Different wave structures in water wave mechanics with two conformable models

This paper investigates the exact wave solutions of the time-fractional modified Liouville equation (mLE) and time-fractional modified regularized long wave equation (mRLWE) which arise in water wave mechanics, via a new version of trial equation method (NVTEM). The present nonlinear models are reduced to nonlinear ordinary differential equations (NLODEs) by the traveling wave transform and the proposed solution by the NVTEM is used to evaluate the solutions of mLE and mRLWE. This analytical method is not applied before to these equations and novel wave solutions are acquired in the form of rational, exponential, hyperbolic, and Jacobi elliptic types. The solitary solutions of the equations under consideration make them essential models in shallow water dynamics, in liquids and gas bubbles, in magneto-hydrodynamics, and in plasma. This fact has become a motivation for this research.