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On the conservation laws and invariant analysis for time-fractional coupled Fitzhugh-Nagumo equations using the Lie symmetry analysis

  • S. Sahoo
  • S. Saha RayEmail author
Regular Article
  • 51 Downloads

Abstract.

In this paper, the application of the fractional Lie symmetry method has been used for similarity reduction of the nonlinear fractional reaction-diffusion model. Also, it has been utilized for analyzing the conservation laws of the nonlinear fractional reaction-diffusion model viz. time fractional coupled Fitzhugh-Nagumo (FHN) equations. Foremost, the proposed method has been utilized to generate the infinitesimal generators for the time fractional coupled FHN equations. Then, with the help of Erdélyi-Kober differential operators, the fractional coupled differential equations have been reduced to fractional ordinary differential equations. Here, the Erdélyi-Kober differential operators have been defined via the Riemann-Liouville derivative. The new conserved vectors have been derived with the help of the proposed conservation theorem and formal Lagrangian.

References

  1. 1.
    L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers (Birkhäusher, Boston, MA, 2012)Google Scholar
  2. 2.
    H. Triki, A.M. Wazwaz, Appl. Math. Model. 37, 3821 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Saha Ray, Appl. Math. Comput. 175, 1046 (2006)MathSciNetGoogle Scholar
  4. 4.
    A. Bekir, Ö. Güner, A.C. Cevikel, Abstr. Appl. Anal. 2013, 426 (2013)CrossRefGoogle Scholar
  5. 5.
    S. Sahoo, S. Saha Ray, Nonlinear Dyn. 85, 1167 (2016)CrossRefGoogle Scholar
  6. 6.
    A.K. Gupta, S. Saha Ray, Appl. Math. Model. 39, 5121 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Atangana, D. Baleanu, Abstr. Appl. Anal. 2013, 1 (2013)Google Scholar
  8. 8.
    A.R. Seadawy, Comput. Math. Appl. 67, 172 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    E.P. Zemskov, I.R. Epstein, Phys. Rev. E 82, 026207 (2010)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    R. FitzHugh, Biophys. J. 1, 445 (1961)ADSCrossRefGoogle Scholar
  11. 11.
    J. Nagumo, S. Arimoto, S. Yoshizawa, Proc. IRE 50, 2061 (1962)CrossRefGoogle Scholar
  12. 12.
    A.L. Hodgkin, A.F. Huxley, J. Physiol. 117, 500 (1952)CrossRefGoogle Scholar
  13. 13.
    M. Armanyos, A.G. Radwan, Fractional-order Fitzhugh-Nagumo and Izhikevich neuron models, in 13th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON) 28 June--1 July 2016, Chiang Mai, Thailand (IEEE 2016), pp. 1--5Google Scholar
  14. 14.
    W. Bu, Y. Tang, Y. Wu, J. Yang, Appl. Math. Comput. 257, 355 (2015)MathSciNetGoogle Scholar
  15. 15.
    F. Liu, P. Zhuang, I. Turner, V. Anh, K. Burrage, J. Comput. Phys. 293, 252 (2015)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    F. Liu, I. Turner, V. Anh, Q. Yang, K. Burrage, ANZIAM J. 54, 608 (2013)CrossRefGoogle Scholar
  17. 17.
    M. Merdan, Int. J. Phys. Sci. 7, 2317 (2012)Google Scholar
  18. 18.
    Y. Pandir, Y.A. Tandogan, AIP Conf. Proc. 1558, 1919 (2013)ADSCrossRefGoogle Scholar
  19. 19.
    I.C. Sungu, H. Demir, Math. Probl. Eng. 2015, 1 (2015)CrossRefGoogle Scholar
  20. 20.
    S.Z. Rida, A.M.A. El-Sayed, A.A.M. Arafa, Commun. Nonlinear Sci. Numer. Simul. 15, 3847 (2010)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    F. Tchier, M. Inc, Z.S. Korpinar, D. Baleanu, Adv. Mech. Eng. 8, 1 (2016)CrossRefGoogle Scholar
  22. 22.
    S. Lie, Theorie der Transformationsgruppen 2 (Teubner, Leipzig, 1890) (in German)Google Scholar
  23. 23.
    P.J. Olver, Applications of Lie Groups to Differential Equations (Springer Nature, New York, 1993)Google Scholar
  24. 24.
    F. Oliveri, Symmetry 2, 658 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    H. Liu, J. Li, Q. Zhang, J. Comput. Appl. Math. 228, 1 (2009)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    S. Sahoo, G. Garai, S. Saha Ray, Nonlinear Dyn. 87, 1995 (2017)CrossRefGoogle Scholar
  27. 27.
    V. Dorodnitsyn, P. Winternitz, Nonlinear Dyn. 22, 49 (2000)CrossRefGoogle Scholar
  28. 28.
    G. Baumann, Symmetry Analysis of Differential Equations with Mathematica (Springer, New York, 2000)Google Scholar
  29. 29.
    S. Sahoo, S. Saha Ray, Comput. Math. Appl. 73, 253 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    J. Hu, Y. Ye, S. Shen, J. Zhang, Appl. Math. Comput. 233, 439 (2014)MathSciNetGoogle Scholar
  31. 31.
    Q. Huang, R. Zhdanov, Physica A 409, 110 (2014)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    C.Y. Qin, S.F. Tian, X.B. Wang, T.T. Zhang, Waves Random Complex Media 27, 308 (2017)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    G.W. Wang, T.Z. Xu, T. Feng, PLoS ONE 9, e88336 (2014)ADSCrossRefGoogle Scholar
  34. 34.
    H. Jafari, N. Kadkhoda, D. Baleanu, Nonlinear Dyn. 81, 1569 (2015)CrossRefGoogle Scholar
  35. 35.
    G.W. Wang, X.Q. Liu, Y.Y. Zhang, Commun. Nonlinear Sci. Numer. Simul. 18, 2321 (2013)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    R.K. Gazizov, N.H. Ibragimov, S.Y. Lukashchuk, Commun. Nonlinear Sci. Numer. Simul. 23, 153 (2015)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    N.H. Ibragimov, J. Math. Anal. Appl. 333, 311 (2007)MathSciNetCrossRefGoogle Scholar
  38. 38.
    E. Yaşar, J. Math. Anal. Appl. 363, 174 (2010)MathSciNetCrossRefGoogle Scholar
  39. 39.
    W. Rui, X. Zhang, Commun. Nonlinear Sci. Numer. Simul. 34, 38 (2016)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    R.A. Leo, G. Sicuro, P. Tempesta, Fract. Calc. Appl. Anal. 20, 212 (2017)MathSciNetCrossRefGoogle Scholar
  41. 41.
    N.H. Ibragimov, J. Phys. A 44, 432002 (2011)ADSCrossRefGoogle Scholar
  42. 42.
    A.M. Vinogradov, Acta Appl. Math. 15, 3 (1989)MathSciNetCrossRefGoogle Scholar
  43. 43.
    I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999)Google Scholar
  44. 44.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, New York, 2006)Google Scholar
  45. 45.
    S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Taylor and Francis, London, 2002)Google Scholar
  46. 46.
    S. Saha Ray, Fractional Calculus with Applications for Nuclear Reactor Dynamics (CRC Press, Boca Raton, FL, 2015)Google Scholar
  47. 47.
    E. Buckwar, Y. Luchko, J. Math. Anal. Appl. 227, 81 (1998)MathSciNetCrossRefGoogle Scholar
  48. 48.
    V.D. Djordjevic, T.M. Atanackovic, J. Comput. Appl. Math. 222, 701 (2008)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashchuk, Phys. Scr. T136, 014016 (2009)ADSCrossRefGoogle Scholar
  50. 50.
    R. Sahadevan, T. Bakkyaraj, J. Math. Anal. Appl. 393, 341 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Kalinga Institute of Industrial TechnologyDepartment of MathematicsBhubaneswarIndia
  2. 2.National Institute of Technology, Department of MathematicsRourkelaIndia

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