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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Some Characteristics of Time-Memory Embedded into a Time-Fractional Version of the Boussinesq System: Graphical Analysis

Journal of Applied Nonlinear Dynamics 9(1) (2020) 47--56 | DOI:10.5890/JAND.2020.03.005

Marwan Alquran, Adnan Jarrah, Motaz Alnaimat

Department of Mathematics and Statistics, Jordan University of Science and Technology, P.O.Box (3030), Irbid 22110, Jordan

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Abstract

The aim of this work is to study the effect of the fractional-time derivative acting on a fractional version of the Boussinesq system that reads 0 = Dαt u(x, t)+Hx(x, t)+u(x, t)ux(x, t), 0 = Dαt H(x, t)+(u(x, t)H(x, t))x+uxxx(x, t), subject to the initial conditions f (x) = u(x,0), g(x) = H(x,0). Dαt is the Caputo fractional operator with α ∈ (0,1] and f (x), g(x) ∈C∞[ℜ]. To achieve our goal, we solved analytically the proposed model using a new technique called modified residual power series method (RPS). The reliability of RPS technique has been verified using tabular and graphical analysis which reveal the fact when the time-memory index “time-fractional order” is close to zero “full memory”, the solution bifurcate and produce a wave-like pattern, whereas the pattern vanishes when the memory is close to 1 “no memory”.

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