The investigation of this research article is the development of studying the dynamical behavior of the drinking population through the fractional drinking model in the sense of Caputo-Fabrizio (CF) arbitrary order operator along with the special non-singular kernel. The proposed system is analyzed for existence result and uniqueness of solution by applying fixed point theory and Picard’s technique. Also on utilizing Adams-Bashforth method (ABM) of numerical analysis to interpret the approximate results through plots to observe dynamical behavior corresponding to different fractional order. For the mentioned simulation some real initial and parameter data are used.

This research focuses on the fascinating exploration of the $ (2+1) $-dimensional complex modified Korteweg-de Vries (CmKdV) system, exhibiting its complex dynamics and solitary wave solutions. This system is a versatile mathematical model that finds applications in various branches of physics, including fluid dynamics, plasma physics, optics, and nonlinear dynamics. Two newly developed methodologies, namely the auxiliary equation (AE) method and the Hirota bilinear (HB) method, are implemented for the construction of novel solitons in various formats. Numerous novel soliton solutions are synthesised in distinct formats, such as dark, bright, singular, periodic, combo, $ W $-shape, mixed trigonometric, exponential, hyperbolic, and rational, based on the proposed methods. Furthermore, we also find some lump solutions, including the periodic cross rational wave, the homoclinic breather (HB) wave solution, the periodic wave solution, the $ M $-shaped rational wave solution, the $ M $-shaped interaction with one kink wave, and the multiwave solution, which are not documented in the literature. In addition, we employ the Galilean transformation to derive the dynamic framework for the presented equation. Our inquiry includes a wide range of topics, including bifurcations, chaotic flows, and other intriguing dynamic properties. Also, for the physical demonstration of the acquired solutions, 3D, 2D, and contour plots are provided. The resulting structure of the acquired results can enrich the nonlinear dynamical behaviors of the given system and may be useful in many domains, such as mathematical physics and fluid dynamics, as well as demonstrate that the approaches used are effective and worthy of validation.

In this manuscript, we study the fractional compartmental model of drinking behavior with Atangana-Baleanu-Caputo (ABC) derivative. The fractional order model is divided into three different groups regarding their consumption in alcohol susceptible (nonconsumer), moderated drinkers and risk drinkers. We show the existence results and uniqueness of the solution by using the fixed point theory under the ABC derivative with fractional order. By using the modified Adams-Bashforth method we obtain the numerical solution of the fractional-order model. We use different values to obtain the numerical results of the fractional-order parameter to show its importance and provide important information about the drinking model.

Smoking is a social trend that is prevalent around the world, particularly in places of learning and at some significant events. The World Health Organization defines smoking as the most important preventable cause of disease and the third major cause of death in humans. In order to analyze this matter, this study typically emphasizes analyzing the dynamics of the fractional order quitting smoking model via the Caputo-Fabrizio differential operator. For the numerical solution of the considered model, the Laplace transform with the Adomian decomposition method (LADM) and Homotopy perturbation method (HPM) is applied, and the comparison of both the achieved numerical solutions is presented. Moreover, numerical simulation for the suggested scheme has been presented in various fractional orders with the aid of Matlab and the numerical results are supported by illustrative graphics. The simulation reveals the aptness of the considered model.