New Soliton Solutions to the Space-Time Fractional Boussinesq Equation Using a Reliable Method

Fractional partial Differential Equations (FPDEs) play an essential role in interpreting a broad variety of events in various scientific disciplines, including physics, engineering, biology, and economics. One remarkable feature of PDEs is the presence and investigation of traveling wave solutions, which are solutions that propagate with a constant speed and preserve their shape. The main purpose of this work is to obtain the traveling wave solutions of the space-time fractional Boussinesq equation using a relatively new mechanism which is the improved modified extended tanh-function approach. Fractional derivatives based on Jumarie’s modified Riemann–Liouville are utilized to deal with the fractional derivatives which appear in the fractional Boussinesq problem. Some periodic and solitary wave solutions are presented in the form of trigonometric, hyperbolic, complex, and rational functions. In addition, the effectiveness of the employed methodology is compared with the performances of other techniques such as the fractional sub-equation approach, (G’/G2)-expansion process, and the modified Kudryashov method.