Time-Fractional Camassa–Holm Equation: Formulation and Solution Using Variational Methods

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
Youwei Zhang
Keyword(s):  
Solitary Wave ◽  
Variational Methods ◽  
Iteration Method ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Variational Iteration Method ◽  
Variational Technique ◽  
Holm Equation ◽  
Liouville Derivative ◽  
He’S Variational Iteration Method

This paper presents the formulation of the time-fractional Camassa–Holm equation using the Euler–Lagrange variational technique in the Riemann–Liouville derivative sense and derives an approximate solitary wave solution. Our results witness that He's variational iteration method was a very efficient and powerful technique in finding the solution of the proposed equation.

THE EXPLICIT NONLINEAR WAVE SOLUTIONS AND THEIR BIFURCATIONS OF THE GENERALIZED CAMASSA–HOLM EQUATION

2011 ◽  
Vol 21 (11) ◽  
pp. 3119-3136 ◽  
Author(s):  
ZHENGRONG LIU ◽  
YONG LIANG
Keyword(s):  
Solitary Wave ◽  
Nonlinear Wave ◽  
Wave Solution ◽  
Wave Speed ◽  
Solitary Wave Solution ◽  
Wave Solutions ◽  
Holm Equation ◽  
Bifurcation Phenomena ◽  
Singular Wave ◽  
Nonlinear Wave Solutions

In this paper, we study the explicit nonlinear wave solutions and their bifurcations of the generalized Camassa–Holm equation [Formula: see text]Not only are the precise expressions of the explicit nonlinear wave solutions obtained, but some interesting bifurcation phenomena are revealed.Firstly, it is verified that k = 3/8 is a bifurcation parametric value for several types of explicit nonlinear wave solutions.When k < 3/8, there are five types of explicit nonlinear wave solutions, which are(i) hyperbolic peakon wave solution,(ii) fractional peakon wave solution,(iii) fractional singular wave solution,(iv) hyperbolic singular wave solution,(v) hyperbolic smooth solitary wave solution.When k = 3/8, there are two types of explicit nonlinear wave solutions, which are fractional peakon wave solution and fractional singular wave solution.When k > 3/8, there is not any type of explicit nonlinear wave solutions.Secondly, it is shown that there are some bifurcation wave speed values such that the peakon wave and the anti-peakon wave appear alternately.Thirdly, it is displayed that there are other bifurcation wave speed values such that the hyperbolic peakon wave solution becomes the fractional peakon wave solution, and the hyperbolic singular wave solution becomes the fractional singular wave solution.

EXPLICIT PERIODIC WAVE SOLUTIONS AND THEIR BIFURCATIONS FOR GENERALIZED CAMASSA–HOLM EQUATION

2010 ◽  
Vol 20 (08) ◽  
pp. 2507-2519 ◽  
Author(s):  
ZHENGRONG LIU ◽  
HAO TANG
Keyword(s):  
Solitary Wave ◽  
Wave Solution ◽  
Blow Up ◽  
Solitary Wave Solution ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Periodic Wave Solutions ◽  
Bifurcation Phenomenon ◽  
Wave Solutions ◽  
Holm Equation

In this paper, through qualitative analysis and integration, we study the explicit periodic wave solutions and their bifurcations for the generalized Camassa–Holm equation [Formula: see text] When the parameter k satisfies k < 3/8 and the constant wave speed c satisfies [Formula: see text], we obtain two types of explicit periodic wave solutions, elliptic smooth periodic wave solution and elliptic periodic blow-up solutions. These solutions include a bifurcation parameter α which has four bifurcation values αi(i = 1, 2, 3, 4). When α tends to the bifurcation values, the elliptic periodic wave solutions become three types of other solutions, the hyperbolic smooth solitary wave solution, the hyperbolic blow-up solution and the trigonometric periodic blow-up solution. Especially, a new bifurcation phenomenon is found, that is, the periodic blow-up solution can become a smooth solitary wave solution when α varies. When k > 3/8, we guess that there is no other explicit solution except the explicit periodic blow-up solution.

scholarly journals Application of He’s Variational Iteration Method to Nonlinear Integro-Differential Equations

2010 ◽  
Vol 65 (5) ◽  
pp. 418-430 ◽  
Author(s):  
Ahmet Yildirim
Keyword(s):  
Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Mathematical Tool ◽  
Variational Iteration ◽  
He’S Variational Iteration Method ◽  
Powerful Mathematical Tool

In this paper, an application of He’s variational iteration method is applied to solve nonlinear integro-differential equations. Some examples are given to illustrate the effectiveness of the method. The results show that the method provides a straightforward and powerful mathematical tool for solving various nonlinear integro-differential equations

scholarly journals Abundant Explicit and Exact Solutions for the Variable Coefficient mKdV Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xiaoxiao Zheng ◽  
Yadong Shang ◽  
Yong Huang
Keyword(s):  
Solitary Wave ◽  
Elliptic Function ◽  
Constant Coefficient ◽  
Wave Solution ◽  
Integral Method ◽  
Solitary Wave Solution ◽  
First Integral ◽  
Mkdv Equation ◽  
Variable Coefficients ◽  
First Integral Method

This paper is concerned with the variable coefficients mKdV (VC-mKdV) equation. First, through some transformation we convert VC-mKdV equation into the constant coefficient mKdV equation. Then, using the first integral method we obtain the exact solutions of VC-mKdV equation, such as rational function solutions, periodic wave solutions of triangle function, bell-shape solitary wave solution, kink-shape solitary wave solution, Jacobi elliptic function solutions, and Weierstrass elliptic function solution. Furthermore, with the aid of Mathematica, the extended hyperbolic functions method is used to establish abundant exact explicit solution of VC-mKdV equation. By the results of the equation, the first integral method and the extended hyperbolic function method are extended from the constant coefficient nonlinear evolution equations to the variable coefficients nonlinear partial differential equation.

Sign in / Sign up

Export Citation Format

Share Document