Skip to main content
SpringerLink
Log in

A method for solving nonlinear time-dependent drainage model

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

The purpose of this paper is to present a method for solving nonlinear time-dependent drainage model. This method is based on the perturbation theory and Laplace transformation. The proposed technique allows us to obtain an approximate solution in a series form. The computed results are in good agreement with the results of Adomian decomposition method. Results are presented graphically and in tabulated forms to study the efficiency and accuracy of method. The present approach provides a reliable technique, which avoids the tedious work needed by classical techniques and existing numerical methods. The nonlinear time-dependent drainage model is solved without linearizing or discretizing the nonlinear terms of the equation. The method does not require physically unrealistic assumptions, linearization or discretization in order to find the solutions of the given problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Weaire D, Hutzler S (2000) The physic of foams. Oxford University Press, Oxford

    Google Scholar 

  2. Weaire D, Hutzler S, Cox S, Alonso MD, Drenckhan D (2003) The fluid dynamics of foams. J Phys Condens Matter 15:65–72

    Article  Google Scholar 

  3. Helal MA, Mehanna MS (2007) The tanh method and Adomian decomposition method for solving the foam drainage equation. Appl Math Comput 190:599–609

    Article  MathSciNet  MATH  Google Scholar 

  4. Hilgenfeldt S, Koehler SA, Stone HA (2001) Dynamics of coarsening foams: accelerated and self-limiting drainage. Phys Rev Lett 86:4704–4707

    Article  Google Scholar 

  5. Verbist G, Weaire D, Kraynik AM (1996) The foam drainage equation. J Phys Condens Matter 83:715–3731

    Google Scholar 

  6. Koehler SA, Stone HA, Brenner MP, Eggers J (1998) Dynamics of foam drainage. Phys Rev E 58:2097–2106

    Article  Google Scholar 

  7. Dahmani Z, Mesmoudi MM, Bebbouchi R (2008) The foam drainage equation with time- and space-fractional derivatives solved by the Adomian method. Electron J Qual Theory Differ Equ 30:1–10

    MathSciNet  Google Scholar 

  8. Mirmoradi SH, Hosseinpour I, Barari A, Ghotbi AR (2009) Analysis of foam drainage problem using variational iteration method. Adv Stud Theor Phys 3:283–292

    Google Scholar 

  9. Dahmani Z, Anber A (2010) The variational iteration method for solving the fractional foam drainage equation. Int J Nonlinear Sci 10:39–45

    MathSciNet  MATH  Google Scholar 

  10. Fadravi HH, Nik HS, Buzhabadi R (2011) Homotopy analysis method for solving foam drainage equation with space- and time-fractional derivatives. Int J Differ Equ 2011:12, Art ID 237045

  11. Khan Y, Wu Q (2011) Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput Math Appl 61:1963–1967

    Article  MathSciNet  MATH  Google Scholar 

  12. Khan Y, Mohyud-Din ST (2010) Coupling of He’s polynomials and Laplace transformation for MHD viscous flow over a stretching sheet. Int J Nonlinear Sci Num Simul 11:1103–1107

    Article  Google Scholar 

  13. He JH (2000) A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int J Nonlinear Mech 35:37–43

    Article  MATH  Google Scholar 

  14. Abbasbandy S (2006) Application of He’s homotopy perturbation method for Laplace transform. Chaos Solitons Fractals 30:1206–1212

    Article  MATH  Google Scholar 

  15. Mohyud-Din ST, Noor MA, Noor KI, Hosseini MM (2010) On the coupling of He’s polynomials and Laplace transformation. Int J Nonlinear Sci Num Simul 11:93–96

    Google Scholar 

  16. Madani M, Fathizadeh M, Khan Y, Yildirim A (2011) On the coupling of the homotopy perturbation method and Laplace transformation. Math Comput Model 53:1937–1945

    Article  MathSciNet  MATH  Google Scholar 

  17. Khan Y, Wu Q, Faraz N, Yildirim A (2011) The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet. Comput Math Appl 61:3391–3399

    Article  MathSciNet  MATH  Google Scholar 

  18. Khan Y, Faraz N, Yildirim A, Wu Q (2011) A series solution of the long porous slider. Tribol Trans 54:187–191

    Article  Google Scholar 

  19. He JH (2011) Analytical methods for thermal science-an elementary introduction. Therm Sci 15:S1–S3

    Article  Google Scholar 

  20. Hesameddini E, Latifzadeh H (2009) Reconstruction of variational iteration algorithm using the Laplace transform. Int J Nonlinear Sci Num Simul 10:1377–1382

    Google Scholar 

  21. Turkyilmazoglu M (2011) An optimal variational iteration method. Appl Math Lett 24:762–765

    Article  MathSciNet  MATH  Google Scholar 

  22. Mohyud-Din ST, Yildirim A (2010) Variation of parameters method using Laplace transformation method. World Appl Sci J 9:300–302

    Google Scholar 

  23. Khan Y (2009) An effective modification of the Laplace decomposition method for nonlinear equations. Int J Nonlinear Sci Num Simul 10:1373–1376

    Google Scholar 

  24. Khan M, Gondal MA (2010) A new analytical solution of foam drainage equation by Laplace decomposition method. J Adv Res Differ Equ 2:53–64

    Google Scholar 

  25. Khan Y, Austin F (2010) Application of the Laplace decomposition method to nonlinear homogeneous and non-homogenous advection equations. Z Naturforsch 65a:849–853

    Google Scholar 

  26. Šmarda Z, Archalousova O (2010) Adomian decomposition method for certain singular initial value problems II. J Appl Math 3:91–98

    Google Scholar 

  27. Turkyilmazoglu M (2011) An optimal analytic approximate solution for the limit cycle of Duffing-van der Pol equation. J Appl Mech Trans ASME 78:021005

    Article  Google Scholar 

  28. Turkyilmazoglu M (2011) Some issues on HPM and HAM methods: a convergence scheme. Math Comput Model 53:1929–1936

    Article  MathSciNet  MATH  Google Scholar 

  29. Diblík J, Šmarda Z, Berezansky L (2010) Positive solutions of a second-order delay differential equations with a damping term. Comput Math Appl 62:1332–1352

    Google Scholar 

  30. Bastinec J, Diblík J, Šmarda Z (2011) An explicit criterion for the existence of positive solutions of the linear delayed equation \( \dot{x}(t) = - c(t)x(t - \tau (t)). \) Abstract Appl Anal 2011:12, Art ID 561902

  31. Bastinec J, Berezansky L, Diblík J, Šmarda Z (2011) A final result on the oscillation of solutions of the linear discrete delayed equation ∆x(n) = −p(n)x(n – k) with a positive coefficient. Abstract Appl Anal 2011:28, Art ID 586328

  32. Ghorbani A (2009) Beyond Adomian’s polynomials: He polynomials. Chaos Solitons Fractals 39:1486–1492

    Article  MathSciNet  MATH  Google Scholar 

  33. Zayed EME, Abdel Rahman HM (2012) On using the He’s polynomials for solving the nonlinear coupled evolution equations in mathematical physics. WSEAS Trans Math 11:294–302

    Google Scholar 

Download references

Acknowledgments

The author would like to thank the referees for their valuable suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    Yasir Khan

Authors
  1. Yasir Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yasir Khan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Khan, Y. A method for solving nonlinear time-dependent drainage model. Neural Comput & Applic 23, 411–415 (2013). https://doi.org/10.1007/s00521-012-0933-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-012-0933-2

Keywords

Search

Navigation