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Mathematica Slovaca

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Volume 68, Issue 5


Application of tan(φ(ξ)/2)-expansion method to burgers and foam drainage equations

Nematollah Kadkhoda
  • Corresponding author
  • Department of Mathematics Faculty of Basic Sciences Bozorgmehr University Of Qaenat Qaenat Iran
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/ Michal Fečkan
  • Department of Mathematical Analysis and Numerical Mathematics Faculty of Mathematics, Physics and Informatics Comenius University in Bratislava Mlynská dolina, 842 48 Bratislava Slovakia
  • Mathematical Institute of Slovak Academy of Sciences Štefánikova 49, 814 73 Bratislava Slovakia
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Published Online: 2018-10-20 | DOI: https://doi.org/10.1515/ms-2017-0167


In this paper, we consider a new direct approach namely the tan(φ(ξ)/2)-Expansion Method to obtain analytical solutions of Burgers and foam drainage equations. With this method, further solutions can be obtained comparing with other techniques and approaches. We use of Mathematica for obtaining these solutions.

MSC 2010: Primary 35L05; Secondary 35L75, 35N05

Keywords: Burgers equation; foam drainage equation; tan(φ(ξ)/2)-Expansion Method; nonlinear evolution equations


  • [1]

    BAKODAH, H. O., AL-ZAID, N. A., MIRZAZADEH, M., ZHOU, Q.: Decomposition method for solving Burgers equation with Dirichlet and Neumann boundary conditions, Optik 130 (2017), 1339–1346.Google Scholar

  • [2]

    BEKIR, A., TASCAN, F., UNSAL, O.: Exact solutions of the Zoomeron and Klein Gordon Zahkharov equations, J. Assoc. Arab Univ. Basic Appl. Sci. 17 (2015), 1–5.Google Scholar

  • [3]

    BIAZAR, J., AMINIKHAH, H.: Exact and numerical solutions for non-linear Burgers equation by VIM, Math. Comput. Modelling 49 (2009), 1394–1400.Google Scholar

  • [4]

    BURGERS, J. M.: A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948), 171–199.Google Scholar

  • [5]

    DURANDA, M., LANGEVIN, D.: Physicochemical approach to the theory of foam drainage, Eur. Phys. J. E. 7 (2002), 35–44.Google Scholar

  • [6]

    GOL DFARB, I. I., KANN, K.B., SHREIBER, I. R.: Liquid flow in foams, Fluid Dyn. 23 (1988), 244–249.Google Scholar

  • [7]

    HE, J. H., WU, X. H.: Exp-function method for nonlinear wave equations, Chaos. Solitons. Fractals. 30 (2006), 700–708.Google Scholar

  • [8]

    HELAL, M. A., MEHANNA, M. S.: The Tanh method and Adomian decomposition method for solving the foam drainage equation, Appl. Math. Comput. 190 (2007), 599–609.Google Scholar

  • [9]

    HILGENFELDT, S., KOEHLER, S. A., STONE, H. A.: Dynamics of coarsening foams: accelerated and self-limiting drainage, Phys. Rev. Lett. 86 (2001), 4704-4707.Google Scholar

  • [10]

    HIROTA, R.: Exact solution of the KdV equation for multiple collisions of solutions, Phys. Rev. Lett. 27 (1971), 1192–1194.Google Scholar

  • [11]

    JAFARI, H., KADKHODA, N., BISWAS, A.: The G′/G-expansion method for solutions of evolution equations from isothermal magnetostatic atmospheres, J. King. Saud. Univ. Sci. 25 (2013), 57–62.Google Scholar

  • [12]

    JAFARI, H., KADKHODA, N., KHALIQUE, C. M.: Travelling wave solutions of nonlinear evolution equations using the simplest equation method, Comput. Math. Appl. 64 (2012), 2084–2088.Google Scholar

  • [13]

    JAFARI, H., TAJADODI, H., KADKHODA, N., BALEANU, D.: Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations, Abstr. Appl. Anal., http://dx.doi.org/10.1155/2013/587179.Google Scholar

  • [14]

    KADKHODA, N., JAFARI, H.: Analytical solutions of the Gerdjikov-Ivanov equation by using exp (-φ(ξ))-expansion method, Optik 139 (2017), 72–76.Google Scholar

  • [15]

    KUDRYASHOV, N. A.: On types of nonlinear integrable equations with exact solutions, Phys. Lett. A 155 (1991), 269–275.Google Scholar

  • [16]

    LI, C., CHEN, A., YE, J.: Numerical approaches to fractional calculus and fractional ordinary differential equation, J. Comput. Phys. 230 (2011), 3352–3368.Google Scholar

  • [17]

    MANAFIAN, J., LAKESTANI, M.: Optical soliton solutions for the Gerdjikov-Ivanov model via tan (φ/2)-expansion method, Optik 127 (2016), 9603–9620.Google Scholar

  • [18]

    MOMANI, S., ODIBAT, Z.: A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor s formula, J. Comput. Appl. Math. 220 (2008), 85–95.Google Scholar

  • [19]

    PRUDHOMME, R. K., KHAN, S. A.: Foams: Theory, Measurements and Applications, New York, Dekker, 1996.Google Scholar

  • [20]

    SABATIER, J., OUSTALOUP, A., TRIGEASSON, J. C., MAAMRI, N.: A Lyapunov approach to the stability of fractional differential equations, Signal Process. 91 (2011), 437–445.Google Scholar

  • [21]

    WANG, D. S., REN, Y. J., ZHANG, H. Q.: Further extended sinh-cosh and sin-cos methods and new non traveling wave solutions of the 2+1 -dimensional dispersive long wave equations, Appl. Math. E-Notes. 5 (2005), 157–163.Google Scholar

  • [22]

    WAZWAZ, A. M.: A sine cosine method for handling nonlinear wave equations, Math. Comput. Modelling 40 (2004), 499–508.Google Scholar

  • [23]

    WAZWAZ, A. M.: The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations, Chaos. Solitons. Fractals. 25 (2005), 55–63.Google Scholar

  • [24]

    WAZWAZ, A. M.: Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations, Appl. Math. Comput. 169 (2005), 639–656.Google Scholar

  • [25]

    WEAIRE, D., HUTZLER, S.: The Physics of Foams, Oxford University Press, Oxford, 2000.Google Scholar

  • [26]

    YANG, X.: Local fractional integral transforms, Prog. Nonlinear Sci. 4 (2011), 221–225.Google Scholar

  • [27]

    ZHANG, J., JIANG, F., ZHAO, X.: An improved (G′/G)-expansion method for solving nonlinear evolution equations, Int. J. Comput. Math. 87 (2010), 1716–1725.Google Scholar

About the article

Received: 2017-05-25

Accepted: 2017-09-13

Published Online: 2018-10-20

Published in Print: 2018-10-25

Communicated by Jozef Džurina

M. Fečckan acknowledges the support by the Slovak Grant Agency VEGA No. 2/0153/16 and No. 1/0078/17, and by the Slovak Research and Development Agency under the contract No. APVV-14-0378.

Citation Information: Mathematica Slovaca, Volume 68, Issue 5, Pages 1057–1064, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0167.

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