A non-linear analysis and fractionalized dynamics of Langmuir waves and ion sound as an application to acoustic waves

There is no refusing fact that ion sound and Langmuir waves generate complex instabilities during oscillations of electrons. In this study, the sinh-Gordon function (ShGFM) and 1/G'-expansion methods have been successfully applied to the fractional ion sound and Langmuir waves (FISALWs) equation. Using both analytical methods, trigonometric and hyperbolic type traveling wave solutions have been produced. The motion of a particle in the electromagnetic field is represented by the generated traveling wave solutions. In these applications, we consider the comformable fractional operator to which the chain rule is applied. Special values were given to the constants in the solution while drawing graphs representing the stationary wave. These two analytical methods used to obtain analytical solutions of the FISALWs equation have been analysed in detail by comparing their respective states. By using symbolic calculations, these methods have been shown to be the powerful and reliable mathematical tool for the solution of fractional non-linear partial differential equations.