Dynamic study of new soliton solutions of time-fractional longitudinal wave equation using an analytical approach

Abstract

In this study, the modified exponential approach is used to investigate the fractional order longitudinal wave equation in a magneto-elastic circular rod, which represents the nonlinear interplay between dispersion and longitudinal wave velocity depending on the rod’s material and geometry. The time-fractional order is used in the standpoint of conformable derivative. Comprehensive and descriptive waveform solutions, including kink-shaped, bell-shaped, and bright bell-shaped solitons, are obtained. The produced soliton solutions are advanced, unique, developed, and crucial. To demonstrate the physical appearance of the obtained solutions, 2D, 3D, and contour graphs of some of the obtained solutions are plotted. The extracted solutions demonstrate the physical significance of the model and express the pressure and electrostatic potential for the longitudinal wave equation and can be used to elucidate more complicated nonlinear models of classical and fractional order. The obtained results are helpful in finding the amplitude of tsunami waves which is of great interest to researchers in the field of ocean engineering. If a tsunami extends across a large area, it becomes a manifestation of solitons. The comparison of the obtained solutions is also given in the form of remarks 1 and 2 which shows the novelty of our work.