An improved modified simplest equation method for exact solitary wave solutions of the three-dimensional nonlinear fractional wazwaz-benjamin-bona-mahony model

Abstract

In this study, we suggest an improved modified simplest equation (IMSE) approach to obtain distinct analytical solutions of the three-dimensional nonlinear fractional Wazwaz-Benjamin-Bona-Mahony (FWBBM) model associated with the conformable derivative. The suggested IMSE approach extends the classical simplest equation approach by introducing four new solutions, enhancing its efficiency and generality for solving nonlinear fractional partial differential equations. This approach yields fifteen solitary wave solutions for each equation of the three-dimensional FWBBM model, including trigonometric, hyperbolic, algebraic, and mixed-function types. The obtained solutions describe diverse soliton shapes such as bell-shaped, anti-bell-shaped, singular, and periodic solitons, which illustrate the rich dynamical behavior of nonlinear dispersive waves. The physical implications of these solutions are analyzed through three-, two-dimensional, and contour plots depicted through Mathematica, showing that the fractional-order parameter significantly affects soliton amplitude, shape, and stability. Comparisons with existing analytical methods, including the tanh-coth and $\text{exp}\left(-\varphi \left(\omega \right)\right)$-expansion techniques, confirm the precedence and broader applicability of the IMSE method. This approach provides deeper insights into nonlinear wave propagation and soliton dynamics and provides a powerful analytical tool for multidimensional fractional models in plasma physics, fluid mechanics, and optical systems.