The extended -expansion method and its applications to the Whitham–Broer–Kaup–Like equations and coupled Hirota–Satsuma KdV equations

Guo, Shenyang; Zhou, Youhe

doi:10.1016/j.amc.2009.10.008

Cited by 119 publications

(54 citation statements)

References 33 publications

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“…We present an expansion method by inspiring the (G /G)-expansion method which was introduced by Wang et al [22] and the other similar methods [7,24]. We implement this method to find several analytic solutions of the time fractional Burgers' equation in form of rational trigonometric solutions, but we consider one of these solutions for the numerical examination.…”

Section: Introductionmentioning

confidence: 99%

Numerical and exact solutions for time fractional Burgers' equation

Yokuş¹,

Kaya²

2017

J. Nonlinear Sci. Appl.

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The main purpose of this paper is to find an exact solution of the traveling wave equation of a nonlinear time fractional Burgers' equation using the expansion method and the Cole-Hopf transformation. For this purpose, a nonlinear time fractional Burgers' equation with the initial conditions considered. The finite difference method (FDM for short) which is based on the Caputo formula is used and some fractional differentials are introduced. The Burgers' equation is linearized by using the ColeHopf transformation for a stability of the FDM. It shows that the FDM is stable for the usage of the Fourier-Von Neumann technique. Accuracy of the method is analyzed in terms of the errors in L 2 and L ∞ . All of obtained results are discussed with an example of the Burgers' equation including numerical solutions for different situations of the fractional order and the behavior of potentials u is investigated with graphically. All the obtained numerical results in this study are presented in tables. We used the Mathematica software package in performing this numerical study.

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Section: Introductionmentioning

confidence: 99%

Numerical and exact solutions for time fractional Burgers' equation

Yokuş¹,

Kaya²

2017

J. Nonlinear Sci. Appl.

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“…Subsequently, a group of scientists studied various nonlinear PDEs to construct many travelling wave solutions (Hamad et al, 2011;Naher and Abdullah, 2012d to f;. Guo and Zhou (2010) introduced the extended (G'/G)-expansion method for obtaining travelling wave solutions of the Whitham-Broer-Kaup-like equation and couple Hirota-Satsuma KdV equations. Consequently, many researchers implemented this method to obtain analytical solutions of various nonlinear partial differential equations (Zayed and Al-Joudi, 2010;Zayed and El-Malky, 2011).…”

Section: Introductionmentioning

confidence: 99%

New Approach of (<em>G\'/G</em>)-expansion Method for RLW Equation

Naher¹,

Abdullah²

2014

RJASET

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Abstract:In this study, new extension of the (G'/G)-expansion method with nonlinear ODE has been introduced to investigate the generalized regularized long wave equation. Therefore, many new travelling wave solutions with many parameters are generated. The obtained solutions are more general and solutions include the solutions, the hyperbolic function, the trigonometric function and the rational forms. In addition, some of our solutions are identical with already published results, which validate newly generated solutions and others solutions have not been reported in the previous literature.

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“…This method can also be applied to fractional ODEs 11 . To extend the range of its applicability, the generalized G /G-expansion method 12 , the extended G /G-expansion method 13 and the improved G /Gexpansion method 14 have been developed. The nonlinear shallow water surface waves satisfy the Korteweg-de Vries (KdV) equation…”

Section: Introductionmentioning

confidence: 99%