scholarly journals On Differential Equations Derived from the Pseudospherical Surfaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hongwei Yang ◽  
Xiangrong Wang ◽  
Baoshu Yin
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Curvature Scalar ◽  
Riemann Curvature ◽  
Tensor Fields ◽  
Nonlinear Superposition ◽  
Metric Tensor ◽  
New Exact Solutions ◽  
Pseudospherical Surfaces

We construct two metric tensor fields; by means of these metric tensor fields, sinh-Gordon equation and elliptic sinh-Gordon equation are obtained, which describe pseudospherical surfaces of constant negative Riemann curvature scalarσ= −2,σ= −1, respectively. By employing the Bäcklund transformation, nonlinear superposition formulas of sinh-Gordon equation and elliptic sinh-Gordon equation are derived; various new exact solutions of the equations are obtained.

Application of the Subequation Method to Some Differential Equations of Time-Fractional Order

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Ahmet Bekir ◽  
Esin Aksoy
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Order ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
New Exact Solutions

The main goal of this paper is to develop subequation method for solving nonlinear evolution equations of time-fractional order. We use the subequation method to calculate the exact solutions of the time-fractional Burgers, Sharma–Tasso–Olver, and Fisher's equations. Consequently, we establish some new exact solutions for these equations.

scholarly journals New Exact Solutions for a Generalized Double Sinh-Gordon Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Gabriel Magalakwe ◽  
Chaudry Masood Khalique
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Fluid Dynamics ◽  
Exact Solutions ◽  
Function Method ◽  
Gordon Equation ◽  
Quantum Field ◽  
Integrable Quantum ◽  
New Exact Solutions ◽  
Integrable Quantum Field Theory

We study a generalized double sinh-Gordon equation, which has applications in various fields, such as fluid dynamics, integrable quantum field theory, and kink dynamics. We employ the Exp-function method to obtain new exact solutions for this generalized double sinh-Gordon equation. This method is important as it gives us new solutions of the generalized double sinh-Gordon equation.

scholarly journals Comparison between the (G’/G) - expansion method and the modified extended tanh method

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 88-94 ◽  
Author(s):  
Şamil Akçaği ◽  
Tuğba Aydemir
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
The Other ◽  
Extended Tanh Method ◽  
New Exact Solutions ◽  
Tanh Method

AbstractIn this paper, firstly, we give a connection between well known and commonly used methods called the $\left( {{{G'} \over G}} \right)$ -expansion method and the modified extended tanh method which are often used for finding exact solutions of nonlinear partial differential equations (NPDEs). We demonstrate that giving a convenient transformation and formula, all of the solutions obtained by using the $\left( {{{G'} \over G}} \right)$ - expansion method can be converted the solutions obtained by using the modified extended tanh method. Secondly, contrary to the assertion in some papers, the $\left( {{{G'} \over G}} \right)$-expansion method gives neither all of the solutions obtained by using the other method nor new solutions for NPDEs. Namely, while the modified extended tanh method gives more solutions in a straightforward, concise and elegant manner without reproducing a lot of different forms of the same solution. On the other hand, the $\left( {{{G'} \over G}} \right)$-expansion method provides less solutions in a rather cumbersome form. Lastly, we obtain new exact solutions for the Lonngren wave equation as an illustrative example by using these methods.

scholarly journals New Generalized Hyperbolic Functions to Find New Exact Solutions of the Nonlinear Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yusuf Pandir ◽  
Halime Ulusoy
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Wave Equations ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Hyperbolic Function ◽  
Hyperbolic Functions ◽  
Partial Differential ◽  
New Exact Solutions

We firstly give some new functions called generalized hyperbolic functions. By the using of the generalized hyperbolic functions, new kinds of transformations are defined to discover the exact approximate solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function transformation of the generalized KdV equation and the coupled equal width wave equations (CEWE), we find new exact solutions of two equations and analyze the properties of them by taking different parameter values of the generalized hyperbolic functions. We think that these solutions are very important to explain some physical phenomena.

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