Abstract.
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.
Similar content being viewed by others
References
I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, CA, 1999)
K.B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York and London, 1974)
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)
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)
M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 1, 73 (2015)
M. Caputo, M. Fabrizio, Progr. Fract. Differ. Appl. 2, 1 (2016)
M. Caputo, M. Fabrizio, J. Comput. Phys. 293, 400 (2015)
H.M. Baskonus, T. Mekkaoui, Z. Hammouch, H. Bulut, Entropy 17, 5771 (2015)
H.M. Baskonus, H. Bulut, Open Math. 13, 547 (2015)
H.M. Baskonus, H. Bulut, AIP Conf. Proc. 1738, 290004 (2016)
H.M. Baskonus, Z. Hammouch, T. Mekkaoui, H. Bulut, AIP Conf. Proc. 1738, 290005 (2016)
H. Bulut, G. Yel, H.M. Baskonus, Turk. J. Math. Comput. Sci. 5, 1 (2016)
M.T. Gencoglu, H.M. Baskonus, H. Bulut, AIP Conf. Proc. 1798, 020103 (2017)
R. Najafi, G.D. Küçük, E. Çelik, Math. Methods Appl. Sci. 23, 939 (2016)
E. Çelik, E. Sefidgar, B. Shiri, Int. J. Appl. Math. Stat. 56, 23 (2017)
G.D. Küçük, M. Yigider, E. Çelik, Brit. J. Appl. Sci. Technol. 4, 3653 (2014)
N.M. Yagmurlu, O. Tasbozan, Y. Ucar, A. Esen, Appl. Math. Inf. Sci. Lett. 4, 19 (2016)
A. Esen, F. Bulut, Ö. Oruç, Eur. Phys. J. Plus 131, 116 (2016)
A. Atangana, B. Dumitru, Therm. Sci. 20, 763 (2016)
A. Atangana, I. Koca, Chaos, Solitons Fractals 89, 447 (2016)
Badr Saad T. Alkahtani, Solitons Fractals 89, 547 (2016)
A. Atangana, Eur. Phys. J. Plus 131, 373 (2016)
W. Thomson, Proc. R. Soc. London 7, 382 (1854)
S. Vitali, G. Castellani, F. Mainardi, Chaos, Solitons Fractals 102, 467 (2017)
F. Bashforth, J.C. Adams, An Attempt to Test the Theories of Capillary Action (Cambridge University Press, London, 1883)
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. 9.11
R.G. Batogna, A. Atangana, Numer. Methods Partial Differ. Equ. (2017) https://doi.org/10.1002/num.22216
G.G. Dahlquist, BIT Numer. Math. 3, 27 (1963)
W.E. Milne, Am. Math. Mon. Math. Assoc. Am. 33, 455 (1926)
T. Von. Karman, M.A. Biot, Mathematical Methods in Engineering: An Introduction to the Mathematical Treatment of Engineering Problems (McGraw-Hill, New York, 1940)
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)
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. 14
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Karaagac, B. Analysis of the cable equation with non-local and non-singular kernel fractional derivative. Eur. Phys. J. Plus 133, 54 (2018). https://doi.org/10.1140/epjp/i2018-11916-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2018-11916-1