ARTICLE IN PRESS JID: CHAOS [m5G;October 24, 2019;10:27] Chaos, Solitons and Fractals xxx (xxxx) xxx Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Review On the traveling waves in nonlinear electrical transmission lines with intrinsic fractional-order using discrete tanh method Emmanuel Fendzi Donfack a,b , Jean Pierre Nguenang a,c,∗ , Laurent Nana a a Pure Physics Laboratory, Group of Nonlinear Physics and Complex Systems, Department of Physics, Faculty of Sciences, University of Douala, P.O. Box 24157, Douala, Cameroon b Nonlinear Physics and Complex Systems Group, Department of Physics, The Higher Teacher’s Training College, University of Yaounde I, P.O. Box 47, Yaounde, Cameroon c The Abdus Salam ICTP, Strada Costiera 11, Trieste I-34151, Italy a r t i c l e i n f o Article history: Received 18 July 2019 Revised 29 August 2019 Accepted 10 October 2019 Available online xxx Keywords: Traveling waves Fractional NETL Fractional complex transform Discrete tanh method Modified Riemann–Liouville derivatives Fractional partial differential-difference equation, a b s t r a c t In this paper we investigate the solutions of the fractional partial differential-difference equations gov- erning the dynamics of the voltage wave flowing inside two fractional (low pass and pass band) nonlinear electrical transmission lines (NETL). Through the discrete Tanh method, we derive the traveling wave so- lutions i.e (kink, dark, singular kink solitons) of two fractional partial differential-difference equations by using the fractional complex transform. The impact of the fractional order on the dynamical behavior of the analytical solutions in both models is highlighted. From our findings the perfect matching between the analytical and the numerical solutions is justified by the stability seen as the resulting wave prop- agates safely inside the fractional low pass NETL including real inductor. We get for the fractional pass band NETL a rise of nonlinear periodic waves and a train of multi periodic solitons during the numerical simulations. We show that, according to the modified Riemann–Liouville derivative properties, the find- ings can describe physical systems such as electrical systems and nonlinear electrical transmission lines with transient effect. © 2019 Elsevier Ltd. All rights reserved. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Description of the fractional complex transform and the discrete Tanh method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. Description of the fractional complex transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2. The discrete Tanh method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Applications of the discrete Tanh method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.1. The fractional low pass nonlinear electrical transmission line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2. The fractional pass band nonlinear electrical transmission line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Declaration of Competing Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1. Introduction Since a decade, fractional calculus has regained interest of re- searchers in different domains as biology, chemistry, physics, and ∗ Corresponding author. E-mail addresses: nguenang@yahoo.com, nguenangjp12@gmail.com (J.P. Ngue- nang). engineering [1]. More recently, some researchers have focused their investigations on realistic modeling in biology [2], in elec- tromechanics [3] and also in physics [4,5]. From the foregoing, those realistic models could reveal new phenomena that were hided in such systems. Henceforth, nowadays the investigations of exact traveling waves solutions in nonlinear equations have re- gained increasingly interest, and even more in fractional nonlinear equations [6–8]. However, since several fractional nonlinear equa- https://doi.org/10.1016/j.chaos.2019.109486 0960-0779/© 2019 Elsevier Ltd. All rights reserved. Please cite this article as: E. Fendzi Donfack, J.P. Nguenang and L. Nana, On the traveling waves in nonlinear electrical transmission lines with intrinsic fractional-order using discrete tanh method, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109486