Skip to main content

On Invariant Analysis, Symmetry Reduction and Conservation Laws of Nonlinear Buckmaster Model

  • Conference paper
  • First Online:
Proceedings of International Conference on Trends in Computational and Cognitive Engineering

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1169))


In this work, a systematic investigation on invariant analysis of Buckmaster model, raised in mathematical physics, is performed. A general set of symmetries and corresponding reductions of the considered equation are obtained. Also, by employing nonclassical approach, it is concluded that no supplementary, nonclassical-type symmetries are admitted by the analysed model. Further, it is also observed that multiplier of any order in the direct construction method, suggested by Anco and Bluman, gives only one local conservation law for this model. Moreover, the nontrivial local conservation laws are constructed by new conservation theorem.

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

Access this chapter

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others


  1. Olver PJ (2012) Applications of Lie groups to differential equations, vol 107, Springer Science & Business Media

    Google Scholar 

  2. Bluman G, Anco S (2008) Symmetry and integration methods for differential equations, vol 154, Springer Science & Business Media

    Google Scholar 

  3. Bluman GW, Cole JD (2012) Similarity methods for differential equations. Vol 13. Springer Science & Business Media

    Google Scholar 

  4. Clarkson PA, Mansfield EL (1994) Algorithms for the nonclassical method of symmetry reductions. SIAM J Appl Math 54(6):1693–1719

    Article  MathSciNet  Google Scholar 

  5. Gandarias ML, Bruzon MS (1998) Classical and nonclassical symmetries of a generalized Boussinesq equation. J Nonlin Math Phys 5(1):8–12

    Article  MathSciNet  Google Scholar 

  6. Fan E, Zhang J (2002) Applications of the Jacobi elliptic function method to special-type nonlinear equations. Phys Lett A 305(6):383–392

    Article  MathSciNet  Google Scholar 

  7. Fu Z, Liu S, Liu S, Zhao Q (2001) New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys Lett A 290(1–2):72–76

    Article  MathSciNet  Google Scholar 

  8. Jafari H, Zabihi M, Saidy M (2008) Application of homotopy perturbation method for solving gas dynamics equation. Appl Math Sci 2(48):2393–2396

    MathSciNet  MATH  Google Scholar 

  9. Khan K, Akbar MA (2013) Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and the modified KdV-Zakharov-Kuznetsov equations using the modified simple equation method. Ain Shams Eng J 4(4):903–909

    Article  Google Scholar 

  10. Noether E (1971) Invariante Variationsprobleme, Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse, 235–257 (1918): English translation MA Travel. Transport Theory and Statistical Physics 1(3)

    Google Scholar 

  11. Anco SC, Bluman G (2002) Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications. Eur J Appl Math 13(5):545–566

    Article  Google Scholar 

  12. Ibragimov NH (2007) A new conservation theorem. J Math Anal Appl 333(1):311–328

    Article  MathSciNet  Google Scholar 

  13. Buckmaster J (1977) Viscous sheets advancing over dry beds. J Fluid Mech 81(4):735–756

    Article  MathSciNet  Google Scholar 

  14. Jain S (2018) Numerical analysis for the fractional diffusion and fractional Buckmaster equation by the two-step Laplace Adam-Bashforth method. Eur Phys J Plus 133(1):19

    Article  Google Scholar 

  15. Chanthrasuwan M, Asri NAM, Hamid NNA, Majid AA, Azmi A (2017) Solving Buckmaster equation using cubic B-spline and cubic trigonometric B-spline collocation methods. In: AIP conference proceedings (Vol 1870, No 1, p. 040027). AIP Publishing

    Google Scholar 

  16. Hussain EA, Alwan ZM (2013) The finite volume method for solving Buckmaster’s Equation, Fisher’s Equation and Sine Gordon’s equation for PDE’s. Int Math Forum 8(13):599–617

    Article  MathSciNet  Google Scholar 

  17. Jafari M (2015) Group analysis via nonclassical symmetries for two-dimensional Ricci flow equation. Gen 29(1):22–29

    Google Scholar 

  18. Gupta RK, Singh M (2017) Nonclassical symmetries and similarity solutions of variable coefficient coupled KdV system using compatibility method. Nonlin Dyn 87(3):1543–1552

    Google Scholar 

  19. Bluman GW, Cheviakov AF, Anco SC (2009) Construction of conservation laws: how the direct method generalizes Noether’s theorem. In: Proceedings of 4th Workshop “Group Analysis of Differential Equations & Integrability, vol 1, pp. 1–23 (2009)

    Google Scholar 

  20. Naz R (2012) Conservation laws for some systems of nonlinear partial differential equations via multiplier approach. J Appl Math

    Google Scholar 

  21. Zhang L, Xu F (2018) Conservation laws, symmetry reductions, and exact solutions of some Keller-Segel models. Adv Dif Equ 2018(1):327

    Article  MathSciNet  Google Scholar 

Download references


It is hereby acknowledged that the author (Pinki Kumari) is grateful to the University Grant Commission for assisting her financially (Ref. ID 19/06/2016(i)EU-V).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Pinki Kumari .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kumari, P., Gupta, R.K., Kumar, S. (2021). On Invariant Analysis, Symmetry Reduction and Conservation Laws of Nonlinear Buckmaster Model. In: Singh, P., Gupta, R.K., Ray, K., Bandyopadhyay, A. (eds) Proceedings of International Conference on Trends in Computational and Cognitive Engineering. Advances in Intelligent Systems and Computing, vol 1169. Springer, Singapore.

Download citation

Publish with us

Policies and ethics