scholarly journals Some Applications of the (G′/G,1/G)-Expansion Method for Finding Exact Traveling Wave Solutions of Nonlinear Fractional Evolution Equations

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 952
Author(s):  
Sekson Sirisubtawee ◽  
Sanoe Koonprasert ◽  
Surattana Sungnul
Keyword(s):  
Partial Differential Equations ◽  
Solitary Wave ◽  
Differential Equations ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Solitary Wave Solutions ◽  
Partial Differential ◽  
Wave Solutions

In this paper, the ( G ′ / G , 1 / G ) -expansion method is applied to acquire some new, exact solutions of certain interesting, nonlinear, fractional-order partial differential equations arising in mathematical physics. The considered equations comprise the time-fractional, (2+1)-dimensional extended quantum Zakharov-Kuznetsov equation, and the space-time-fractional generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) system in the sense of the conformable fractional derivative. Applying traveling wave transformations to the equations, we obtain the corresponding ordinary differential equations in which each of them provides a system of nonlinear algebraic equations when the method is used. As a result, many analytical exact solutions obtained of these equations are expressed in terms of hyperbolic function solutions, trigonometric function solutions, and rational function solutions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions, solitary wave solutions of kink type, and so on. The method is very efficient, powerful, and reliable for solving the proposed equations and other nonlinear fractional partial differential equations with the aid of a symbolic software package.

scholarly journals Exact Traveling Wave Solutions of Certain Nonlinear Partial Differential Equations Using the G′/G2-Expansion Method

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Sekson Sirisubtawee ◽  
Sanoe Koonprasert
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Rational Solutions ◽  
Partial Differential ◽  
Wave Solutions

We apply the G′/G2-expansion method to construct exact solutions of three interesting problems in physics and nanobiosciences which are modeled by nonlinear partial differential equations (NPDEs). The problems to which we want to obtain exact solutions consist of the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The obtained exact solutions of the problems via using the method are categorized into three types including trigonometric solutions, exponential solutions, and rational solutions. The applications of the method are simple, efficient, and reliable by means of using a symbolically computational package. Applying the proposed method to the problems, we have some innovative exact solutions which are different from the ones obtained using other methods employed previously.

Solitary wave solutions of (2+1)-dimensional Maccari system

2021 ◽  
pp. 2150391
Author(s):  
Ghazala Akram ◽  
Naila Sajid
Keyword(s):  
Nonlinear Optics ◽  
Solitary Wave ◽  
Plasma Physics ◽  
Exact Solutions ◽  
Function Method ◽  
Expansion Method ◽  
Periodic Wave ◽  
Hyperbolic Function ◽  
Solitary Wave Solutions ◽  
Wave Solutions

In this article, three mathematical techniques have been operationalized to discover novel solitary wave solutions of (2+1)-dimensional Maccari system, which also known as soliton equation. This model equation is usually of applicative relevance in hydrodynamics, nonlinear optics and plasma physics. The [Formula: see text] function, the hyperbolic function and the [Formula: see text]-expansion techniques are used to obtain the novel exact solutions of the (2+1)-dimensional Maccari system (arising in nonlinear optics and in plasma physics). Many novel solutions such as periodic wave solutions by [Formula: see text] function method, singular, combined-singular and periodic solutions by hyperbolic function method, hyperbolic, rational and trigonometric solutions by [Formula: see text]-expansion method are obtained. The exact solutions are shown through 3D graphics which present the movement of the obtained solutions.

scholarly journals Application of the extended exp(-φ(ξ))-expansion method to the nonlinear conformable time-fractional partial differential equations

2019 ◽  
Vol 7 (2) ◽  
pp. 81
Author(s):  
Dipankar Kumar ◽  
Samir Chandra Ray
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Expansion Method ◽  
Point Of View ◽  
Hyperbolic Function ◽  
Fractional Partial Differential Equations ◽  
Kawahara Equation ◽  
Partial Differential ◽  
New Exact Solutions

This paper investigates the new exact solutions of the three nonlinear time fractional partial differential equations namely the nonlinear time fractional Clannish Random Walker’s Parabolic (CRWP) equation, the nonlinear time fractional modified Kawahara equation, and the nonlinear time fractional BBM-Burger equation by utilizing an extended form of exp(-φ(ξ))-expansion method in the sense of conformable fractional derivative. As outcomes, some new exact solutions are obtained and signified by hyperbolic function solutions, trigonometric function solutions, and rational function solutions. Some solutions have been plotted by MATLAB software to show the physical significance of our studied equations. In the point of view of our executed method and generated results, we may conclude that extended exp (-φ(ξ))-expansion method is more efficient than exp(-φ(ξ))-expansion method to extract the new exact solutions for solving any types of integer and fractional differential equations arising in mathematical physics.   

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