Skip to main content
SpringerLink
Log in

On the fractional Jaulent-Miodek equation associated with energy-dependent Schrödinger potential: Lie symmetry reductions, explicit exact solutions and conservation laws

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract.

In this study, the Lie symmetry analysis is performed on a coupled system of nonlinear time-fractional Jaulent-Miodek equations associated with energy-dependent Schrödinger potential. The underlying problem is similarity reduced to a system of nonlinear ordinary differential equations with Erdelyi-Kober fractional derivatives. Employing the invariant subspace method, a set of explicit solutions for the problem has been well constructed. In addition, the new conservation theorem is used to construct the conservation laws of the problem.

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

Similar content being viewed by others

References

  1. B. Ross (Editor), Fractional calculus and its applications, in Proceedings of the International Conference Held at the University of New Haven, June 1974 (Springer-Verlag, 1975)

  2. K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order (Academic Press, New York, 1974)

  3. K.S. Miller, B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)

  4. L. Debnath, Int. J. Math. Math. Sci. 2003, 3413 (2003)

    Article  Google Scholar 

  5. A. Loverro, Fractional Calculus: History, Defnitions and Applications for the Engineer (2004)

  6. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in North-Holland Mathematics Studies, Vol. 204 (Elsevier, Amsterdam, 2006)

  7. Vladimir V. Uchaikin, Fractional Derivatives for Physicists and Engineers (Springer, Berlin, 2013)

  8. A. Atangana, D. Baleanu, Therm. Sci. 20, 763 (2016)

    Article  Google Scholar 

  9. A. Atangana, I. Koca, Chaos, Solitons Fractals 89, 447 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  10. O.J.J. Algahtani, Chaos, Solitons Fractals 89, 552 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  11. B.S.T. Alkahtani, Chaos, Solitons Fractals 89, 547 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  12. A. Atangana, Chaos, Solitons Fractals 102, 396 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  13. S. Abbasbandy, J. Comput. Appl. Math. 207, 53 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  14. S. Kumar, D. Kumar, S. Abbasbandy, M.M. Rashidi, Ain Shams Eng. J. 5, 569 (2014)

    Article  Google Scholar 

  15. Mohammad Mirzazadeh et al., Eur. Phys. J. Plus 131, 166 (2016)

    Article  Google Scholar 

  16. K. Parand, P. Mazaheri, H. Yousefi, M. Delkhosh, Eur. Phys. J. Plus 132, 77 (2017)

    Article  Google Scholar 

  17. Vahid Reza Hosseini, Elyas Shivanian, Wen Chen, Eur. Phys. J. Plus 130, 33 (2015)

    Article  Google Scholar 

  18. Mohammad Aslefallah, Elyas Shivanian, Eur. Phys. J. Plus 130, 47 (2015)

    Article  Google Scholar 

  19. E. Shivanian, S. Abbasbandy, M.S. Alhuthali, H.H. Alsulami, Eng. Anal. Bound. Elem. 56, 98 (2015)

    Article  MathSciNet  Google Scholar 

  20. H. Roohani Ghehsareh, S.H. Bateni, A. Zaghian, Eng. Anal. Bound. Elem. 61, 52 (2015)

    Article  MathSciNet  Google Scholar 

  21. H.R. Ghehsareh, A. Zaghian, S.M. Zabetzadeh, Neural Comput. Appl., https://doi.org/10.1007/s00521-016-2595-y (2017)

  22. A. Atangana, D. Baleanu, Abstr. Appl. Anal. 2013, 8 (2013)

    Google Scholar 

  23. A.K. Gupta, S. Saha Ray, Appl. Math. Comput. 270, 458 (2015)

    MathSciNet  Google Scholar 

  24. S. Sahoo, S. Saha Ray, Nonlinear Dyn. 85, 1167 (2016)

    Article  Google Scholar 

  25. M. Jaulent, I. Miodek, Lett. Math. Phys. 1, 243 (1976)

    Article  ADS  Google Scholar 

  26. Dogan Kaya, Salah M. El-Sayed, Phys. Lett. A 318, 345 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  27. J.B. Chen, X.G. Geng, Chaos, Solitons Fractals 30, 797 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  28. D.D. Ganji, M. Jannatabadi, E. Mohseni, J. Comput. Appl. Math. 207, 35 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  29. A.M. Wazwaz, Phys. Lett. A 366, 85 (2007)

    Article  ADS  Google Scholar 

  30. Ji-Huan He, Li-Na Zhang, Phys. Lett. A 372, 1044 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  31. L. Kavitha, P. Sathishkumar, T. Nathiyaa, D. Gopi, Phys. Scr. 79, 035403 (2009)

    Article  ADS  Google Scholar 

  32. Jafar Biazar, Mostafa Eslami, J. King Saud Univ. Sci. 23, 133 (2011)

    Article  Google Scholar 

  33. Akbar Mohebbi, Zohreh Asgari, Mehdi Dehghan, Commun. Nonlinear Sci. Numer. Simul. 17, 4602 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  34. Anjan Biswas, A.H. Kara, Appl. Math. Comput. 217, 944 (2010)

    MathSciNet  Google Scholar 

  35. Mehdi Nadjafikhah, Mostafa Hesamiarshad, Abstr. Appl. Anal. 2014, 476025 (2014)

    Google Scholar 

  36. H. Wang, T. Xia, Commun. Nonlinear Sci. Numer. Simul. 18, 2859 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  37. P.J. Olver, Application of Lie groups to Differential Equations (Springer-Verlag, 1986)

  38. G.W. Bluman, A.F. Cheviakov, S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations (Springer, 2010)

  39. V. Dorodnitsyn, Applications of Lie Groups to Difference Equations (CRC Press, 2011)

  40. E. Buckwar, Y. Luchko, J. Math. Anal. Appl. 227, 81 (1998)

    Article  MathSciNet  Google Scholar 

  41. R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashchuk, Vestn. USATU 9, 125 (2007)

    Google Scholar 

  42. R.K. Gazizov, A.A. Kasatkin, S. Yu Lukashchuk, Phys. Scr. 2009, 014016 (2009)

    Article  Google Scholar 

  43. G.F. Jefferson, J. Carminati, Comput. Phys. Commun. 185, 430 (2014)

    Article  ADS  Google Scholar 

  44. Hossein Jafari, Nematollah Kadkhoda, Dumitru Baleanu, Nonlinear Dyn. 81, 1569 (2015)

    Article  Google Scholar 

  45. M.S. Hashemi, Physica A 417, 141 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  46. Emrullah Yasar, Yakup Yildirim, Chaudry Masood Khalique, Results Phys. 6, 322 (2016)

    Article  ADS  Google Scholar 

  47. M.S. Hashemi, F. Bahrami, R. Najafi, Optik 138, 240 (2017)

    Article  ADS  Google Scholar 

  48. M.S. Hashemi, D. Baleanu, J. Optoelectron. Adv. Mater. 18, 383 (2016)

    Google Scholar 

  49. R. Najafi, F. Bahrami, M.S. Hashemi, Nonlinear Dyn. 87, 1785 (2017)

    Article  Google Scholar 

  50. S. Pashayi, M.S. Hashemi, S. Shahmorad, Commun. Nonlinear Sci. Numer. Simul. 51, 66 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  51. Komal Singla, R.K. Gupta, J. Math. Phys. 57, 101504 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  52. Komal Singla, R.K. Gupta, J. Math. Phys. 58, 054101 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  53. Komal Singla, R.K. Gupta, J. Math. Phys. 58, 051503 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  54. Komal Singla, R.K. Gupta, J. Math. Phys. 58, 061501 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  55. Victor A. Galaktionov, Sergey R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics (CRC Press, 2006)

  56. Raphael K. Gazizov, Alexey A. Kasatkin, Comput. Math. Appl. 66, 576 (2013)

    Article  MathSciNet  Google Scholar 

  57. R. Sahadevan, P. Prakash, Commun. Nonlinear Sci. Numer. Simul. 42, 158 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  58. R. Sahadevan, P. Prakash, Chaos, Solitons Fractals 104, 107 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  59. A.V. Bocharov, V.N. Chetverikov, S.V. Duzhin, N.G. Chorkova, A.M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics (AMS, 1999)

  60. E. Noether, Nachr. Math. Phys. Kl. 2, 235 (1918) English translation: Transp. Theory Stat. Phys. 1

    Google Scholar 

  61. O.P. Agrawal, J. Math. Anal. Appl. 272, 368 (2002)

    Article  MathSciNet  Google Scholar 

  62. G.S.F. Frederico, D.F.M. Torres, J. Math. Anal. Appl. 334, 834 (2007)

    Article  MathSciNet  Google Scholar 

  63. T.M. Atanackovic, S. Konjik, S. Pilipovic, S. Simic, Nonlinear Anal. 71, 1504 (2009)

    Article  MathSciNet  Google Scholar 

  64. L. Bourdin, J. Cresson, I. Greff, Commun. Nonlinear Sci. Numer. Simul. 18, 878 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  65. N.H. Ibragimov, J. Math. Anal. Appl. 333, 311 (2007)

    Article  MathSciNet  Google Scholar 

  66. R.K. Gazizov, N.H. Ibragimov, SYu. Lukashchuk, Commun. Nonlinear Sci. Numer. Simul. 23, 153 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  67. Komal Singla, R.K. Gupta, Commun. Nonlinear Sci. Numer. Simul. 53, 10 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  68. Komal Singla, R.K. Gupta, Nonlinear Dyn. 89, 321 (2017)

    Article  Google Scholar 

  69. V. Kiryakova, Generalized Fractional Calculus and Applications, in Research Notes in Mathematics Series (Longman Sci. & Tech., Longman Group, 1994)

  70. S.F. Shen, C.Z. Qu, Y.Y. Jin, L.N. Ji, Chin. Ann. Math. Ser. B 33, 161 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Malek Ashtar University Of Technology, 34149-16818, Shahin Shahr, Iran

    A. Majlesi, H. Roohani Ghehsareh & A. Zaghian

Authors
  1. A. Majlesi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Roohani Ghehsareh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Zaghian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Roohani Ghehsareh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Majlesi, A., Roohani Ghehsareh, H. & Zaghian, A. On the fractional Jaulent-Miodek equation associated with energy-dependent Schrödinger potential: Lie symmetry reductions, explicit exact solutions and conservation laws. Eur. Phys. J. Plus 132, 516 (2017). https://doi.org/10.1140/epjp/i2017-11793-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/i2017-11793-0

Search

Navigation