Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics

Abstract

In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the K (m, n) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.

On the Solution of Nonlinear Time-Fractional Generalized Burgers Equation by Homotopy Analysis Method and Modified Trial Equation Method

“…Using a complete discrimination system for polynomial to classify the roots of ( ) u F , we solve (11) with the help of Mathematica 9 and classify the exact solutions to (6). In addition, we can write the exact traveling wave solutions of (4).…”

Symmetric Fibonacci Function Solutions of some Nonlinear Partial Differential Equations

“…Using the generalized balance formula (15) for the nonlinear terms VV ′′ and V ′ V 2 in Eq. (18), we find…”

“…In literature, there are also a lot of methods that are used to solve the nonlinear partial differential equations such as the ansatz method [7][8][9][10][11], the exp-function method [12,13], the trial equation method [14,15], the (G ′ /G)-expansion method [16], the Hirota's method [17,18], the three wave method [19], extended Jacobi elliptic function expansion method [20], Kudyrashov method [21], semi-inverse variational principle [22], the multiplier method using the Lie symmetry [23] and many more. Key idea of this paper is that traditional base e of the exponential function is replaced by an arbitrary base a = 1.…”

Solving nonlinear evolution equation system using two different methods

“…Obtaining their explicit solutions is quite di cult. So far, with the development of soliton theory, many e cient methods for obtaining these exact solutions have been presented, such as Hirota bilinear tranformation [1], Darboux and Backlund transform [2], Weierstrass function method [3], symmetry method [4,5], Painleve analysis method [6], theta function method [7], Wronskian technique [8], homogeneous balance method [9,10], F-expansion method [11], sine-cosine method [12,13], exponential function method [14,15], inverse scattering method [16,17], functional variable method [18], extended tanh method [19], modi ed simple equation method [20,21], trial equation method [22], (G ′ /G)-expansion method [23,24], sub-equation method [25], auxiliary equation method [26] and so on. When we nd the exact solutions of nonlinear partial di erential equations by using the (G ′ /G)-expansion method, we obtain the solution in the terms of (G ′ /G).…”