Closed-form wave solutions to the van der Waals model arising in nature and the longitudinal wave motion equation in a magneto-electro-elastic circular rod

Closed-form wave solutions of nonlinear evolution equations perform a crucial task to show the intrinsic device of complex physical phenomena. In this study, we acquire new closed-form wave solutions to the van der Waals standard form for the fluidized granular matter, which is one of the famous equations in industrial and nature and the Longitudinal Wave Motion Equation in a Magneto-electro-elastic (MEE) Circular Rod through the variation of [Formula: see text]-expansion method. In this regard, the first equation is one of the famous equations in industrial and nature. Understanding of static and dynamic property for the model is critical in numerous perspectives of geophysics, industrial applications, pharmaceutical, civil engineering, as well as the model illustrates the phase separation phenomenon. Therefore, this model is physically meaningful in their own right. By utilizing our new rule, we get not only new exact traveling wave answers but also exact no-traveling wave answers in the expressed of hyperbolic functions. And the second equation that arises in applied mathematics, physical sciences and mathematical physics with dispersion affected over the transverse Poisson’s impact in a broad magneto-electro-elastic circular rod in which a set of closed-form wave solutions concerning the considered equation is presented. To interpret the physical aspect of the obtained solutions, we show the [Formula: see text] and [Formula: see text]-dimensional surfaces and the corresponding contour plot surfaces of the closed-form wave solutions by using Maple. Additionally, it has been discovered that the considered methods contribute a further substantial mathematical mechanism for finding exact solutions to nonlinear PDEs. Furthermore, this technique is proficient of significantly decreasing the size of computational work compared to other existing methods.