Solving unsteady Korteweg–de Vries equation and its two alternatives

Abstract

Communicated by S. A. MessaoudiThis paper aims to present the generalized Kudryashov method to find the exact traveling wave solutions transmutable to the solitary wave solutions of the ubiquitous unsteady Korteweg-de Vries equation and its two famed alternatives, namely, the regularized long-wave equation and the time regularized long-wave equation. The exact analytic solutions of the studied equations are constructed explicitly in three forms, namely, hyperbolic, trigonometric, and rational function. The validity of our solutions is verified with MAPLE by putting them back into the original equation and found correct. Moreover, it has shown that the generalized Kudryashov method is an easy and reliable technique over the existing methods.

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

“…The working steps of the Generalized Kudryashov and improved F-expansion methods are described in the following two subsections. [30][31][32] The main steps of Generalized Kudryashov method are as follows:…”

“…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].…”

Two Different Systematic Techniques to Seek Analytical Solutions of the Higher-Order Modified Boussinesq Equation

“…into Eq. ( 18) respectively to yield the systems of algebraic equations, and solve the algebraic equations to obtain elliptic function solutions, rational function solutions and simply periodic solutions with the pole at z = 0, where β −ij are determined by (19),…”

“…For example, the singular behaviors [2], [3] and impulsive phenomena [4], [5] often show some blow-up properties [6], [7] which occur in lots of complex physical processes. In order to solve various differential equations, symbolic calculation techniques as well as some analytical tools were established, such as sine-Gordon expansion method [8]- [10], modified simple equation method [11], modified extended tanh method [12]- [15], Kudryashov method [16]- [19], generalized (G /G)-expansion method [20]- [23], improved F-expansion method [24], exp(−ψ(z))-expansion method [25]- [29], complex method [30]- [35], fixed point method [36]- [39], and topological degree method [40]- [43].…”

Exact solutions of the classical Boussinesq system

“…To seek the exact solutions of the nonlinear partial differential equations, many methods have been proposed, for instance the inverse scattering transformation (IST) (Ablowitz & Segur, 2000), B€ acklund transformation (Rogers & Schief, 2002), Painlev e analysis method (Chowdhury, 1999), Darboux transformation (DT) (Gu, Hu, & Zhou, 2005), Hirota direct method (Hirota, 2004), the tanh-function method (Parkes & Duffy, 1996;Zhang, Xu, & Li, 2002), the improved F-expansion method (Islam, Khan, Akbar, & Mastroberardino, 2014;Wang & Zhang, 2005), the modified simple equation method (Akter & Akbar, 2015;Khan, Akbar, & Alam, 2013), the (G 0 =G)-expansion and extend (G 0 =G)-expansion method (Akbar & Ali, 2011;Akbar, Ali, & Mohyud-Din, 2013;Alam, Hafez, Belgacem, & Akbar, 2015b;Zayed & Shorog, 2010), the Exp-function method (He & Abdou, 2007;, the generalized Kudryashov method (Khan & Akbar, 2016), the expðÀUðgÞÞ-expansion and expðUðgÞÞ method (Alam, Hafez, Akbar, & Roshid, 2015a;Roshid & Rahman, 2014), the extended three-wave method Li, Dai, & Liu, 2011;Singh & Gupta, 2016;Wang, Dai, & Liang, 2010). In this paper, based on the bilinear form, we consider exact solutions including solitary wave solution, periodic solitary solution and rational solution of the classical Boussinesq (CB) system (see Wu & Zhang, 1996) …”