I the present paper, we construct the traveling wave solutions involving parameters of the combined Korteweg-de Vries–modified Korteweg-de Vries equation, the reaction-diffusion equation, the compound KdV–Burgers equation, and the generalized shallow water wave equation by using a new approach, namely, the (G′/G)-expansion method, where G=G(ξ) satisfies a second order linear ordinary differential equation. When the parameters take special values, the solitary waves are derived from the traveling waves. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions.

a b s t r a c tThe fractional derivatives in the sense of Caputo, and the homotopy perturbation method are used to construct approximate solutions for nonlinear Kolmogorov-PetrovskiiPiskunov (KPP) equations with respect to time and space fractional derivatives. Also, we apply complex transformation to convert a time and space fractional nonlinear KPP equation to an ordinary differential equation and use the homotopy perturbation method to calculate the approximate solution. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.

In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved (G ′ /G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation. This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.