Construction of shock, periodic and solitary wave solutions for fractional-time Gardner equation by Jacobi elliptic function method

The investigation revolved around the study of the time fractional Gardner equation, which was examined in terms of the conformable derivative. The reduction of the Gardner equation to an integer order nonlinear ordinary differential equation was carried out, and subsequently, the resulting equations were solved using the Jacobi elliptic function method. The construction of exact solutions, including solitary wave, periodic, and shock wave solutions, for the fractional order of the Gardner equation was performed. A comparison between the exact solutions and the fractional solutions was presented. This work is important because the suggested technique offers a simple and efficient way to examine a wide range of nonlinear fractional differential equations. By employing this approach, it becomes possible to solve several nonlinear time-fractional differential equations that involve conformable derivatives. The graphical representation of the resulting data simplifies the process of determining the physical significance of the equation.