Fractional Wave Dynamics and Soliton Solutions in Magneto-Electro-Elastic Circular Rods

This paper includes an extensive research into the analytical and numerical solutions for the beta space-time fractional longitudinal wave equation describing a magneto-electro-elastic circular rod. The above equation governs a number of different phenomena related to scientific disciplines and technological sectors, ranging from plasma physics, magnetohydrodynamics, and aerodynamics to a generalized study on the magneto-electro-elastic materials. Those materials play the central role in the future of energy conversion units, actuators of precision engineering, and exquisitely sensitive sensors. In order to investigate the analytical solutions of the beta space-time fractional longitudinal wave equation, we use two efficient methods: the Generalized Riccati Equation Mapping Method and the Modified Extended Tanh Method. Meanwhile, we also use the Differential Transform Method as a useful numerical method to obtain approximate solutions when exact forms are difficult or inconvenient to obtain. These integrated approaches allow us to build a range of new soliton solutions, such as dark solitons, bright solitons, periodic waveforms, and singular solitons. All these types of solutions are significant in explaining the dynamics of energy and information transmission in magneto-electro-elastic circular rods. Furthermore, a detailed parametric analysis is carried out to explore how various physical parameters affect the behavior of the solutions in question. To facilitate visualization and understanding of the findings, we produce three-dimensional plots, two-dimensional projections, and density distributions for a representative set of parameters. The principal goal of the present study is to determine exact and approximate soliton solutions for the fractional wave model. The new dark, bright, periodic, and singular solitons with thorough parametric effects are the main results. These results enhance the knowledge of wave dynamics in magneto-electro-elastic rods for future smart material applications.