scholarly journals The Extended Fractional (DξαG/G)-Expansion Method and Its Applications to a Space-Time Fractional Fokas Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Yunmei Zhao ◽  
Yinghui He
Keyword(s):  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Space Time ◽  
Hyperbolic Function ◽  
Fractional Partial Differential Equations ◽  
Exact Analytical Solutions ◽  
Wave Solutions ◽  
Liouville Fractional Derivative ◽  
Negative Exponential

Based on a fractional subequation and the properties of the modified Riemann-Liouville fractional derivative, we propose a new analytical method named extended fractional (DξαG/G)-expansion method for seeking traveling wave solutions of fractional partial differential equations. To illustrate the effectiveness of the method, we discuss a space-time fractional Fokas equation, many types of exact analytical solutions are obtained, and the solutions include hyperbolic function and trigonometric and negative exponential solutions.

scholarly journals Traveling Wave Solutions of Space-Time Fractional Generalized Fifth-Order KdV Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Dianchen Lu ◽  
Chen Yue ◽  
Muhammad Arshad
Keyword(s):  
Wave Propagation ◽  
Traveling Wave ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Space Time ◽  
Fractional Partial Differential Equations ◽  
One Dimensional ◽  
Wave Solutions ◽  
Fifth Order

The Korteweg-de Vries (KdV) equation, especially the fractional higher order one, provides a relatively accurate description of motions of long waves in shallow water under gravity and wave propagation in one-dimensional nonlinear lattice. In this article, the generalizedexp⁡(-Φ(ξ))-expansion method is proposed to construct exact solutions of space-time fractional generalized fifth-order KdV equation with Jumarie’s modified Riemann-Liouville derivatives. At the end, three types of exact traveling wave solutions are obtained which indicate that the method is very practical and suitable for solving nonlinear fractional partial differential equations.

scholarly journals New Exact Traveling Wave Solutions of the Time Fractional Complex Ginzburg-Landau Equation via the Conformable Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Zhao Li ◽  
Tianyong Han
Keyword(s):  
Traveling Wave ◽  
Landau Equation ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Wave Transformation ◽  
Hyperbolic Function ◽  
Ginzburg Landau ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Ginzburg Landau Equation

In this study, the exact traveling wave solutions of the time fractional complex Ginzburg-Landau equation with the Kerr law and dual-power law nonlinearity are studied. The nonlinear fractional partial differential equations are converted to a nonlinear ordinary differential equation via a traveling wave transformation in the sense of conformable fractional derivatives. A range of solutions, which include hyperbolic function solutions, trigonometric function solutions, and rational function solutions, is derived by utilizing the new extended G ′ / G -expansion method. By selecting appropriate parameters of the solutions, numerical simulations are presented to explain further the propagation of optical pulses in optic fibers.

scholarly journals Exploration on traveling wave solutions to the 3rd-order klein–fock-gordon equation (KFGE) in mathematical physics

2020 ◽  
Vol 8 (1) ◽  
pp. 14 ◽  
Author(s):  
Nur Hasan Mahmud Shahen ◽  
Foyjonnesa . ◽  
Md. Habibul Bashar
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Dynamic Models ◽  
Gordon Equation ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Analytic Solutions ◽  
Hyperbolic Function ◽  
Wave Solutions

In this paper, the -expansion method has been applied to find the new exact traveling wave solutions of the nonlinear evaluation equations (NLEEs) by utilizing 3rd-order Klein–Gordon Equation (KFGE). With the collaboration of symbolic commercial software maple, the competence of this method for inventing these exact solutions has been more exhibited. As an upshot, some new exact solutions are obtained and signified by hyperbolic function solutions, different combinations of trigonometric function solutions, and exponential function solutions. Moreover, the -expansion method is a more efficient method for exploring essential nonlinear waves that enrich a variety of dynamic models that arises in nonlinear fields. All sketching is given out to show the properties of the innovative explicit analytic solutions. Our proposed method is directed, succinct, and reasonably good for the various nonlinear evaluation equations (NLEEs) related treatment and mathematical physics also. 

scholarly journals Traveling Wave Solutions of Nonlinear Evolution Equation via Enhanced(G’/G)- Expansion Method

2014 ◽  
Vol 33 ◽  
pp. 83-92 ◽  
Author(s):  
Md. Ekramul Islam ◽  
Kamruzzaman Khan ◽  
M Ali Akbar ◽  
Rafiqul Islam
Keyword(s):  
Rational Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Wave Solutions

In this article, the Enhanced (G'/G)-expansion method has been projected to find the traveling wave solutions for nonlinear evolution equations(NLEEs) via the (2+1)-dimensional Burgers equation. The efficiency of this method for finding these exact solutions has been demonstrated with the help of symbolic computation software Maple. By this method we have obtained many new types of complexiton soliton solutions, such as, various combinations of trigonometric periodic function and rational function solutions, various combination of hyperbolic function and rational function solutions. The proposed method is direct, concise and effective, and can be used for many other nonlinear evolution equations. GANIT J. Bangladesh Math. Soc. Vol. 33 (2013) 83-92 DOI: http://dx.doi.org/10.3329/ganit.v33i0.17662

scholarly journals Traveling Wave Solutions of Two Nonlinear Wave Equations by (G′/G)-Expansion Method

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Yazhou Shi ◽  
Xiangpeng Li ◽  
Ben-gong Zhang
Keyword(s):  
Nonlinear Wave ◽  
Traveling Wave ◽  
Wave Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Nonlinear Wave Equations ◽  
Hyperbolic Function ◽  
Exact Traveling Wave Solutions ◽  
Function Solution ◽  
Wave Solutions

We employ the (G′/G)-expansion method to seek exact traveling wave solutions of two nonlinear wave equations—Padé-II equation and Drinfel’d-Sokolov-Wilson (DSW) equation. As a result, hyperbolic function solution, trigonometric function solution, and rational solution with general parameters are obtained. The interesting thing is that the exact solitary wave solutions and new exact traveling wave solutions can be obtained when the special values of the parameters are taken. Comparing with other methods, the method used in this paper is very direct. The (G′/G)-expansion method presents wide applicability for handling nonlinear wave equations.

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