scholarly journals Novel Exact Solutions of the Extended Shallow Water Wave and the Fokas Equations

2018 ◽  
Vol 22 ◽  
pp. 01022
Author(s):  
Serbay DURAN ◽  
Berat KARAAGAC ◽  
Alaattin ESEN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Wave ◽  
Expansion Method ◽  
Travelling Wave ◽  
Wave Transformation ◽  
Partial Differential ◽  
Shallow Water Wave

In this study, a Sine-Gordon expansion method for obtaining novel exact solutions of extended shallow water wave equation and Fokas equation is presented. All of the equations which are under consideration consist of three or four variable. In this method, first of all, partial differential equations are reduced to ordinary differential equations by the help of variable change called as travelling wave transformation, then Sine Gordon expansion method allows us to obtain new exact solutions defined as in terms of hyperbolic trig functions of considered equations. The newly obtained results showed that the method is successful and applicable and can be extended to a wide class of nonlinear partial differential equations.

scholarly journals A Fresh Look To Exact Solutions of Some Coupled Equations

2018 ◽  
Vol 22 ◽  
pp. 01006
Author(s):  
Berat Karaagac ◽  
Nuri Murat Yagmurlu ◽  
Alaattin Esen ◽  
Selcuk Kutluay
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Expansion Method ◽  
Travelling Wave ◽  
Wave Transformation ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Coupled Partial Differential Equations ◽  
Coupled Equations ◽  
New Exact Solutions

This manuscript is going to seek travelling wave solutions of some coupled partial differential equations with an expansion method known as Sine- Gordon expansion method. Primarily, we are going to employ a wave transformation to partial differential equation to reduce the equations into ordinary differential equations. Then, the solution form of the handled equations is going to be constructed as polynomial of hyperbolic trig or trig functions. Finally, with the aid of symbolic computation, new exact solutions of the partial differentials equations will have been found.

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.

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.   

scholarly journals Optical Solutions of Fokas-Lenells Equation via (m-1/G') - Expansion Method

2020 ◽  
Vol 7 ◽  
pp. 20-24
Author(s):  
Hasan Bulut ◽  
Khalid ◽  
Ban Jamal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Efficient Method ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Expansion Method ◽  
Three Dimensions ◽  
Partial Differential ◽  
Wide Range

In this research paper, we investigate some novel soliton solutions to the perturbed Fokas-Lenells equation by using the (m + 1/G') expansion method. Some new solutions are obtained and they are plotted in two and three dimensions. This technique appears as a suitable, applicable, and efficient method to search for the exact solutions of nonlinear partial differential equations in a wide range. All gained optical soliton solutions are substituted into the FokasLenells equation and they verify it. The constraint conditions are also given.

scholarly journals Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method

2015 ◽  
Vol 34 (2) ◽  
pp. 213-229 ◽  
Author(s):  
Hossein Aminikhah ◽  
Amir Hosein Refahi Sheikhani ◽  
Hadi Rezazadeh
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Travelling Wave ◽  
Variable Method ◽  
Boussinesq System ◽  
Wave Transformation ◽  
Partial Differential ◽  
Functional Variable ◽  
Functional Variable Method

In this paper, we will use the functional variable method to construct exact solutions of some nonlinear systems of partial differential equations, including, the (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation, the WhithamBroer-Kaup-Like systems and the Kaup-Boussinesq system. This approach can also be applied to other nonlinear systems of partial differential equations which can be converted to a second-order ordinary differential equation through the travelling wave transformation.

New Exact Solutions of (2+1)-Dimensional Generalization of Shallow Water Wave Equation by (G′/G)-Expansion Method

2010 ◽  
Vol 20-23 ◽  
pp. 1516-1521 ◽  
Author(s):  
Bang Qing Li ◽  
Mei Ping Xu ◽  
Yu Lan Ma
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Wave ◽  
Evolution Equations ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Hyperbolic Function ◽  
Function Solution ◽  
Shallow Water Wave

Extending a symbolic computation algorithm, namely, (G′/G)-expansion method, a new series of exact solutions are constructed for (2+1)-dimensional generalization of shallow water wave equation. These solutions included hyperbolic function solution, trigonometric function solution and rational function solution. The procedure can illustrate that the new algorithm is concise, powerful and straightforward, and it can also be applied to find exact solutions for other high dimensional nonlinear evolution equations.

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