Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter October 20, 2018

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

  • Nematollah Kadkhoda EMAIL logo and Michal Fečkan
From the journal Mathematica Slovaca

Abstract

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.

  1. Communicated by Jozef Džurina

  2. 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.

References

[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), 1339–1346.10.1016/j.ijleo.2016.11.140Search in 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.Search in Google Scholar

[3] BIAZAR, J., AMINIKHAH, H.: Exact and numerical solutions for non-linear Burgers equation by VIM, Math. Comput. Modelling 49 (2009), 1394–1400.10.1016/j.mcm.2008.12.006Search in Google Scholar

[4] BURGERS, J. M.: A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948), 171–199.10.1016/S0065-2156(08)70100-5Search in Google Scholar

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

[6] GOL DFARB, I. I., KANN, K.B., SHREIBER, I. R.: Liquid flow in foams, Fluid Dyn. 23 (1988), 244–249.10.1007/BF01051894Search in Google Scholar

[7] HE, J. H., WU, X. H.: Exp-function method for nonlinear wave equations, Chaos. Solitons. Fractals. 30 (2006), 700–708.10.1016/j.chaos.2006.03.020Search in 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.Search in 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), 4704-4707.10.1103/PhysRevLett.86.4704Search in Google Scholar PubMed

[10] HIROTA, R.: Exact solution of the KdV equation for multiple collisions of solutions, Phys. Rev. Lett. 27 (1971), 1192–1194.10.1103/PhysRevLett.27.1192Search in 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), 57–62.10.1016/j.jksus.2012.02.002Search in 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), 2084–2088.10.1016/j.camwa.2012.04.004Search in 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/10.1155/2013/587179.10.1155/2013/587179Search in Google Scholar

[14] KADKHODA, N., JAFARI, H.: Analytical solutions of the Gerdjikov-Ivanov equation by using exp (-φ(ξ))-expansion method, Optik 139 (2017), 72–76.10.1016/j.ijleo.2017.03.078Search in Google Scholar

[15] KUDRYASHOV, N. A.: On types of nonlinear integrable equations with exact solutions, Phys. Lett. A 155 (1991), 269–275.10.1016/0375-9601(91)90481-MSearch in Google Scholar

[16] LI, C., CHEN, A., YE, J.: Numerical approaches to fractional calculus and fractional ordinary differential equation, J. Comput. Phys. 230 (2011), 3352–3368.10.1016/j.jcp.2011.01.030Search in Google Scholar

[17] MANAFIAN, J., LAKESTANI, M.: Optical soliton solutions for the Gerdjikov-Ivanov model via tan (φ/2)-expansion method, Optik 127 (2016), 9603–9620.10.1016/j.ijleo.2016.07.032Search in 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), 85–95.10.1016/j.cam.2007.07.033Search in Google Scholar

[19] PRUDHOMME, R. K., KHAN, S. A.: Foams: Theory, Measurements and Applications, New York, Dekker, 1996.Search in 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), 437–445.10.1016/j.sigpro.2010.04.024Search in 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.Search in Google Scholar

[22] WAZWAZ, A. M.: A sine cosine method for handling nonlinear wave equations, Math. Comput. Modelling 40 (2004), 499–508.10.1016/j.mcm.2003.12.010Search in 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), 55–63.10.1016/j.chaos.2004.09.122Search in 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.Search in Google Scholar

[25] WEAIRE, D., HUTZLER, S.: The Physics of Foams, Oxford University Press, Oxford, 2000.Search in Google Scholar

[26] YANG, X.: Local fractional integral transforms, Prog. Nonlinear Sci. 4 (2011), 221–225.Search in 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), 1716–1725.10.1080/00207160802450166Search in Google Scholar

Received: 2017-05-25
Accepted: 2017-09-13
Published Online: 2018-10-20
Published in Print: 2018-10-25

© 2018 Mathematical Institute Slovak Academy of Sciences

Downloaded on 29.3.2024 from https://www.degruyter.com/document/doi/10.1515/ms-2017-0167/html
Scroll to top button