Journal of Applied Nonlinear Dynamics
Some Characteristics of TimeMemory Embedded into a TimeFractional Version of the Boussinesq System: Graphical Analysis
Journal of Applied Nonlinear Dynamics 9(1) (2020) 4756  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 fractionaltime 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 timememory index “timefractional order” is close to zero “full memory”, the solution bifurcate and produce a wavelike pattern, whereas the pattern vanishes when the memory is close to 1 “no memory”.
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