By using the two distinct methods known as the \(\exp (-\phi (\eta ))\) and the \(Exp_a\) -function methods, various forms of soliton solutions of the modified equal width wave equation (MEWE) with beta time derivative (BTD) are produced in this study. This model is used as a model in partial differential equations for the simulation of one-dimensional wave transmission in nonlinear media with dispersion processes. The obtained solutions are in the form of rational, trigonometry and hyperbolic trigonometry functions. Using Mathematica software, the resulting solitons are validated. Graphs are also used at the conclusion to explain the findings. These soliton solutions imply that these two methods are more dependable, simple and efficient than other methods. The findings can be used to explain how studious structures and other comparable non-linear physical structures are substantially understood. The obtained results are very helpful in the fields of optics, kinetics solid-state physics and hydro-magnetic waves in cold plasma.
