Advertisement

Fractional analysis for nonlinear electrical transmission line and nonlinear Schroedinger equations with incomplete sub-equation

  • Emmanuel Fendzi-Donfack
  • Jean Pierre NguenangEmail author
  • Laurent Nana
Regular Article

Abstract.

We use the fractional complex transform with the modified Riemann-Liouville derivative operator to establish the exact and generalized solutions of two fractional partial differential equations. We determine the solutions of fractional nonlinear electrical transmission lines (NETL) and the perturbed nonlinear Schroedinger (NLS) equation with the Kerr law nonlinearity term. The solutions are obtained for the parameters in the range (\(0<\alpha\le 1\)) of the derivative operator and we found the traditional solutions for the limiting case of \(\alpha =1\). We show that according to the modified Riemann-Liouville derivative, the solutions found can describe physical systems with memory effect, transient effects in electrical systems and nonlinear transmission lines, and other systems such as optical fiber.

References

  1. 1.
    A. Saadatmandi, M. Deghghan, Comput. Math. Appl. 59, 1326 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Y. Zhou, F. Jiao, J. Li, Nonlinear Anal. 71, 2724 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    L. Galeone, R. Garrappa, J. comput. Appl. Math. 228, 548 (2009)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    J.C. Trigeassou, N. Maamri, J. Sabatier, A. Oustaloup, Signal Process. 91, 437 (2011)CrossRefGoogle Scholar
  5. 5.
    W. Deng, Nonlinear Anal. 72, 1768 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    F. Ghoreishi, S. Yazdani, Comput. Math. Appl. 61, 30 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J.T. Edwards, N.J. Ford, A.C. Simpson, J. Comput. Appl. Math. 148, 401 (2002)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Muslim, Math. Comput. Model. 49, 1164 (2009)CrossRefGoogle Scholar
  9. 9.
    A.M.A. El-Sayed, M. Gaber, Phys. Lett. A 359, 175 (2006)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    A.M.A. El-Sayed, S.H. Behiry, W.E. Raslan, Comput. Math. Appl. 59, 1759 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J.H. He, Commun. Nonlinear Sci. Numer. Simul. 2, 230 (1997)ADSCrossRefGoogle Scholar
  12. 12.
    G. Wu, E.W.M. Lee, Phys. Lett. A 374, 2506 (2010)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    S. Guo, L. Mei, Phys. Lett. A 375, 309 (2011)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    J.H. He, Comput. Methods Appl. Mech. Eng. 178, 257 (1999)ADSCrossRefGoogle Scholar
  15. 15.
    J.H. He, Int. J. Non-Linear Mech. 35, 37 (2000)ADSCrossRefGoogle Scholar
  16. 16.
    Z. Odibat, S. Momani, Appl. Math. Lett. 21, 194 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Cui, J. Comput. Phys. 228, 7792 (2009)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Q. Huang, G. Huang, H. Zhan, Adv. Water Resour. 31, 1578 (2008)ADSCrossRefGoogle Scholar
  19. 19.
    S. Zhang, H.Q. Zhang, Phys. Lett. A 375, 1069 (2011)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    S.M. Guo, L.Q. Mei, Y. Li, Y.F. Sun, Phys. Lett. A 376, 407 (2012)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    B. Lu, Phys. Lett. A 376, 2045 (2012)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    G. Jumarie, Comput. Math. Appl. 51, 1367 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    I. Petráš, Chaos, Solitons Fractals 38, 140 (2008)ADSCrossRefGoogle Scholar
  24. 24.
    M.L. Wang, X.Z. Li, J.L. Zhang, Phys. Lett. A 372, 417 (2008)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    E.M.E. Zayed, M. Abdelaziz, WSEAS Trans. Math. 10, 115 (2011)Google Scholar
  26. 26.
    M.L. Wang, J.L. Zhang, X.Z. Li, Appl. Math. Comput. 206, 321 (2008)MathSciNetGoogle Scholar
  27. 27.
    E.M.E. Zayed, WSEAS Trans. Math. 10, 56 (2011)Google Scholar
  28. 28.
    I. Aslan, Appl. Math. Comput. 215, 3140 (2009)MathSciNetGoogle Scholar
  29. 29.
    B. Ayhan, A. Bekir, Commun. Nonlinear Sci. Numer. Simul. 17, 3490 (2012)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    E.M.E. Zayed, K.A. Gepreel, WSEAS Trans. Math. 10, 270 (2011)Google Scholar
  31. 31.
    B. Tang, Y. He, L. Wei, S. Wang, Phys. Lett. A 375, 3355 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    M.B. Hubert, G. Betchewe, S.Y. Doka, T.C. Kofane, Appl. Math. Comput. 239, 299 (2014)MathSciNetGoogle Scholar
  33. 33.
    E. Tala-Tebue, D.C. Tsobgni-Fozap, A. Kenfack-Jiotsa, T.C. Kofane, Eur. Phys. J. Plus 129, 136 (2014)CrossRefGoogle Scholar
  34. 34.
    Z. Zai-Yun, G. Xiang-Yang, Y. De-Min, Z. Ying-Hui, L. Xin-Ping, Commun. Theor. Phys. 57, 764 (2012)ADSCrossRefGoogle Scholar
  35. 35.
    Ji-Huan He, S.K. Elagan, Z.B. Li, Phys. Lett. A 376, 257 (2012)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Vasily E. Tarasov, Commun. Nonlinear Sci. Numer. Simul. 30, 1 (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    H. Ertik, A.E. Çalik, H. Sirin, M. Sen, B. Öder, Rev. Mex. Fís. 61, 58 (2015)MathSciNetGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Pure Physics Laboratory, Group of Nonlinear Physics and Complex Systems, Department of Physics Faculty of SciencesUniversity of DoualaDoualaCameroon
  2. 2.Nonlinear Physics and Complex Systems Group, Department of Physics, The Higher Teacher’s Training CollegeUniversity of Yaounde IYaoundeCameroon
  3. 3.Abdus Salam ICTPTriesteItaly

Personalised recommendations