Extraction of soliton waves from the longitudinal wave equation with local M-truncated derivatives

We have extracted some soliton solutions of the fractional longitudinal wave equation with the M-truncated derivative (M-LWE), which emerges in a magneto electro-elastic circular rod. To obtain new results of this model, the unified Riccati equation expansion and new Kudryashov methods have been utilized for the first time. The presented methods have been productively implemented to the considered model. With the help of these two methods, new soliton waves of the M-LWE have been obtained successfully. For a better understanding of the subject and analysis of the results, 3D, contour, and 2D graphs of some soliton solutions have been presented. The interesting part of our work is that both methods, named unified Riccati equation expansion and the new Kudryashov methods have successfully been applied for the first time to get new soliton solutions of M-LWE.