(untitled) (Journal of Mathematical Modeling and Fractional Calculus, 2025)
In this paper, a fractal fractional derivative is used to examine an environmentally friendly approach of understanding the workings of smoking in humans. We proposed a fractional differential equation system to represent a time-fractional order smoking model with illness effects. Studies are conducted using methodologies. Lipschitz circumstances and linear growth are utilized to demonstrate the presence and distinction of the suggested model in relation to the impact of the global offset. It is confirmed that the fractional order model’s solutions are bounded and positive. When conducting the initial and subsequent derivative assessments, the Lyapunov function is employed to verify analysis of global stability. In order to examine the influence of smoking on humans, the fractional operator is studied. To do this, solutions are constructed applying the extended version of the Mittag-Leffler kernel using a two-step Lagrange polynomial method. Numerical simulation is performed to see how the fractional order smoking model behaves. Such research will aid in understanding the behavior of smoking and in developing human defenses.