0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

He, J., 1997, “A New Approach to Nonlinear Partial Differential Equations,” Commun. Nonlinear Sci. Numer. Simul., 2(4), pp. 230–235. [CrossRef]
He, J., 1998, “Approximate Analytical Solution for Seepage Flow With Fractional Derivatives in Porous Media,” Comput. Meth. Appl. Mech. Eng., 167, pp. 57–68. [CrossRef]
Molliq, R. Y., Noorani, M. S. M., and Hashim, I., 2009, “Variational Iteration Method for Fractional Heat- and Wave-Like Equations,” Nonlinear Anal. Real World Appl., 10, pp. 1854–1869. [CrossRef]
Momani, S., Odibat, Z., and Alawnah, A., 2008, “Variational Iteration Method for Solving the Space-and Time-Fractional KdV Equation,” Numer. Meth. Part. Differ. Equ., 24(1), pp. 261–271. [CrossRef]
Camassa, R., Holm, D., and Hyman, J., 1994, “A New Integrable Shallow Water Equation,” Adv. Appl. Mech., 31, pp. 1–33. [CrossRef]
Johnson, R. S., 2002, “Camassa–-Holm, Korteweg-deVries and Related Models for Water Waves,” J. Fluid Mech., 455, pp. 63–82. [CrossRef]
Fokas, A., and Fuchssteiner, B., 1981, “Symplectic Structures, Their Bäcklund Transformation and Hereditary Symmetries,” Phys. D, 4, pp. 47–66. [CrossRef]
Lenells, J., 2005, “Conservation Laws of the Camassa–-Holm Equation,” J. Phys. A, 38, pp. 869–880. [CrossRef]
Camassa, R., and Holm, D., 1993, “An Integrable Shallow Water Equation With Peaked Solutions,” Phys. Rev. Lett., 71, pp. 1661–1664. [CrossRef]
El-Wakil, S., Abulwafa, E., Zahran, M., and Mahmoud, A., 2011, “Time-Fractional KdV Equation: Formulation and Solution Using Variational Methods,” Nonlinear Dyn., 65, pp. 55–63. [CrossRef]
Gorenflo, R., Mainardi, F., Scalas, E., and Raberto, M., 2001, “Fractional Calculus and Continuous-Time Finance III. The Diffusion Limit,” Mathematical Finance (Konstanz, 2000), (Trends in Mathematics), Birkhäuser, Basel, pp. 171–180.
Hilfer, R., 2000, Applications of Fractional Calculus in Physics, World Scientific, Singapore.
Lundstrom, B., Higgs, M., Spain, W., and FairhallA., 2008, “Fractional Differentiation by Neocortical Pyramidal Neurons,” Nature Neurosci., 11, pp. 1335–1342. [CrossRef]
Malinowska, A. B., and Torres, D. F. M., 2012, Introduction to the Fractional Calculus of Variations, Imperial College Press, London.
Metzler, R., and Klafter, J., 2004, “The Restaurant at the End of the Random Walk: Recent Developments in the Description of Anomalous Transport by Fractional Dynamics,” J. Phys. A, 37, pp. R161–R208. [CrossRef]
Rossikhin, Y. A., and Shitikova, M. V., 1997, “Application of Fractional Derivatives to the Analysis of Damped Vibrations of Viscoelastic Single Mass Systems, Acta Mech., 120, pp. 109–125. [CrossRef]
Sabatelli, L., Keating, S., Dudley, J., and Richmond, P., 2002, “Waiting Time Distributions in Financial Markets,” Eur. Phys. J. B., 27, pp. 273–275.
Schumer, R., Benson, D. A., Meerschaert, M. M., and Baeumer, B., 2003, “Multiscaling Fractional Advection-Dispersion Equations and Their Solutions,” Water Resour. Res., 39, pp. 1022–1032.
Schumer, R., Benson, D. A., Meerschaert, M. M., and Wheatcraft, S.W., 2001, “Eulerian Derivation of the Fractional Advection-Dispersion Equation,” J. Contamin. Hydrol., 48, pp. 69–88. [CrossRef]
Tavazoei, M. S., and Haeri, M., 2009, “Describing Function Based Methods for Predicting Chaos in a Class of Fractional Order Differential Equations,” Nonlinear Dyn., 57(3), pp. 363–373. [CrossRef]
Cresson, J., 2007, “Fractional Embedding of Differential Operators and Lagrangian Systems,” J. Math. Phys., 48, p. 033504. [CrossRef]
Herzallah, M. A. E., and Baleanu, D., 2012, “Fractional Euler–Lagrange Equations Revisited,” Nonlinear Dyn., 69(3), pp. 977–982. [CrossRef]
Malinowska, A. B., 2012, “A Formulation of the Fractional Noether-Type Theorem for Multidimensional Lagrangians,” Appl. Math. Lett., 25, pp. 1941–1946. [CrossRef]
Riewe, F., 1996, “Nonconservative Lagrangian and Hamiltonian Mechanics,” Phys. Rev. E, 53(2), pp. 1890–1899. [CrossRef]
Riewe, F., 1997, “Mechanics With Fractional Derivatives,” Phys. Rev. E, 55(3), pp. 3581–3592. [CrossRef]
Wu, G. C., and Baleanu, D., 2013, “Variational Iteration Method for the Burgers' Flow With Fractional Derivatives—New Lagrange Multipliers,” Appl. Math. Model., 37(9), pp. 6183–6190. [CrossRef]
Odzijewicz, T., Malinowska, A. B., and Torres, D. F M., 2012, “Generalized Fractional Calculus With Applications to the Calculus of Variations,” Comput. Math. Appl., 64(10), pp. 3351–3366. [CrossRef]
Odzijewicz, T., Malinowska, A. B., and Torres, D. F. M., 2012, “Fractional Variational Calculus With Classical and Combined Caputo Derivatives,” Nonlinear Anal. TMA, 75(3), pp. 1507–1515. [CrossRef]
Agrawal, O. P., 2002, “Formulation of Euler–Lagrange Equations for Fractional Variational Problems,” J. Math. Anal. Appl., 272(1), pp. 368–379. [CrossRef]
Agrawal, O. P., 2004, “A General Formulation and Solution Scheme for Fractional Optimal Control Problems,” Nonlinear Dyn., 38(4), pp. 323–337. [CrossRef]
Agrawal, O. P., 2007, “Fractional Variational Calculus in Terms of Riesz Fractional Derivatives,” J. Phys. A Math. Theor., 40, pp. 62–87.
Baleanu, D., and Muslih, S. I., 2005, “Lagrangian Formulation of Classical Fields Within Riemann–Liouville Fractional Derivatives, Phys Scr., 72, pp. 119–123. [CrossRef]
Inokuti, M., Sekine, H., and Mura, T., 1978, “General Use of the Lagrange Multiplier in Non-Linear Mathematical Physics,” Variational Method in the Mechanics of Solids, S.Nemat-Nasser (ed.), Pergamon Press, Oxford.
Saha Ray, S., and Bera, R., 2005, “An Approximate Solution of a Nonlinear Fractional Differential Equation by Adomian Decomposition Method,” Appl. Math. Comput., 167, pp. 561–571. [CrossRef]
Cang, J., Tan, Y., Xu, H., and LiaoS., 2009, “Series Solutions of Nonlinear Fractional Riccati Differential Equations,” Chaos Solitons Fractals, 40(1), pp. 1–9. [CrossRef]
Liao, S., 1992, “The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,” Ph.D. thesis, Shanghai Jiao Tong University, Shanghai, China.
Sweilam, N. H., Khader, M. M., and Al-Bar, R. F., 2007, “Numerical Studies for a Multi-Order Fractional Differential Equation,” Phys. Lett. A, 371, pp. 26–33. [CrossRef]
Kilbas, A. A., Srivastava, H. M., and TrujilloJ. J., 2006, Theory and Applications of Fractional Differential Equations, Vol. 204 (North-Holland Mathematics Studies), Elsevier, Amsterdam, The Netherlands.
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, San Diego.
Samko, S. G., Kilbas, A. A., and Marichev, O. I., 1993. Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York.
He, J., 1997, “Semi-Inverse Method of Establishing Generalized Variational Principles for Fluid Mechanics With Emphasis on Turbo-Machinery Aerodynamics,” Int. J. Turbo Jet-Engines, 14(1), pp. 23–28.
HeJ., 2004, Variational Principles for Some Nonlinear Partial Differential Equations With Variable Coefficients,” Chaos Solitons Fractals, 19, pp. 847–851. [CrossRef]

Figures

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
Fig. 11

Solitary wave solution of u(x, t)

Grahic Jump Location
Fig. 2

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

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
Fig. 1

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

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In