The Investigation of Dynamical Behavior of Benjamin–Bona–Mahony–Burger Equation with Different Differential Operators Using Two Analytical Approaches

Abstract

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the proposed model. We demonstrate some of the extracted solutions using definitions of the β-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions.

A dynamical behavior of the coupled Broer-Kaup-Kupershmidt equation using two efficient analytical techniques

“…Definition 2 For the function p : [0, ∞) → R of order ς ∈ (0, 1), the M-TD [ 20 ] is defined, as for h > 0. Where is defined as a truncated Mittag-Leffler function with a single parameter as follows : …”

“…The rational, hyperbolic, trigonometric, singular, periodic, and singular solutions also properly expressed the solitary patterns of the BKK equation [ 15 ]. The coupled time-fractional BKK equation [ 16 ] was converted using numerous wave transformations for four distinct operators into an ordinary differential equation, namely the beta derivative (BD) [ 17 ], M-truncated derivatives (M-TD) [ 18 ], Local fractional derivative (L-FD) [ 19 ], and conformable derivative (CD) [ 20 ], each of which produces a non-linear algebraic equation system when the technique is used. This was done in order to look at how fractional parameters affected the equation’s soliton waves in a dynamic reaction.…”