scholarly journals The -Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Hasibun Naher ◽  
Farah Aini Abdullah ◽  
M. Ali Akbar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Mathematical Tool ◽  
Wave Solutions ◽  
Fifth Order

We construct the traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG) equation by the -expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the -expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.

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 Conformal Fractional ((D_ξ^α G(ξ))/G(ξ) )- Expansion Method and Its Applications for Solving the Nonlinear Fractional Sharma-Tasso-Olver equation

2021 ◽  
Vol 2 (5) ◽  
pp. 1-8
Author(s):  
Alaaeddin Amin Moussa ◽  
Lama Abdulaziz Alhakim
Keyword(s):  
Partial Differential Equations ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Traveling Wave Solutions ◽  
Rational Functions ◽  
Expansion Method ◽  
Wave Solutions ◽  
Special Cases

In this article, we generalize the ((G^' (ξ))/G(ξ) )- expansion method which is one of the most important methods to finding the exact solutions of nonlinear partial differential equations. The new generalized method, named conformal fractional ((D_ξ^α G(ξ))/G(ξ) )-expansion method, takes advantage of Katugampola’s fractional derivative to create many useful traveling wave solutions of the nonlinear conformal fractional Sharma-Tasso-Olver equation. The obtained solutions have been articulated by the hyperbolic, trigonometric and rational functions with arbitrary constants. These solutions are algebraically verified using Maple and their physical characteristics are illustrated in some special cases.

Symbolic computations: Dispersive soliton solutions for (3+1)-dimensional Boussinesq and Kadomtsev–Petviashvili dynamical equations and its applications

2019 ◽  
Vol 33 (29) ◽  
pp. 1950342 ◽  
Author(s):  
Aly R. Seadawy ◽  
Kalim U. Tariq ◽  
Jian-Guo Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Equation Method ◽  
Dynamical Equations ◽  
Wave Solutions ◽  
Kp Equations

In this paper, the auxiliary expansion equation method is applied to compute the analytical wave solutions for (3[Formula: see text]+[Formula: see text]1)-dimensional Boussinesq and Kadomtsev–Petviashvili (KP) equations. A simple transformation is carried out to reduce the set of nonlinear partial differential equations (NPDEs) into ODEs. These obtained results hold numerous traveling wave solutions that are of key importance in elucidating some physical circumstance.

scholarly journals Exact Traveling Wave Solutions of the Gardner Equation by the Improved tanΘϑ-Expansion Method and the Wave Ansatz Method

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Hatıra Günerhan
Keyword(s):  
Traveling Wave ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Tool ◽  
Ansatz Method ◽  
Gardner Equation ◽  
Wave Solutions ◽  
Analytical Approaches

Nonlinear partial differential equations (NLPDEs) are an inevitable mathematical tool to explore a large variety of engineering and physical phenomena. Due to this importance, many mathematical approaches have been established to seek their traveling wave solutions. In this study, the researchers examine the Gardner equation via two well-known analytical approaches, namely, the improved tanΘϑ-expansion method and the wave ansatz method. We derive the exact bright, dark, singular, and W-shaped soliton solutions of the Gardner equation. One can see that the methods are relatively easy and efficient to use. To better understand the characteristics of the theoretical results, several numerical simulations are carried out.

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