scholarly journals Exact Solutions of the Generalized Benjamin-Bona-Mahony Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-5 ◽  
Author(s):  
Xun Liu ◽  
Lixin Tian ◽  
Yuhai Wu
Keyword(s):  
Exact Solutions ◽  
Elliptic Function ◽  
Traveling Wave ◽  
Arbitrary Order ◽  
Traveling Wave Solutions ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Weierstrass Elliptic Function ◽  
Wave Solutions

We apply the theory of Weierstrass elliptic function to study exact solutions of the generalized Benjamin-Bona-Mahony equation. By using the theory of Weierstrass elliptic integration, we get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. This method is effective to find exact solutions of many other similar equations which have arbitrary-order nonlinearity.

scholarly journals Exact Solutions of Equation Generated by the Jaulent-Miodek Hierarchy by(G’/G)-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Wafaa M. Taha ◽  
M. S. M. Noorani
Keyword(s):  
Exact Solutions ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Solitary Wave Solutions ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Wave Solutions ◽  
Tanh Method ◽  
Dimensional Equation

The(G’/G)-expansion method is proposed for constructing more general exact solutions of the nonlinear(2+1)-dimensional equation generated by the Jaulent-Miodek Hierarchy. As a result, when the parameters are taken at special values, some new traveling wave solutions are obtained which include solitary wave solutions which are based from the hyperbolic functions, trigonometric functions, and rational functions. We find in this work that the(G’/G)-expansion method give some new results which are easier and faster to compute by the help of a symbolic computation system. The results obtained were compared with tanh method.

scholarly journals The(G′/G)-Expansion Method and Its Application for Higher-Order Equations of KdV (III)

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Huizhang Yang ◽  
Wei Li ◽  
Biyu Yang
Keyword(s):  
Traveling Wave ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Higher Order ◽  
Trigonometric Functions ◽  
Hyperbolic Functions ◽  
Wave Solutions ◽  
Linear Differential

New exact traveling wave solutions of a higher-order KdV equation type are studied by the(G′/G)-expansion method, whereG=G(ξ)satisfies a second-order linear differential equation. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. The property of this method is that it is quite simple and understandable.

scholarly journals Exact Solutions of the Kudryashov-Sinelshchikov Equation Using the MultipleG′/G-Expansion Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yinghui He ◽  
Shaolin Li ◽  
Yao Long
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Rational Functions ◽  
Expansion Method ◽  
Jacobi Elliptic Functions ◽  
Hyperbolic Functions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

Exact traveling wave solutions of the Kudryashov-Sinelshchikov equation are studied by theG′/G-expansion method and its variants. The solutions obtained include the form of Jacobi elliptic functions, hyperbolic functions, and trigonometric and rational functions. Many new exact traveling wave solutions can easily be derived from the general results under certain conditions. These methods are effective, simple, and many types of solutions can be obtained at the same time.

scholarly journals The first integral method and traveling wave solutions to Davey–Stewartson equation

2012 ◽  
Vol 17 (2) ◽  
pp. 182-193 ◽  
Author(s):  
Hossein Jafari ◽  
Atefe Sooraki ◽  
Yahya Talebi ◽  
Anjan Biswas
Keyword(s):  
Exact Solutions ◽  
Soliton Solution ◽  
Commutative Algebra ◽  
Traveling Wave ◽  
Integral Method ◽  
Traveling Wave Solutions ◽  
First Integral ◽  
First Integral Method ◽  
Wave Solutions

In this paper, the first integral method will be applied to integrate the Davey–Stewartson’s equation. Using this method, a few exact solutions will be obtained using ideas from the theory of commutative algebra. Finally, soliton solution will also be obtained using the traveling wave hypothesis.

scholarly journals Approximate Traveling Wave Solutions of Coupled Whitham-Broer-Kaup Shallow Water Equations by Homotopy Analysis Method

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
M. M. Rashidi ◽  
D. D. Ganji ◽  
S. Dinarvand
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Homotopy Analysis Method ◽  
Shallow Water Equations ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Analysis Method ◽  
Wave Solutions ◽  
Homotopy Analysis

The homotopy analysis method (HAM) is applied to obtain the approximate traveling wave solutions of the coupled Whitham-Broer-Kaup (WBK) equations in shallow water. Comparisons are made between the results of the proposed method and exact solutions. The results show that the homotopy analysis method is an attractive method in solving the systems of nonlinear partial differential equations.

Similarity reductions and exact solutions for two-dimensional Euler–Boussinesq equations

2019 ◽  
Vol 33 (27) ◽  
pp. 1950328
Author(s):  
En Gui Fan ◽  
Man Wai Yuen
Keyword(s):  
Exact Solutions ◽  
Stream Function ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Boussinesq Equations ◽  
Two Dimensional ◽  
Similarity Reduction ◽  
Wave Solutions ◽  
Similarity Reductions

In this paper, by introducing a stream function and new coordinates, we transform classical Euler–Boussinesq equations into a vorticity form. We further construct traveling wave solutions and similarity reduction for the vorticity form of Euler–Boussinesq equations. In fact, our similarity reduction provides a kind of linearization transformation of Euler–Boussinesq equations.

Abundant soliton solutions for the Hirota–Maccari equation via the generalized exponential rational function method

2019 ◽  
Vol 33 (09) ◽  
pp. 1950106 ◽  
Author(s):  
Behzad Ghanbari
Keyword(s):  
Exact Solutions ◽  
Rational Function ◽  
Traveling Wave ◽  
Function Method ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Graphical Interpretation ◽  
Wave Solutions ◽  
Complex Models ◽  
Exponential Rational Function Method

In this paper, some new traveling wave solutions to the Hirota–Maccari equation are constructed with the help of the newly introduced method called generalized exponential rational function method. Several families of exact solutions are found corresponding to the equation. To the best of our knowledge, these solutions are new, and have never been addressed in the literature. The graphical interpretation of the solutions is also depicted. Moreover, it is contemplated that the proposed technique can also be employed to another sort of complex models.

scholarly journals New Solutions of Elastic Waves in an Elastic Rod under Finite Deformation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Peng Guo ◽  
Xiang Wu ◽  
Liangbi Wang
Keyword(s):  
Elliptic Function ◽  
Finite Deformation ◽  
Evolution Equations ◽  
Traveling Wave Solutions ◽  
Mapping Method ◽  
Nonlinear Evolution Equations ◽  
Elastic Rod ◽  
Weierstrass Elliptic Function ◽  
Wave Solutions ◽  
Trigonometric Function Solutions

The nonlinear wave equation of an elastic rod under finite deformation is solved by the extended mapping method. Abundant new exact traveling wave solutions for this equation are obtained, which contain trigonometric function solutions, solitary wave solutions, Jacobian elliptic function solutions, and Weierstrass elliptic function solutions. The method can be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics.

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 Further Results about Traveling Wave Exact Solutions of the Drinfeld-Sokolov Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Fu Zhang ◽  
Jian-ming Qi ◽  
Wen-jun Yuan
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Complex Method ◽  
Wave Solutions ◽  
Periodic Traveling Wave

We employ the complex method to obtain all meromorphic exact solutions of complex Drinfeld-Sokolov equations (DS system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all constant and simply periodic traveling wave exact solutions of the equations (DS) are solitary wave solutions, the complex method is simpler than other methods and there exist simply periodic solutionsvs,3(z)which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.

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