Skip to main content
SpringerLink
Log in

New Exact Solutions of the Thomas Equation Using Symmetry Transformations

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

Lie symmetry analysis of differential equations is one of the most effective techniques to find exact solutions to a differential equation or to reduce the number of independent variables or at least reduce the order and nonlinearity of the equation. In this article, we present invariant group solutions of the Thomas equation based on four-dimensional Lie symmetry algebra. Then by using nonzero commutators, we compute discrete symmetry transformations and then use them to find new exact solutions of the Thomas equation by following Hydon’s method.

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

Data Availability

Enquiries about data availability should be directed to the authors.

References

  1. Lie, S.: On the integration by definite integrals of a class of linear partial differential equations. Cammermeyer (1880)

  2. Lie, S.: on the integration by definite integrals of a class of linear partial differential equations. Arch. Math. 6, 328 (2020)

    MATH  Google Scholar 

  3. Rota, G.C.: Group analysis of differential equations: LV Ovsiannikov, vol 416. Academic Press (1983)

  4. Noether, E.: Konig Gesell Wissen. Gottingen. Math. Phys. 1918, 235–257 (1918)

    Google Scholar 

  5. Olver, P.J.: Applications of Lie Groups to Differential Equations (Vol. 107). Springer Science & Business Media (2000)

  6. Ibragimov, N.H.: CRC Handbook of Lie Group Analysis of Differential Equations. 3. CRC Press (1995)

  7. Hydon, P.E., Hydon, P.E.: Symmetry Methods for Differential Equations: A Beginner’s Guide (No. 22). Cambridge University Press (2000)

  8. Hydon, P.E.: Discrete point symmetries of ordinary differential equations. Proc R Soc Lond Ser A Math Phys Eng Sci. 454(1975), 1961–1972 (1998)

    MathSciNet  MATH  Google Scholar 

  9. Bluman, G., Anco, S.: Symmetry and Integration Methods for Differential Equations (Vol. 154). Springer Science & Business Media (2008)

  10. Gaeta, G., Rodríguez, M.A.: Discrete symmetries of differential equations. J. Phys. A: Math. Gen. 29(4), 859 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hydon, P.E.: Discrete point symmetries of ordinary differential equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 454(1975), 1961–1972 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Laine-Pearson, F.E., Hydon, P.E.: Classification of discrete symmetries of ordinary differential equations. Stud. Appl. Math. 111(3), 269–299 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chatibi, Y., El Kinani, E.H., Ouhadan, A.: On the discrete symmetry analysis of some classical and fractional differential equations. Math. Methods Appl. Sci. 44(4), 2868–2878 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Levi, D., Rodríguez, M.A.: Lie discrete symmetries of lattice equations. J. Phys. A: Math. Gen. 37(5), 1711 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Silberberg, G.: Discrete symmetries of the Black-Scholes equation. In: Proceedings of 10th International Conference in Modern Group Analysis. (Vol. 190, p. 197) (2005)

  16. Yang, H., Shi, Y., Yin, B., Dong, H.: Discrete symmetries analysis and exact solutions of the inviscid burgers equation. Discret. Dyn. Nat. Soc. 2012(1), 275–289 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Hamad, M.A.A., Hassanien, I.A., El-Nahary, E.K.H.: Discrete symmetries analysis of Burgers equation with time dependent flux at the origin. World Appl. Sci. J. 12(12), 2291–2300 (2011)

    Google Scholar 

  18. Bibi, K., Feroze, T.: Discrete symmetry group approach for numerical solution of the heat equation. Symmetry 12(3), 359 (2020)

    Article  Google Scholar 

  19. Bibi, K.: Particular solutions of ordinary differential equations using discrete symmetry groups. Symmetry 12(1), 180 (2020)

    Article  Google Scholar 

  20. Hussain, A., Kara, A.H., Zaman, F.D.: An invariance analysis of the Vakhnenko-Parkes equation. Chaos Solit. Fractals. 171, 113423 (2023)

    Article  MathSciNet  Google Scholar 

  21. Thomas, H.C.: Heterogeneous ion exchange in a flowing system. J. Am. Chem. Soc. 66(10), 1664–1666 (1944)

    Article  Google Scholar 

  22. Gray, R.J.: How to calculate all point symmetries of linear and linearizable differential equations. Proc. R. Soc. A: Math. Phys. Eng. Sci. 471(2175), 20140685 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sakovich, S.Y.: On the Thomas equation. J. Phys. A: Math. Gen. 21(23), L1123 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ouhadan, A., El Kinani, E.H.: Lie symmetries and preliminary classification of group-invariant solutions of Thomas equation. ArXiv:math-ph/0412043 (2004)

  25. Malima, P.T., Jamal, S.Y., Manale, J.M.: Solutions of the Thomas equation using the pure lie symmetries approach and manifolds. Adv. Dyn. Syst. Appl. 16(2), 1844–1865 (2021)

    Google Scholar 

  26. Hussain, A., Usman, M., Zaman, F.D., Eldin, S.M.: Double reductions and traveling wave structures of the generalized Pochhammer–Chree equation. Partial Differ. Equ. Appl. Math. 7, 100521 (2023)

    Article  Google Scholar 

  27. Usman, M., Hussain, A., Zaman, F.D.: Invariance analysis of thermophoretic motion equation depicting the wrinkle propagation in substrate-supported Graphene sheets. Phys. Scr. 98(9), 095205 (2023)

    Article  Google Scholar 

  28. Mbusi, S.O., Muatjetjeja, B., Adem, A.R.: Lagrangian formulation, conservation laws, travelling wave solutions: a generalized Benney–Luke equation. Mathematics 9(13), 1480 (2021)

    Article  Google Scholar 

  29. Hussain, A., Usman, M., Al-Sinan, B.R., Osman, W.M., Ibrahim, T.F.: Symmetry analysis and closed-form invariant solutions of the nonlinear wave equations in elasticity using optimal system of Lie subalgebra. Chin. J. Phys. 83, 1–13 (2023)

    Article  MathSciNet  Google Scholar 

  30. Usman, M., Hussain, A., Zaman, F.D., Eldin, S.M.: Group invariant solutions of wave propagation in phononic materials based on the reduced micromorphic model via optimal system of Lie subalgebra. Results Phys. 48, 106413 (2023)

    Article  Google Scholar 

  31. Hussain, A., Usman, M., Zaman, F.D., Ibrahim, T.F., Dawood, A.A.: Symmetry analysis, closed-form invariant solutions and dynamical wave structures of the Benney-Luke equation using optimal system of Lie subalgebras. Chin. J. Phys. 84, 66–88 (2023)

    Article  MathSciNet  Google Scholar 

  32. Hussain, A., Kara, A.H., Zaman, F.D.: Symmetries, associated first integrals and successive reduction of Schr ödinger type and other second order difference equations. Optik 287, 171080 (2023)

    Article  Google Scholar 

  33. Hussain, A., Usman, M., Zaman, F.D., Eldin, S.M.: Symmetry analysis and invariant solutions of Riabouchinsky Proudman Johnson equation using optimal system of Lie subalgebras. Results Phys. 49, 106507 (2023)

    Article  Google Scholar 

  34. Usman, M., Hussain, A., Zaman, F.D., Eldin, S.M.: Symmetry analysis and exact Jacobi elliptic solutions for the nonlinear couple Drinfeld Sokolov Wilson dynamical system arising in shallow water waves. Results Phys. 50, 106613 (2023)

    Article  Google Scholar 

  35. Hydon, P.E.: How to construct the discrete symmetries of partial differential equations. Eur. J. Appl. Math. 11(5), 515–527 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Abdus Salam School of Mathematical Sciences, Government College University, Lahore, 54600, Pakistan

    Akhtar Hussain & F. D. Zaman

  2. School of Mathematics, University of the Witwatersrand, Johannesburg, Wits, 2050, South Africa

    A. H. Kara

Authors
  1. Akhtar Hussain
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. H. Kara
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. D. Zaman
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors made equal contributions to the article.

Corresponding author

Correspondence to F. D. Zaman.

Ethics declarations

Conflict on interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hussain, A., Kara, A.H. & Zaman, F.D. New Exact Solutions of the Thomas Equation Using Symmetry Transformations. Int. J. Appl. Comput. Math 9, 106 (2023). https://doi.org/10.1007/s40819-023-01585-5

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-023-01585-5

Keywords

Search

Navigation