Eur. Phys. J. Plus (2018) 133: 54
https://doi.org/10.1140/epjp/i2018-11916-1

Regular Article

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

Department of Mathematics Education, Faculty of Education, Adıyaman University, 2230, Adıyaman, Turkey

^* e-mail: brtkrg44@gmail.com

Received: 3 November 2017
Accepted: 13 January 2018
Published online: 14 February 2018

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.