The current study explores the Impact of activation energy and melting heat transfer on unsteady Prandtl–Eyring model with variable thermal conductivity induced by a stretched cylinder. The Cattaneo–Christov double diffusion theory is used to study heat and mass transfer phenomena. The foremost coupled partial differential equations (PDEs) of the time-dependent Prandtl–Eyring model is transformed to ordinary ones using appropriate local similarity variables. The deduced system is numerically solved using the shooting iterative technique. The characteristics of key flow parameters against fluid concentration, temperature, velocity, skin friction, Sherwood and local Nusselt numbers are examined graphically with justified physical consequences. The comparison with the previously published work showed a fabulous agreement. This investigation presented that the velocity and concentration of fluid are increased by enhancing melting and curvature parameters, while the fluid temperature is diminished due to the impacts of melting and curvature parameters. By raising reaction rate constant, melting, thermal relaxation time, and temperature difference parameters, the fluid temperature is enhanced while the fluid temperature declines with thermal conductivity, activation energy, and unsteadiness parameters. Furthermore, the unsteadiness and activation energy parameters cause a rise in the concentration of a fluid.

Vaccination against any infectious disease is considered to be an effective way of controlling it. This paper studies a fractional order model with vaccine efficacy and waning immunity. We present the model’s dynamics under vaccine efficacy, the impact of immunization, and the waning of the vaccine on coronavirus infection disease. We analyze the model under their equilibrium points. The model under the equilibrium points is discussed and proven that it is locally asymptotically stable if ${\mathcal{R}}_v$ is lesser than unity. We present the backward bifurcation hypothesis of the model and show that there is a parameter ${\beta }_2$ that causes the backward bifurcation in the imperfect vaccine model. We show certain assumptions when $\psi =1$ for the imperfect vaccine case; the model is both stable globally asymptotically at the disease-free (${\mathcal{R}}_0\le 1$) and endemic cases (${\mathcal{R}}_0>1$). By using infected cases from the recent wave throughout Pakistan, we shall estimate the model parameters and calculate the numerical value of the basic reproductive number ${\mathcal{R}}_0\approx 1.2591$. We present the comprehensive graphical results for the realistic parameter values and show many useful suggestions regarding the elimination of the infection from society. The vaccination efficacy that provides an important role in disease elimination is discussed in detail.

The stability of the power system is a critical issue for the reliable and safe operation of the network. Where maintaining voltage levels constant or within the prescribed permissible limit and robustness against disturbances, while the power system is working near its stability margin due to growth of power consumption, nowadays are great challenges. Chaotic oscillation in power network may lead to system bus voltage collapse, angle divergence and possibly both phenomena simultaneously. These cases directly affect the service quality of the power system. The paper presents the problem of chaos suppressing in a three-bus power system of a six-dimensional model. The dynamics of the power system are investigated through examining the nonlinear system’s behavior analysis tools, such as power spectral density, bicoherence, Poincaré map and the Lyapunov exponents. The chaotic oscillation of the power system is suppressed through a Lyapunov-based adaptive algorithm with synergetic control theory. A nonlinear evolution constraint is used for achieving better transient responses and fast dynamics. The dynamics of the energy storage device and STATCOM compensator are employed within the control loop to restore the synchronous operation and maintain the rated voltage, respectively. Numerical simulations are conducted to verify the effectiveness and robustness of the proposed control algorithm. The stabilization of the chaotic power system dynamics and the fast recovery to the normal state are characterized by a smooth and free-of-chattering controller output.

In this article, we consider an extensive class of monotone nonexpansive mappings and introduce a new iteration algorithm to approximate the fixed point for monotone total asymptotically nonexpansive mappings in the framework of hyperbolic space. Faster convergence and stability results are proved for that iteration; also, fixed point is approximated numerically in a nontrivial example by using MATLAB.