Analysis of the cable equation with non-local and non-singular kernel fractional derivative

  • Berat KaraagacEmail author
Regular Article
Part of the following topical collections:
  1. Focus Point on Modelling Complex Real-World Problems with Fractal and New Trends of Fractional Differentiation


Recently a new concept of differentiation was introduced in the literature where the kernel was converted from non-local singular to non-local and non-singular. One of the great advantages of this new kernel is its ability to portray fading memory and also well defined memory of the system under investigation. In this paper the cable equation which is used to develop mathematical models of signal decay in submarine or underwater telegraphic cables will be analysed using the Atangana-Baleanu fractional derivative due to the ability of the new fractional derivative to describe non-local fading memory. The existence and uniqueness of the more generalized model is presented in detail via the fixed point theorem. A new numerical scheme is used to solve the new equation. In addition, stability, convergence and numerical simulations are presented.


  1. 1.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, CA, 1999)Google Scholar
  2. 2.
    K.B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York and London, 1974)Google Scholar
  3. 3.
    K. Diethelm, The Analysis of Fractional Differential Equations, an Application Oriented, Exposition Using Differential Operators of Caputo type, in Lecture Notes in Mathematics, nr. 2004 (Springer, Heidelbereg, 2010)Google Scholar
  4. 4.
    D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods in Series on Complexity, Nonlinearity and Chaos (World Scientific, Boston, 2012)Google Scholar
  5. 5.
    M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)Google Scholar
  6. 6.
    M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 2, 1 (2016)CrossRefGoogle Scholar
  7. 7.
    M. Caputo, M. Fabrizio, J. Comput. Phys. 293, 400 (2015)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    H.M. Baskonus, T. Mekkaoui, Z. Hammouch, H. Bulut, Entropy 17, 5771 (2015)ADSCrossRefGoogle Scholar
  9. 9.
    H.M. Baskonus, H. Bulut, Open Math. 13, 547 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    H.M. Baskonus, H. Bulut, AIP Conf. Proc. 1738, 290004 (2016)CrossRefGoogle Scholar
  11. 11.
    H.M. Baskonus, Z. Hammouch, T. Mekkaoui, H. Bulut, AIP Conf. Proc. 1738, 290005 (2016)CrossRefGoogle Scholar
  12. 12.
    H. Bulut, G. Yel, H.M. Baskonus, Turk. J. Math. Comput. Sci. 5, 1 (2016)Google Scholar
  13. 13.
    M.T. Gencoglu, H.M. Baskonus, H. Bulut, AIP Conf. Proc. 1798, 020103 (2017)CrossRefGoogle Scholar
  14. 14.
    R. Najafi, G.D. Küçük, E. Çelik, Math. Methods Appl. Sci. 23, 939 (2016)Google Scholar
  15. 15.
    E. Çelik, E. Sefidgar, B. Shiri, Int. J. Appl. Math. Stat. 56, 23 (2017)MathSciNetGoogle Scholar
  16. 16.
    G.D. Küçük, M. Yigider, E. Çelik, Brit. J. Appl. Sci. Technol. 4, 3653 (2014)CrossRefGoogle Scholar
  17. 17.
    N.M. Yagmurlu, O. Tasbozan, Y. Ucar, A. Esen, Appl. Math. Inf. Sci. Lett. 4, 19 (2016)Google Scholar
  18. 18.
    A. Esen, F. Bulut, Ö. Oruç, Eur. Phys. J. Plus 131, 116 (2016)CrossRefGoogle Scholar
  19. 19.
    A. Atangana, B. Dumitru, Therm. Sci. 20, 763 (2016)CrossRefGoogle Scholar
  20. 20.
    A. Atangana, I. Koca, Chaos, Solitons Fractals 89, 447 (2016)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Badr Saad T. Alkahtani, Solitons Fractals 89, 547 (2016)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Atangana, Eur. Phys. J. Plus 131, 373 (2016)CrossRefGoogle Scholar
  23. 23.
    W. Thomson, Proc. R. Soc. London 7, 382 (1854)CrossRefGoogle Scholar
  24. 24.
    S. Vitali, G. Castellani, F. Mainardi, Chaos, Solitons Fractals 102, 467 (2017)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    F. Bashforth, J.C. Adams, An Attempt to Test the Theories of Capillary Action (Cambridge University Press, London, 1883)Google Scholar
  26. 26.
    H. Jeffreys, B.S. Jeffreys, The Adams-Bashforth Method, in Methods of Mathematical Physics, 3rd edition (Cambridge University Press, England, 1988) pp. 292--293, sect. 9.11Google Scholar
  27. 27.
    R.G. Batogna, A. Atangana, Numer. Methods Partial Differ. Equ. (2017)
  28. 28.
    G.G. Dahlquist, BIT Numer. Math. 3, 27 (1963)CrossRefGoogle Scholar
  29. 29.
    W.E. Milne, Am. Math. Mon. Math. Assoc. Am. 33, 455 (1926)CrossRefGoogle Scholar
  30. 30.
    T. Von. Karman, M.A. Biot, Mathematical Methods in Engineering: An Introduction to the Mathematical Treatment of Engineering Problems (McGraw-Hill, New York, 1940)Google Scholar
  31. 31.
    W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in FORTRAN: The Art of Scientific Computing (Cambridge University Press, England, 1992)Google Scholar
  32. 32.
    E.T. Whittaker, G. Robinson, The Numerical Solution of Differential Equations, in The Calculus of Observations: A Treatise on Numerical Mathematics (Dover, New York, 1967) pp. 363--367, chapt. 14Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematics Education, Faculty of EducationAdıyaman UniversityAdıyamanTurkey

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