Some Characteristics of Time-Memory Embedded into a Time-Fractional Version of the Boussinesq System: Graphical Analysis - Alquran, Marwan; Jarrah, Adnan; Alnaimat, Motaz (Journal of Applied Nonlinear Dynamics, 2026)
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”.