Stable and effective traveling wave solutions to the non-linear fractional Gardner and Zakharov–Kuznetsov–Benjamin–Bona–Mahony equations

Abstract

The space–time fractional Gardner and Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equations are used to explain the transmission of shallow water waves inside a water channel of uniform speed and constant depth. It moreover simulates water waves in harbors and shallow water, which is useful for oceanography. These equations are significant nonlinear model equations to illustrate numerous physical structures namely in marine and coastal science, water wave mechanics, control theory, plasma engineering, fluid flow, optical fibers, phenomena of fission and fusion, and so on. In this study, we employed the extended tanh-function approach to generate some fresh and further general closed-form traveling waves solutions of those equations in the lite of conformal derivatives. The achieve results are more applicable to resolve the above mentioned phenomena properly. The fractional differential transform simplifies generate ordinary differential equations from fractional order differential equations. We discovered many types of solutions using the maple, including solitons, kink types, bell types, and other types of solutions that are illustrated using 3D and contour plots. It is important to note that all derived solutions are checked for accuracy by being directly replaced with the original equation. We proposed that the technique be revised to be more realistic, effective, and trustworthy and that we explore more generalized exact results of traveling waves, like solitary wave solutions.

Keywords

• The space–time fractional Gardner equation
• The conformable derivative
• The extended tanh-function technique
• The space–time fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equations
• Non-linear fractional partial differential equation