Families of exact traveling wave solutions to the space time fractional modified KdV equation and the fractional Kolmogorov-Petrovskii-Piskunovequation

The space time fractional modified KdV equation and fractional Kolmogorov-Petrovskii-Piskunov (KPP) equation models the unidirectional and bidirectional waves on shallow water surfaces, long internal wave in a density-stratified ocean, ion acoustic waves in plasma, acoustic waves on a crystal lattice. The fractional derivatives are defined in the modified Riemann-Liouville sense. In this article, we obtain exact solution of these equations by means of the recently established two variables (G'/G, 1/G) - expansion method. The solutions are obtained in the form of hyperbolic, trigonometric and rational functions involving parameters. When the parameters are assigned particular values, the solitary wave solutions are generated from the traveling wave solutions. The method indicates that it is easy to implement, computationally attractive and is the general form of the original (G'/G) - expansion method.