Skip to main content

An efficient numerical method for solving nonlinear foam drainage equation


In this paper, the nonlinear foam drainage equation, which is a famous nonlinear partial differential equation, is solved by using a hybrid numerical method based on the quasilinearization method and the bivariate generalized fractional order of the Chebyshev functions (B-GFCF) collocation method. First, using the quasilinearization method, the equation is converted into a sequence of linear partial differential equations (LPD), and then these LPDs are solved using the B-GFCF collocation method. A very good approximation of solutions is obtained, and comparisons show that the obtained results are more accurate than the results of other researchers.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9



Unknown function

\(\bar{u}(x,t)\) :

The approximate solution

\(\psi (x,t)\) :

Plateau border

\(_\eta FT_n^\alpha \) :

Basis functions

m :

The number of basis functions

\(E_A\) :

The maximum of the absolute error

t :

Time position coordinate

x :

Scaled coordinate

\(\eta \) :

Length of the domain of function

\(\alpha \) :

The order of basis functions

\(\mathbf{K }\) :

Unknown coefficients matrix

\(\varPhi (t)\) :

The vector of basis functions


  1. J A F Plateau Statique Experimentale et Theorique des Liquides Soumis aux Seules Forces Moleculaires (Paris, Gauthier-Villars) (1873)

  2. R K Prudhomme and S A Khan Foams: Theory, Measurements and Applications (New York, Dekker) (1996)

  3. D L Weaire and S Hutzler The Physics of Foams (Oxford, Oxford University Press) (2000)

    Google Scholar 

  4. H A Stone, S A Koehler, S Hilgenfeldt and M Durand J. Phys. Condens Matter. 15 S283 (2003)

    Article  ADS  Google Scholar 

  5. S Hilgenfeldt, S A Koehler and H A Stone Phys. Rev. Lett. 20 4704 (2001)

    Article  ADS  Google Scholar 

  6. L J Gibson and M F Ashby Cellular Solids: Structure and Properties (Cambridge Univ. Press) (1997)

  7. J Banhart and W Brinkers J. Mater. Sci. Lett. 18 617 (1999)

    Article  Google Scholar 

  8. M F Ashby, A G Evans, N A Fleck, L J Gibson, J W Hutchinson and H N G Wadley Metal Foams: A Design Guide (Boston, Society of Automotive Engineers) (2000)

  9. M A Helal and M S Mehanna Appl. Math. Comput. 190 599 (2007)

    MathSciNet  Google Scholar 

  10. M T Darvishi and F Khani Comput. Math. Appl. 58 360 (2009)

    Article  MathSciNet  Google Scholar 

  11. F Khani, S Hamedi-Nezhad, M T Darvishi and S W Ryu Nonlinear Anal. Real World Appl. 10 1904 (2009)

    Article  MathSciNet  Google Scholar 

  12. S H Mirmoradi, I Hosseinpour, A Barari and A R Ghotbi Adv. Stud. Theor. Phys. 3 283 (2009)

    Google Scholar 

  13. A H Fereidoon, H Yaghoobi and M R Davoudabadi Int. J. Differ. Equ. 2011 864023 (2011)

    Google Scholar 

  14. M T Darvishi, M Najafi and M Najafi Food Public Health 2 6 (2012)

    Article  Google Scholar 

  15. Y Khan Neural Comput. Appl. 23 411 (2013)

    Article  Google Scholar 

  16. M Khan and M A Gondal J. Adv. Res. Differ. Equ. 2 53 (2010)

    Google Scholar 

  17. S Arbabi, A Nazari and M T Darvishi Optik - Int. J. Light Electron Optics 127 5443 (2016).

    Article  Google Scholar 

  18. E M E Zayed and A G Al-Nowehy Indian J. Phys. 91 209 (2017)

    Article  ADS  Google Scholar 

  19. M Gubes, Y Keskin and G Oturanc Comput. Method. Diff. Equ. 3 111 (2015)

    Google Scholar 

  20. M Sheikholeslami and D D Ganji J. Mol. Liq. 224 526 (2016)

    Article  Google Scholar 

  21. M Sheikholeslami, D D Ganji and M M Rashidi J. Magn. Magn. Mater. 416 164 (2016)

    Article  ADS  Google Scholar 

  22. M Sheikholeslami and D D Ganji Comput. Methods Appl. Mech. Engrg. 283 651 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  23. J A Rad and K Parand Appl. Numer. Math. 115 252 (2017)

    Article  MathSciNet  Google Scholar 

  24. J A Rad, S Kazem and K Parand Commun. Nonlinear Sci. Numer. Simu. 19 2559 (2014)

    Article  ADS  Google Scholar 

  25. M Sheikholeslami and D D Ganji Powder Technol. 235 873 (2013)

    Article  Google Scholar 

  26. M Sheikholeslami, H R Ashorynejad, D D Ganji and A Yildirim Scientia Iranica B 19 437 (2012)

    Article  Google Scholar 

  27. M Sheikholeslami, R Ellahi, H R Ashorynejad, G Domairry and T Hayat J. Comput. Theor. Nanosci. 11 486 (2014)

    Article  Google Scholar 

  28. M Sheikholeslami, D D Ganji, H R Ashorynejad and H B Rokni Appl. Math. Mech. -Engl. Ed. 33 25 (2012)

    Article  Google Scholar 

  29. M Sheikholeslami, D D Ganji and H R Ashorynejad Powder Technol. 239 259 (2013)

    Article  Google Scholar 

  30. M Sheikholeslami and D D Ganji Scientia Iranica B 21 203 (2014)

    Google Scholar 

  31. M Sheikholeslami, H R Ashorynejad, D Domairry and I. Hashim Sains Malaysiana 41 1281 (2012)

    Google Scholar 

  32. S Yuzbasi Int. J. Comput. Method. 14 1 (2017).

  33. M Delkhosh, M Delkhosh and M Jamali Middle-East J. Scient. Res. 11 974 (2012)

    Google Scholar 

  34. K Parand, M Nikarya and JA Rad Celes. Mech. Dyn. Astr. 116 97 (2013)

    Article  ADS  Google Scholar 

  35. K Parand, H Yousefi, M Delkhosh and A Ghaderi Euro. Phys. J. Plus 131 1 (2016)

    Article  Google Scholar 

  36. S Yuzbasi Math. Method. Appl. Sci. 38 3033 (2015)

    Article  ADS  Google Scholar 

  37. K Parand and M Delkhosh J. Comput. Appl. Math. 317 624 (2017)

    Article  MathSciNet  Google Scholar 

  38. S Yuzbasi Appl. Math. Comput. 219 6328 (2013)

    MathSciNet  Google Scholar 

  39. S Yuzbasi Appl. Math. Modll. 37 3216 (2013)

  40. S Yuzbasi Math. Method. Appl. Sci. 36 300 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  41. A H Bhrawy and A S Alofi Appl. Math. Lett. 26 25 (2013)

    Article  MathSciNet  Google Scholar 

  42. K Parand, S Abbasbandy, S Kazem and A R Rezaei Phys. Scr. 83 015011 (2011)

    Article  ADS  Google Scholar 

  43. E H Doha, A H Bhrawy and S S Ezz-Eldien Comput. Math. Appl. 62 2364 (2011)

    Article  MathSciNet  Google Scholar 

  44. A Saadatmandi and M Dehghan Num. Meth. Partial Diff. Equ. 26 239 (2010)

    Article  Google Scholar 

  45. K Parand, A Taghavi and M Shahini Acta Phys. Polo. B 40 1749 (2009)

    ADS  Google Scholar 

  46. K Parand, A R Rezaei and A Taghavi Math. Method. Appl. Sci. 33 2076 (2010)

    Article  ADS  Google Scholar 

  47. K Parand and S Khaleqi Euro. Phys. J. Plus 131 1 (2016)

    Article  Google Scholar 

  48. K Parand, M Dehghan and A Taghavi Int. J. Numer. Method. H. 20 728 (2010)

    Article  Google Scholar 

  49. J P Boyd Chebyshev and Fourier Spectral Methods (Mineola, Dover Publications) (2000)

  50. K Parand and M Delkhosh Ricerche Mat. 65 307 (2016)

    Article  MathSciNet  Google Scholar 

  51. K Parand and M Delkhosh Afr. Mat. 28 459 (2017)

    Article  MathSciNet  Google Scholar 

  52. K Parand and M Delkhosh J. Korean Soc. Indus. Appl. Math. 20 261 (2016)

    Article  Google Scholar 

  53. S D Conte and C De Boor Elementary Numerical Analysis (McGraw-Hill International Editions) (1981)

  54. A Ralston and P Rabinowitz A First Course in Numerical Analysis (McGraw-Hill International Editions) (1988)

  55. R E Bellman and R E Kalaba Quasilinearization and Nonlinear Boundary-Value Problems (New York, Elsevier Publishing Company) (1965)

  56. V B Mandelzweig and F Tabakinb Comput. Phys. Commun. 141 268 (2001)

    Article  ADS  Google Scholar 

  57. R Kalaba On Nonlinear Differential Equations, the Maximum Operation and Monotone Convergence (RAND Corporation) (1957)

  58. V Lakshmikantham and A S Vatsala Generalized Quasilinearization for Nonlinear Problems, Mathematics and its Applications (Kluwer Academic Publishers) (1998)

  59. K Parand, M Delkhosh Commun. Theor. Phys. 67 483 (2017)

    Article  Google Scholar 

  60. R Krivec and V B Mandelzweig Comput. Phys. Comm. 179 865 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  61. E Z Liverts and V B Mandelzweig Ann. Phys-New York 324 388 (2009)

    Article  ADS  Google Scholar 

  62. A Rezaei, F Baharifard and K Parand Int. J. Comp. Elect. Auto. Cont. Info. Eng. 5 194 (2011)

    Google Scholar 

  63. S S Motsa, V M Magagula and P Sibanda: Scient. World J. 2014 581987 (2014)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Kourosh Parand.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Parand, K., Delkhosh, M. An efficient numerical method for solving nonlinear foam drainage equation. Indian J Phys 92, 231–243 (2018).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Foam drainage equation
  • Fractional order of the Chebyshev functions
  • Quasilinearization method
  • Collocation method
  • Nonlinear PDE


  • 02.30.Jr
  • 02.70.Hm
  • 82.70.Rr