Constitutive thermal laws and the exact solutions of Timoshenko systems

  • S. JamalEmail author
  • A. G. Johnpillai
Original Paper


We consider a Timoshenko-type thermoelastic beam model that is defined by a system of partial differential equations, coupled with several thermal laws of heat conduction. Namely, we consider laws that characterize the dynamics of the system where any loss of energy is due to thermal effects in the absence of mechanical dissipation. Fourier law, as a consequence of parabolicity, describes an infinite propagation speed of thermal signals. We find that it possesses advantages over Cattaneo and Green–Naghdi Type III thermal law. This advantage can be seen from symmetry classifications of the optimal system of one-dimensional subalgebras. We show that Fourier law leads to a greater number of exact solutions when compared to any other thermal law studied. Finally, the evolution of the exact solutions is graphically presented.


Exact solutions Lie symmetries Fourier thermal law Timoshenko beam 


02.30.Jr 02.30.Xx 02.20.Sv 46.25.Hf 



  1. [1]
    S P Timoshenko Lond. Edinb. Dublin Philos. Mag. J. Sci. 43 125–131 (1922)Google Scholar
  2. [2]
    S P Timoshenko Philos. Mag. 41 744–746 (1921)CrossRefGoogle Scholar
  3. [3]
    S Jamal Quaest. Math. 41 409–421 (2018)MathSciNetCrossRefGoogle Scholar
  4. [4]
    N Dimakis, A Giacomini, S Jamal, G Leon and A Paliathanasis Phys. Rev. D 95 064031 (2017)ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A G Johnpillai, K S Mahomed, C Harley and F M Mahomed Z. Naturforsch. 71(5)a 447–456 (2016)ADSGoogle Scholar
  6. [6]
    S Jamal Gen. Relativ. Grav. 49 1–14 (2017)CrossRefGoogle Scholar
  7. [7]
    P J Olver Application of Lie Groups to Differential Equations (New York: Springer) (1993)CrossRefzbMATHGoogle Scholar
  8. [8]
    H Stephani, Differential Equations: Their Solutions using Symmetry (Cambridge: Cambridge University Press) (1989)zbMATHGoogle Scholar
  9. [9]
    G W Bluman and S Kumei Symmetries and Differential Equations (New York: Springer ) (1989)CrossRefzbMATHGoogle Scholar
  10. [10]
    N H Ibragimov (Ed.) CRC Handbook of Lie Group Analysis of Differential Equations 1–3 (Boca Raton: CRC Press) (1994–1996)Google Scholar
  11. [11]
    L V Ovsiannikov Group Analysis of Differential Equations (New York: Academic Press) (1982)zbMATHGoogle Scholar
  12. [12]
    S M Al-Omari , F D Zaman and H Azad Mathematics 5(34) (2017)Google Scholar
  13. [13]
    P A Djondjorov Int. J. Eng. Sci. 33(4) 2103–2114 (1995)MathSciNetCrossRefGoogle Scholar
  14. [14]
    S M Han, H Benaroya and T Wei J. Sound Vib. 225(5) 935–988 (1999)ADSCrossRefGoogle Scholar
  15. [15]
    F Dell’oro and V Pata. arXiv:1304.0985v1
  16. [16]
    A E Green and P M Naghdi J. Elast. 31 189–208 (1993)CrossRefGoogle Scholar
  17. [17]
    T M Rocha Filio and A Figueiredo Comput. Phys. Commun. 182 467 (2011)ADSCrossRefGoogle Scholar
  18. [18]
    S Dimas and D Tsoubelis SYM: A New Symmetry-Finding Package for Mathematica. In Group Analysis of Differential Equations (Cyprus: University of Cyprus) pp. 64–70 (2005)Google Scholar

Copyright information

© Indian Association for the Cultivation of Science 2019

Authors and Affiliations

  1. 1.School of MathematicsUniversity of the WitwatersrandJohannesburg, WitsSouth Africa
  2. 2.Department of MathematicsEastern University, Sri LankaChenkaladySri Lanka

Personalised recommendations