Investigation of rogue wave and dynamic solitary wave propagations of the $$\mathbf{M}$$ -fractional (1 + 1)-dimensional longitudinal wave equation in a magnetic-electro-elastic circular rod

The longitudinal wave equation (LWE) is crucial for understanding the dynamic behavior of the material in a magneto-electro-elastic (MEE) circular rod. This equation describes the propagation of longitudinal waves along the rod’s length, accounting for the interactions between mechanical, electrical, and magnetic fields within the material. By considering properties such as mass density and stress–strain relationships, the equation elucidates how longitudinal waves propagate through the MEE rod, influenced by its magneto-electro-elastic nature. In this study, we employ two robust techniques using a truncated \(M\)-fractional derivative to integrate the time-fractional longitudinal wave equation (LWE). The simplest equation (SE) technique and the novel modified Kudryashov (NMK) technique are used to obtain additional solitary wave solutions to the LWE. These solutions are analyzed using the NMK and SE techniques, forming trigonometric, hyperbolic, and exponential functional solutions. We demonstrate new phenomena from the numerical conditions of the derived soliton solutions of the time \(M\)-fractional LWE model. We verify the behavior of the \(M\)-fractional parameter with two-dimensional charts and compare the effects of the \(M\)-fractional derivative with the classical derivative form using various graphical systems. The NMK technique reveals bright and dark bell-shaped waves, periodic waves, linked periodic rogue waves, and periodic rogue waves with curved bell-shaped wave patterns. Similarly, the SE technique produces bright and dark bell-shaped waves, linked periodic rogue waves, periodic rogue waves with curved bell-shaped waves, and periodic wave patterns. As a result, these techniques are demonstrated to be useful tools for producing distinct, accurate solitary wave solutions for various applications. These applications are critical in fields such as materials science, ecology, sociology, and urban planning, where understanding the collective behavior of individuals within a spatial context is essential.