Yasser A. Amer

The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics

Abstract In this article, the modified extended tanh-function method is employed to solve fractional partial differential equations in the sense of the modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into nonlinear ordinary differential equations of integer orders. For illustrating the validity of this method, we apply it to four nonlinear equations namely, the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space–time fractional nonlinear Whitham–Broer–Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations.

Comment on " Application of Improved - Expansion Method to Traveling Wave Solutions of Two Nonlinear Evolution Equations, Adv. Appl. Math. Mech. 4(2012) 122-130"

The authors of the above article proposed the improved ( / ) G G  expansion method and found some traveling wave solutions for each of two nonlinear evolution equations in mathematical physics, namely the Regularized Long Wave (RLW) equation and the Symmetric Regularized Long Wave (SRLW) equation. In the present article, we have noted that if we use a suitable transformation, the improved (G’/G)expansion method can be reduced into the well -known generalized Riccati equation mapping method which provides us with much more traveling wave solutions, namely twenty seven solutions for each of these two nonlinear evaluation equations. Comparison between the results of these two methods is presented. General Terms 02.30.Jr, 05.45.Yv, 02.30.Ik