A New Analytical Study of Modified  Camassa-Holm and Degasperis-Procesi  Equations

Majeed A. Yousif; Bewar A. Mahmood; Fadhil H. Easif

doi:10.4236/ajcm.2015.53024

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AJCM> Vol.5 No.3, September 2015

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A New Analytical Study of Modified Camassa-Holm and Degasperis-Procesi Equations

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DOI: 10.4236/ajcm.2015.53024 4,282 Downloads 4,753 Views Citations

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Majeed A. Yousif¹, Bewar A. Mahmood², Fadhil H. Easif¹

Affiliation(s)

¹Department of Mathematics, University of Zakho, Zakho, Iraq.
²Department of Mathematics, University of Duhok, Duhok, Iraq.

ABSTRACT

In this letter, variational homotopy perturbation method (VHPM) has been studied to obtain solitary wave solutions of modified Camassa-Holm and Degasperis-Procesi equations. The results show that the VHPM is suitable for solving nonlinear differential equations with fully nonlinear dispersion term. The travelling wave solution for above equation compared with VIM, HPM, and exact solution. Also, it was shown that the present method is effective, suitable, and reliable for these types of equations.

KEYWORDS

Homotopy Perturbation Method, Modified Camassa-Holm Equation, Modified Degasperis-Procesi Equation

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Yousif, M. , Mahmood, B. and Easif, F. (2015) A New Analytical Study of Modified Camassa-Holm and Degasperis-Procesi Equations. American Journal of Computational Mathematics, 5, 267-273. doi: 10.4236/ajcm.2015.53024.

References

[1]	Adomian, G. (1976) Nonlinear Stochastic Differential Equations. Journal of Mathematical Analysis and Applications, 55, 441-452. http://dx.doi.org/10.1016/0022-247X(76)90174-8

[2]	Adomian, G. (1991) A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations. Computers & Mathematics with Applications, 21, 101-127. http://dx.doi.org/10.1016/0898-1221(91)90220-X

[3]	Liao, S.J. (2004) On the Homotopy Analysis Method for Nonlinear Problems. Applied Mathematics and Computation, 147, 499-513. http://dx.doi.org/10.1016/S0096-3003(02)00790-7

[4]	Abbasbandy, S. (2009) Solitary Wave Solutions to the Modified Form of Camassa Holm Equation by Means of the Homotopy Analysis Method. Chaos, Solitons and Fractals, 39, 428-435. http://dx.doi.org/10.1016/j.chaos.2007.04.007

[5]	He, J.H. (1998) Approximate Solution of Nonlinear Differential Equations with Convolution Product Nonlinearities. Computer Methods in Applied Mechanics and Engineering, 167, 69-73. http://dx.doi.org/10.1016/S0045-7825(98)00109-1

[6]	He, J.H. (1999) Variational Iteration Method a Kind of Non-Linear Analytical Technique: Some Examples. International Journal of Non-Linear Mechanics, 34, 699-708. http://dx.doi.org/10.1016/S0020-7462(98)00048-1

[7]	He, J.H. (1999) Homotopy Perturbation Technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262. http://dx.doi.org/10.1016/S0045-7825(99)00018-3

[8]	He, J.H. (2005) Application of Homotopy Perturbation Method to Nonlinear Wave Equations. Chaos, Solitons Fractals, 26, 695-700. http://dx.doi.org/10.1016/j.chaos.2005.03.006

[9]	Rashidi, M.M., Ganji, D.D. and Dinarvand, S. (2009) Explicit Analytical Solutions of the Generalized Burger and Burger Fisher Equations by Homotopy Perturbation Method. Numerical Methods for Partial Differential Equations, 25, 409-417. http://dx.doi.org/10.1002/num.20350

[10]	Rashidi, M.M., Freidoonimehr, N., Hosseini, A., Anwar Bg, O. and Hung, T.K. (2014) Homotopy Simulation of Nanofluid Dynamics from a Non-Linearly Stretching Isothermal Permeable Sheet with Transpiration. Meccanica, 49, 469-482. http://dx.doi.org/10.1007/s11012-013-9805-9

[11]	Fadhil, H.E., Manaa, S.A., Bewar, A.M. and Majeed, A.Y. (2015) Variational Homotopy Perturbation Method for Solving Benjamin-Bona-Mahony Equation. Applied Mathematics, 6, 675-683. http://dx.doi.org/10.4236/am.2015.64062

[12]	Olusola, E. (2013) New Improved Variational Homotopy Perturbation Method for Bratu-Type Problems. American Journal of Computational Mathematics, 3, 110-113. http://dx.doi.org/10.4236/ajcm.2013.32018

[13]	Wazwaz, A.M. (2006) Solitary Wave Solutions for Modified Forms of Degasperis-Procesi and Camassa-Holm Equations. Physics Letter A, 352, 500-504.

[14]	Wu, G.C. (2013) Challenge in the Variational Iteration Method—A New Approach to Identification of the Lagrange Multi-Pliers. Journal of King Saud University—Science, 25, 175-178. http://dx.doi.org/10.1016/j.jksus.2012.12.002

[15]	He, J.H. (1997) A New Approach to Nonlinear Partial Differential Equations. Communications in Nonlinear Science and Numerical Simulation, 2, 230-235. http://dx.doi.org/10.1016/S1007-5704(97)90007-1

[16]	Zhang, B.-G. and Li, S.-Y. (2008) Homotopy Perturbation Method for Modified Camassa-Holm and Degasperis-Procesi Equation. Physics Letter A, 372, 1867-1872.

[17]	Yildirim, A. (2010) Variational Iteration Method for Modified Camassa-Holm and Degasperis-Procesi Equations. International Journal for Biomedical Engineering, 26, 266-272.