Search

Starting flows of a viscous incompressible fluid, modeled by the time-fractional derivatives, within a rotating channel due to an impulsive pressure gradient are studied. Using the eigenfunction expansion, the analytic solutions in series form are obtained. The flow of the ordinary fluid is studied as a special case of the time-fractional problem. The convergence of series solutions is proved. In addition, using the classical analytical method, coupled with the Laplace transform and Stehfest's algorithm, an approximate solution is found. The flow rates in x- and y-directions are determined. In the case of the ordinary fluid, the steady-state and transient components of velocities are obtained. The numerical calculations are carried out by using the Mathcad software. It is found that, for fractional fluids, the reversal flow is much attenuated if the values of the fractional parameter are less than 1. [ABSTRACT FROM AUTHOR]

Realizing the behavior of the solution in the asymptotic situations is essential for repetitive applications in the control theory and modeling of the real-world systems. This study discusses a robust and definitive attitude to find the interval approximate asymptotic solutions of fractional differential equations (FDEs) with the Atangana-Baleanu (A-B) derivative. In fact, such critical tasks require to observe precisely the behavior of the noninterval case at first. In this regard, we initially shed light on the noninterval cases and analyze the behavior of the approximate asymptotic solutions, and then, we introduce the A-B derivative for FDEs under interval arithmetic and develop a new and reliable approximation approach for fractional interval differential equations with the interval A-B derivative to get the interval approximate asymptotic solutions. We exploit Laplace transforms to get the asymptotic approximate solution based on the interval asymptotic A-B fractional derivatives under interval arithmetic. The techniques developed here provide essential tools for finding interval approximation asymptotic solutions under interval fractional derivatives with nonsingular Mittag-Leffler kernels. Two cases arising in the real-world systems are modeled under interval notion and given to interpret the behavior of the interval approximate asymptotic solutions under different conditions as well as to validate this new approach. This study highlights the importance of the asymptotic solutions for FDEs regardless of interval or noninterval parameters. 1 --> a --> [ABSTRACT FROM AUTHOR]

A drug is any substance capable of altering the functioning of a person's body and mind. In this paper, a deterministic nonlinear model was adapted to investigate the behavior of drug abuse and addiction that incorporates intervention in the form of awareness and rehabilitation. In the mathematical analysis part, the positivity and boundedness of the solution and the existence of drug equilibria have been ascertained, which shows that the model consists of two equilibria: a drug-free equilibrium and a drug endemic equilibrium point. The drug-free equilibrium was found to be both globally and locally asymptotically stable if the effective reproduction number is less than or equal to one (Rc≤1). Furthermore, we were able to show the existence of a unique drug endemic equilibrium whenever Rc>1. Global asymptotic stability of a drug endemic equilibrium point has been ascertained using a nonlinear Lyapunov function of Go-Volterra type, which reveals that the drug endemic equilibrium point is globally asymptotically stable if an effective reproduction number is greater than unity and if there is an absence of a reversion rate of mended individuals (i.e., ω=0). In addition, an optimal control problem was formulated to investigate the optimal strategy for curtailing the spread of the behavior using control variables. The control variables are massive awareness and rehabilitation intervention of both public and secret addicted individuals. The optimal control simulation shows that massive awareness control is the best to control drug addiction in a society. In sensitivity analysis section, the proportion of those who are exposed publicly shows to be a must sensitive parameter that can reduce the reproduction number, and the effective contact rate shows to be a must sensitive parameter to increase the reproduction number. Numerical simulations reveal that the awareness rate of exposed publicly and the rehabilitation rate of addicted publicly are very important parameters to control drug addiction in a society., (© 2024 Author(s). Published under an exclusive license by AIP Publishing.)