Abstract
The two variable (G'⁄G, 1⁄G)-expansion method is significant for finding the exact traveling wave solution to nonlinear evolution equations (NLEEs) in mathematical physics, applied mathematics and engineering. In this article, we exert the two variable (G'⁄G, 1⁄G)-expansion method for investigating the fractional generalized reaction Duffing model and density dependent fractional diffusion reaction equation and obtain exact solutions containing parameters. When the parameters are taken particular values, traveling wave solutions are transferred into the solitary wave solutions. The two variable (G'⁄G, 1⁄G)-expansion method is the generalization of the original (G'⁄G)-expansion method established by Wang et al [21].
Highlights
The significance of nonlinear evolution equations is well established
As a key problem, finding their exact solutions is of great importance and it is executed through various efficient and powerful method, such as, the Hirota method [1], the Backlund transform method [2, 3], the inverse scattering transform method [4], the Jacobi elliptic function expansion method [5,6,7], the truncated Painleve expansion method [8,9,10,11], the tanh function method [12,13,14,15], the Exp-function method [16,17,18,19,20,21,22], the ( ⁄ ) -expansion method [23,24,25,26,27,28,29,30], the improved ( ⁄ ) -expansion method [31,32], the two variable ( ⁄, 1⁄ ) -expansion method [33, 34], the first integral method [35] etc
The main concept of the ( ⁄ ) -expansion method is the exact solution of nonlinear nonlinear evolution equations (NLEEs) are revealed by a polynomial in one variable ( ⁄ ) in which = ( ) satisfies the second order ordinary differential equation (ODE) ( ) +
Summary
The significance of nonlinear evolution equations is well established. In the last three decades nonlinear phenomena are one of the most impressive fields of research. The main concept of the ( ⁄ ) -expansion method is the exact solution of nonlinear NLEEs are revealed by a polynomial in one variable ( ⁄ ) in which = ( ) satisfies the second order ordinary differential equation (ODE) ( ) +. The main concept of the two variable ( ⁄ , 1⁄ )expansion method is the exact traveling wave solutions of nonlinear NLEEs can be written as a polynomials in two variables ( ⁄ ) and (1⁄ ), in which = ( ) satisfies a second order linear ODE ( ) + ( ) = , where and are constants. Solving the algebraic equation with the aid of Maple or Mathematica, we obtain the following results: 7< = 7
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