New applications of the fractional derivative to extract abundant soliton solutions of the fractional order PDEs in mathematics physics

The motive of this research work is to unravel the mysteries of nature through fractional order partial differential equations (PDEs). Here, we focus on two important fractional order nonlinear PDEs, namely the fractional order (4+1)-dimensional Fokas equation, which is used to give the model of many physical phenomena and dynamical processes, and the other one is the fractional order (2+1)-dimensional breaking soliton equation which is used to analyze the nonlinear problems like optical fiber communications, ocean engineering, etc. Recently, it has been an essential topic to extract the new soliton solutions which are used to investigate the hidden physical conditions of the nonlinear fractional PDEs. For this reason, it is essential to solve those nonlinear fractional PDEs which have a physical impact in the fields of science and modern engineering. In our investigation, we attempt to provide nonlinear wave propagation patterns and investigate the equations, as mentioned earlier, through a computational method. A computing operating software called Mathematica has been applied to get a clear visualization of our gained outcomes, and we ascertain such types of shapes as the bell shape soliton, the anti-bell shape soliton, the singular bell shape soliton, the periodic solution, and the singular periodic solution. All in all, our obtained results can keep an indispensable role in explaining various physical phenomena of nature in the near future, and the applied method are very cogent, efficient, and relatively latest method to extract such types of soliton solutions.