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Research Papers

Traveling Wave Solutions to Riesz Time-Fractional Camassa–Holm Equation in Modeling for Shallow-Water Waves

[+] Author and Article Information
S. Saha Ray

Department of Mathematics,
National Institute of Technology,
Rourkela 769008, India
e-mail: santanusaharay@yahoo.com

S. Sahoo

Department of Mathematics,
National Institute of Technology,
Rourkela 769008, India

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 22, 2014; final manuscript received February 4, 2015; published online June 25, 2015. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(6), 061026 (Nov 01, 2015) (5 pages) Paper No: CND-14-1296; doi: 10.1115/1.4029800 History: Received November 22, 2014; Revised February 04, 2015; Online June 25, 2015

In the present paper, we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics, viz., Riesz time-fractional Camassa–Holm (CH) equation by modified homotopy analysis method (MHAM). As a result, new types of solutions are obtained. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. The main aim of this paper is to employ a new approach, which enables us successful and efficient derivation of the analytical solutions for the Riesz time-fractional CH equation.

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References

Zhang, Y., 2013, “Time-Fractional Camassa–Holm Equation: Formulation and Solution Using Variational Methods,” ASME J. Comput. Nonlinear Dyn., 8(4), p. 041020-1-8. [CrossRef]
Boyd, J. P., 1997, “Peakons and Coshoidal Waves: Travelling Wave Solutions of the Camassa–Holm Equation,” Appl. Math. Comput., 81(2–3), pp. 173–187. [CrossRef]
Camassa, R., and Holm, D. D., 1993, “An Integrable Shallow Water Equation With Peaked Solitons,” Phys. Rev. Lett., 71(11), pp. 1661–1664. [CrossRef] [PubMed]
Qian, T., and Tang, M., 2001, “Peakons and Periodic Cusp Waves in a Generalized Camassa–Holm Equation,” Chaos, Solitons Fractals, 12(7), pp. 1347–1360. [CrossRef]
Liu, Z.-R., Wang, R.-Q., and Jing, Z.-J., 2004, “Peaked Wave Solutions of Camassa–Holm Equation,” Chaos, Solitons Fractals, 19(1), pp. 77–92. [CrossRef]
Liu, Z., and Qian, T., 2001, “Peakons and Their Bifurcation in a Generalized Camassa–Holm Equation,” Int. J. Bifurcation Chaos Appl. Sci. Eng., 11(3), pp. 781–792. [CrossRef]
Tian, L., and Song, X., 2004, “New Peaked Solitary Wave Solutions of the Generalized Camassa–Holm Equation,” Chaos, Solitons Fractals, 19(3), pp. 621–637. [CrossRef]
Kalisch, H., 2004, “Stability of Solitary Waves for a Nonlinearly Dispersive Equation,” Discrete Contin. Dyn. Syst. Ser. A, 10(3), pp. 709–717. [CrossRef]
Liu, Z., and Ouyang, Z., 2007, “A Note on Solitary Waves for Modified Forms of Camassa–Holm and Degasperis–Procesi Equations,” Phys. Lett. A, 366(4–5), pp. 377–381. [CrossRef]
Camassa, R., Holm, D. D., and Hyman, J. M., 1994, “A New Integrable Shallow Water Equation,” Adv. Appl. Mech., 31, pp. 1–33. [CrossRef]
Cooper, F., and Shepard, H., 1994, “Solitons in the Camassa–Holm Shallow Water Equation,” Phys. Lett. A, 194(4), pp. 246–250. [CrossRef]
He, B., Rui, W., Chen, C., and Li, S., 2008, “Exact Travelling Wave Solutions of a Generalized Camassa–Holm Equation Using the Integral Bifurcation Method,” Appl. Math. Comput., 206(1), pp. 141–149. [CrossRef]
Wazwaz, A., 2006, “Solitary Wave Solutions for Modified Forms of Degasperis–Procesi and Camassa–Holm Equations,” Phys. Lett. A, 352(6), pp. 500–504. [CrossRef]
Wazwaz, A., 2007, “New Solitary Wave Solutions to the Modified Forms of Degasperis–Procesi and Camassa–Holm Equations,” Appl. Math. Comput., 186(1), pp. 130–141. [CrossRef]
Tian, L., and Yin, J., 2004, “New Compact on Solutions and Solitary Wave Solutions of Fully Nonlinear Generalized Camassa–Holm Equations,” Chaos, Solitons and Fractals, 20, pp. 289–299. [CrossRef]
Wang, Q., and Tang, M., 2008, “New Exact Solutions for Two Nonlinear Equations,” Phys. Lett. A, 372(17), pp. 2995–3000. [CrossRef]
Yomba, E., 2008, “The Sub-ODE Method for Finding Exact Travelling Wave Solutions of Generalized Nonlinear Camassa–Holm, and Generalized Nonlinear Schrödinger Equations,” Phys. Lett. A, 372(3), pp. 215–222. [CrossRef]
Yomba, E., 2008, “A Generalized Auxiliary Equation Method and Its Application to Nonlinear Klein–Gordon and Generalized Nonlinear Camassa–Holm Equations,” Phys. Lett. A, 372(7), pp. 1048–1060. [CrossRef]
Liu, Z., and Pan, J., 2009, “Coexistence of Multifarious Explicit Nonlinear Wave Solutions for Modified Forms of Camassa–Holm and Degasperis–Procesi Equations,” Int. J. Bifurcation Chaos Appl. Sci. Eng., 19(7), pp. 2267–2282. [CrossRef]
Liu, Z., and Liang, Y., 2011, “The Explicit Nonlinear Wave Solutions and Their Bifurcations of the Generalized Camassa–Holm Equation,” Int. J. Bifurcation Chaos Appl. Sci. Eng., 21(11), pp. 3119–3136. [CrossRef]
Parkes, E. J., and Vakhnenko, V. O., 2005, “Explicit Solutions of the Camassa–Holm Equation,” Chaos, Solitons Fractals, 26(5), pp. 1309–1316. [CrossRef]
Jafari, H., Tajadodi, H., and Baleanu, D., 2014, “Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Fractional Evolution Equations,” ASME J. Comput. Nonlinear Dyn., 9(2), p. 021019. [CrossRef]
Saha Ray, S., 2008, “A New Approach for the Application of Adomian Decomposition Method for the Solution of Fractional Space Diffusion Equation With Insulated Ends,” Appl. Math. Comput., 202(2), pp. 544–549. [CrossRef]
Liao, S., 2003, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, FL.
Jafarian, A., Ghaderi, P., Golmankhaneh, A. K., and Baleanu, D., 2014, “Analytical Treatment of System of Abel Integral Equations by Homotopy Analysis Method,” Rom. Rep. Phys., 66(3), pp. 603–611.
Jafarian, A., Ghaderi, P., Golmankhaneh, A. K., and Baleanu, D., 2014, “Analytical Approximate Solutions of the Zakharov–Kuznetsov Equations,” Rom. Rep. Phys., 66(2), pp. 296–306.
Shen, S., Liu, F., Anh, V., and Turner, I., 2008, “The Fundamental Solution and Numerical Solution of the Riesz Fractional Advection–Dispersion Equation,” IMA J. Appl. Math., 73(6), pp. 850–872. [CrossRef]
Herrmann, R., 2011, Fractional Calculus: An Introduction for Physicists, World Scientific, Singapore.
Samko, S. G., Kilbas, A. A., and Marichev, O. I., 2002, Fractional Integrals and Derivatives: Theory and Applications, Taylor and Francis, London.
Podlubny, I., 1999, Fractional Differential Equation, Academic, New York.
Adomian, G., 1994, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston.

Figures

Grahic Jump Location
Fig. 1

The ℏ-curve for partial derivatives of u(x,t) at (0,0) for the MHAM solution

Grahic Jump Location
Fig. 2

(a) The MHAM method traveling wave solution for u(x,t) and (b) corresponding 2D solution for u(x,t) when t=1

Grahic Jump Location
Fig. 3

(a) The MHAM method traveling wave solution for u(x,t) and (b) corresponding 2D solution for u(x,t) when t=1

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