On the fractional order space-time nonlinear equations arising in plasma physics

Abstract

In this study, the $\exp (-\phi (\xi ))$-expansion function method is considered for solving two classes of space-time fractional partial differential equations of very special interest. The two classes, namely the higher dimensional Kadomtsev–Petviashvili and Boussinesq equations, have a wide range applications in different areas of complex nonlinear physics such as plasma physics, fluid dynamics and nonlinear optics. As a result, the $\exp (-\phi (\xi ))$-expansion function method yields a different class of traveling solutions mapped to trigonometric functions, rational functions and hyperbolic functions. Also, the behavior of these solutions has been significantly affected by changing the values of fractional order where the obtained solutions go back to those obtained previously to the normal case, i.e.,$\alpha =\beta =1$. Finally, our finding may be of wide relevance and helpful to better understand the main features and propagation of the nonlinear waves in fractal medium.