Solutions of Kudryashov - Sinelshchikov equation and generalized Radhakrishnan-Kundu-Lakshmanan equation by the first integral method

Abstract

This paper shows the applicability of the First Integral Method in obtaining solutions of Nonlinear Partial Differential Equations (NLPDEs). The method is applied in constructing solutions of Kudryashov-Sinelshchikov equation (KSE) and Generalized Radhakrishnan-Kundu-Lakshmanan Equation (GRKLE). The First Integral Method, which is based on the Ring Theory of Commutative Algebra, is a direct algebraic method for obtaining exact solutions of NLPDEs. This method is applicable to integrable as well as nonintegrable NLPDEs. The method is an efficient method for obtaining exact solutions of many Nonlinear Evolution Equations (NLEEs).

A variety of exact analytical solutions of extended shallow water wave equations via improved (G’/G) -expansion method

“…In the recent past, many researchers have implemented various analytical methods to seek traveling wave solutions of different nonlinear partial differential equations. Many powerful methods have been successfully developed by diverse groups of mathematicians and physicists, such as, Hirota's bilinear method [1], the tanh-function method [2], [3], the extended than-function method [4], the exp -function method [5][6][7], sine-cosine method [8], the Inverse scattering transforms [9], the Jacobi elliptic function expansion [10], [11], the homogeneous balance method [12], the Homotopy perturbation methods [13], auxiliary equation method [14], the first integral method [15][16][17], the tanh-coth method [18], the Cole-Holf transformation method [19], the   '/ GG-expansion method [20][21][22], the improved   '/ GG -expansion method [23], the Enhance   '/ GG -expansion method [24] and so on. The improved…”

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...…”