Nonlinear regularized long-wave models with a new integral transformation applied to the fractional derivative with power and Mittag-Leffler kernel

Abstract This paper presents a fundamental solution method for nonlinear fractional regularized long-wave (RLW) models. Since analytical methods cannot be applied easily to solve such models, numerical or semianalytical methods have been extensively considered in the literature. In this paper, we suggest a solution method that is coupled with a kind of integral transformation, namely Elzaki transform (ET), and apply it to two different nonlinear regularized long wave equations. They play an important role to describe the propagation of unilateral weakly nonlinear and weakly distributer liquid waves. Therefore, these equations have been noticed by scientists who study waves their movements. Particularly, they have been used to model a large class of physical and engineering phenomena. In this context, this paper takes into consideration an up-to-date method and fractional operators, and aims to obtain satisfactory approximate solutions to nonlinear problems. We present this achievement, firstly, by defining the Elzaki transforms of Atangana–Baleanu fractional derivative (ABFD) and Caputo fractional derivative (CFD) and then applying them to the RLW equations. Finally, numerical outcomes giving us better approximations after only a few iterations can be easily obtained.