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Research Papers

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

[+] Author and Article Information
Youwei Zhang

Department of Mathematics,
Hexi University Zhangye,
Gansu 734000, China
e-mail: ywzhang0288@163.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 20, 2013; final manuscript received July 1, 2013; published online July 29, 2013. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 8(4), 041020 (Jul 29, 2013) (7 pages) Paper No: CND-13-1013; doi: 10.1115/1.4024970 History: Received January 20, 2013; Revised July 01, 2013

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.

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References

Figures

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Fig. 1

The distribution function u(x, t) as a three-dimensional graph for different values of the order α = 1

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Fig. 2

The distribution function u(x, t) as a three-dimensional graph for different values of the order α = 2/3

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Fig. 3

The distribution function u(x, t) as a three-dimensional graph for different values of the order α = 1/2

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Fig. 4

The distribution function u(x, t) as a three-dimensional graph for different values of the order α = 1/4

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Fig. 5

The distribution function u(x, t) as a three-dimensional graph for different values of the order α = 1/6

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Fig. 6

The distribution function u(x, t) as a three-dimensional graph for different values of the order α = 1/8

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Fig. 7

The distribution function u(x, t) as a function of space x at time t = 1 for different values of the order α of three-dimensional graph

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Fig. 8

The distribution function u(x, t) as a function of space x at time t = 1 for different values of the order α of two-dimensional graph

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Fig. 9

The amplitude of the distribution function u(0, t) as a function of the fractional order α at different time values t of three-dimensional graph

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Fig. 10

The amplitude of the distribution function u(0, t) as a function of the fractional order α at different time values t of two-dimensional graph

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Fig. 11

Solitary wave solution of u(x, t)

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