Construction of Mechanically Preserved Optical Travelling Wave Solution for Fractional Generalized Reaction Duffing Model

This research paper presents an efficient analytical method, namely the Sardar Sub-Equation Method (SSM), to construct new families of traveling solitary wave solutions of Fractional Partial Differential Equations (FPDEs), i.e., the Fractional Phi-Four Equation (FPFE) and the Fractional Generalized Duffing Equation (FGDE). By applying fractional wave transformations, these FPDEs are transformed into nonlinear ordinary differential equations. The SSM solutions of the reduced nonlinear ordinary differential equations are further investigated to attain traveling (mechanical) wave solutions to the selected problems in terms of hyperbolic and trigonometric functions. The propagating behavior of some traveling wave solutions is graphically illustrated through 3D, 2D, and contour graphs, which reveal Cuspon solitons as well as dark and bright traveling wave solutions. The obtained soliton solutions play a vital role in the practical discussion of mechanical procedures. We believe that the current results can be utilized to capture many dynamics in mechanical engineering. Moreover, the traveling wave solutions attained by the Sardar Sub-Equation Method are highly efficient and accurate. The results demonstrate that the employed method has a high degree of reliability, with a straightforward methodology, and can thus be used to solve other types of nonlinear equations.