F-Expansion Method and Its Application for Finding New Exact Solutions to the Kudryashov-Sinelshchikov Equation

Abstract

Based on the F-expansion method, and the extended version of F-expansion method, we investigate the exact solutions of the Kudryashov-Sinelshchikov equation. With the aid of Maple, more exact solutions expressed by Jacobi elliptic function are obtained. When the modulus m of Jacobi elliptic function is driven to the limits 1 and 0, some exact solutions expressed by hyperbolic function solutions and trigonometric functions can also be obtained.Journal of Applied Mathematics [29,30], He et al. [31] investigated bifurcations of travelling wave solutions of the Kudryashov-Sinelshchikov equation and proved the existence of the peakon, solitary wave, and smooth and nonsmooth periodic waves. In this paper, we will derive more new exact Jacobian elliptic function solutions of the Kudryashov-Sinelshchikov equation based on the Fexpansion method and its extended version.

Travelling wave solution of Dodd-Bullough-Mikhailov equation: a comparative study between Generalized Kudryashov and improved F-expansion methods

“…A large number of methods have been developed to examine NLPDEs. For example, F-expansion method [1,2], tanh-function method [3,4], the Kudrayshov method [5,6], the modified simple equation method [7][8][9], the Jacobi elliptic-function method [10], the Bernoulli sub-ODE method [11,12], the Expfunction method [13,14], the multiple Exp-function method [15,16], the ¢ ( ) / G G -expansion method [17,18], the variational iteration method [19], the homotopy perturbation method [20], the exp f x -( ( ))-expansion method [21], the homotopy analysis method [22,23], the extended tanh-method [24][25][26], the enhanced ¢ ( ) / G G -expansion method [27][28][29], the Generalized Kudryashov method [30][31][32], and the improved F-expansion method [33][34][35][36], to name a few. Few other methods have been proposed in recent years, for example, Shehu transform [37], Caputo fractional partial derivatives [38] and local fractional homotopy analysis method [39].…”

Closed Form Exact Solutions to the Higher Dimensional Fractional Schrodinger Equation via the Modified Simple Equation Method

“…To know the internal mechanism of complex physical phenomena exact solutions of nonlinear fractional differential equations is very much important. As a result, recently some useful methods have been established and enhanced for obtaining exact solution to the fractional evolution equations such as, the extended direct algebraic function method [3] [4], the F-expansion method [5], the Adomian decomposition method [6], the homotopy perturbation method [7] [8] [9] [10], the tanh-function method [11], the Sine-Cosine method [12], the Jacobi elliptic method [13], the finite difference method [14], the variational iteration method [15] [16], the variational method [17], the Fourier transform technique [18], the modified decomposition method [19], the Laplace transform technique [20], the operational calculus method in [21], the exp-function method [22] [23], the ( ) G G ′ -expansion method [24] [25] [26], the modified simple equation method (MSE) [27]- [34], the ( ) ( ) exp ϕ η − -expansion method [35], the sub equation method [36], the multiple exp-function method [37] [38], the simplest equation method [39], the direct algebraic function method [40] [41] [42] [43], the extended auxiliary equation method [44] etc.…”

Investigation of exact traveling wave solution for the (2+1) dimensional nonlinear evolution equations via modified extended tanh-function method

“…It's prominent that finding exact solutions of nonlinear evolution equations (NLEEs), by using different abundant method plays an important role in the proper understanding of mechanisms of the numerous physical phenomena in mathematical physics and become one of the furthermost exciting and awfully active areas of research investigation for mathematicians, physicist, and engineers. On the basis of the finding new exact solutions of nonlinear evolution equations, many researchers have devoted significant effort to study of exact explicit traveling and solitary wave solutions and several effective techniques have been proposed and developed such as the sine-cosine method [1][2][3], homogeneous balance method [4,5], auxiliary equation method [6,7], the tanhfunction method [8], the extended tanh function method [9,10], the modified extended tanh-function method [11][12][13], the modified simple equation method [14][15][16][17][18], the   G G /  -expansion method [19][20][21][22][23], the Exp-function method [24,25], the )) ( exp(    expansion method [26][27][28], the F-expansion method [29][30][31], ansatz method [32][33] , the first integral method [ 34] and so on. The extended tanh function method, which was developed by Wazwaz [9,10] is a direct and effective algebraic method for handling nonlinear equations and authors [11][1...…”