators

Abstract

In the current research article, the truncated M-fractional derivative and Atangana–Baleanu derivative in Riemann–Liouville sense is practiced in the fractional modeling of the weakly non-linear shallow water wave equation. This paper provides some traveling wave transformations for the conversion of the partial fractional differential equation into an ordinary differential equation concerning fractional operators. In order to portray the wave propagation in weakly non-linear and dispersive media, the weakly non-linear shallow water wave model has been utilized. The proficient approach new extended direct algebraic scheme is used to find the solitonic structures. It is observed that the acquired solutions are purely new and have full novelty. The obtained results are carrying dark, dark singular, bright singular, and bright-dark singular solutions, which are obtained via Mathematica. The graphical comparison is also depicted in the perspective of operating fractional differential definitions with the selection of appropriate parametric values.