Drinking kills a significant proportion of individuals every year, particularly in low‐income communities. An impulsive differential equation system is used to explore the effectiveness of activated charcoal in detoxifying the body after methanol poisoning. Our impression of activated charcoal is shaped by the fractional dynamics of the problem, which leads to speedy and low‐cost first aid. The adsorption capacity of activated charcoal is investigated using impulsive differential equations. The ABC fractional operator’s findings paint a more realistic image of first aid in public and primary health centers, which can assist to reduce methanol poisoning deaths. Numerical simulations are provided using generalized Adams–Bashforth–Moulton method (GABMM).

This study presents a modified simplest equation method (MSEM) to investigate some real and exact solutions of conformable time fractional Benjamin‐Bona‐Mahony (BBM) equation and Chan‐Hilliard (CH) equation. We use traveling wave transformation to obtain the results in the form of series solution. Some calculations are performed through Mathematica software to analyze the accuracy of this approach. Graphical representations are reported for more significant results at different fractional‐order which demonstrates that this approach is very simple, adequate, and legitimate.

The instant system availability $$S_\tau (t)$$ S τ ( t ) of a repairable system with the renewal equation was studied. The starting point monotonicity of $$S_\tau (t)$$ S τ ( t ) was proved and the upper bound of $$S_\tau (t)$$ S τ ( t ) is also derived. It was found that the interval of instant system availability monotonically decreases. In addition, we provide examples that validate the analytically derived properties of $$S_\tau (t)$$ S τ ( t ) based on the Lognormal, Gamma and Weibull distributions and the results show that the value of T is slightly smaller than its value defined in Theorem 2. The procedure of using a bathtub as application for this article is also discussed.

CO₂ emissions are one of the most critical and challenging problems impacting the atmosphere. In order to reduce CO₂ emissions, significant efforts must be put forward in advanced research. In this research, we propose a fractional optimal control approach to effectively deal with the accessible resources and minimize the restriction of variables. The mathematical model is formulated in terms of the Caputo fractional derivative. The formulated problem is numerically solved using a forward-backward sweep approach with the generalized Euler method. We present a numerical problem to obtain the optimal solution and show that the suggested process is very efficient in science and engineering problems.