The aim of this research work is to investigate the innovative concept of magnetohydrodynamic (MHD) three-dimensional rotational flow of nanoparticles (single-walled carbon nanotubes and multi-walled carbon nanotubes). This flow occurs in the presence of non-linear thermal radiation along with heat generation or absorption based on the Casson fluid model over a stretching sheet. Three common types of liquids (water, engine oil, and kerosene oil) are proposed as a base liquid for these carbon nanotubes (CNTs). The formulation of the problem is based upon the basic equation of the Casson fluid model to describe the non-Newtonian behavior. By implementing the suitable non-dimensional conditions, the model system of equations is altered to provide an appropriate non-dimensional nature. The extremely productive Homotopy Asymptotic Method (HAM) is developed to solve the model equations for velocity and temperature distributions, and a graphical presentation is provided. The influences of conspicuous physical variables on the velocity and temperature distributions are described and discussed using graphs. Moreover, skin fraction coefficient and heat transfer rate (Nusselt number) are tabulated for several values of relevant variables. For ease of comprehension, physical representations of embedded parameters such as radiation parameter $\left(Rd\right)$ , magnetic parameter $\left(M\right)$ , rotation parameter ( $K$ ), Prandtl number ( $Pr$ ), Biot number $\left(\lambda \right)$ , and heat generation or absorption parameter $\left(Q_h\right)$ are plotted and deliberated graphically.

We prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, s-convex, and coordinate convex functions. We prove new Montgomery identity and by using this identity we obtain generalized Hermite-Hadamard type inequalities.

In the work, author’s presents a very significant and important issues related to the health of mankind’s. Which is extremely important to realize the complex dynamic of inflected disease. With the help of Caputo fractional derivative, We capture the epidemiological system for the transmission of Novel Coronavirus-19 Infectious Disease (nCOVID-19). We constructed the model in four compartments susceptible, exposed, infected and recovered. We obtained the conditions for existence and Ulam’s type stability for proposed system by using the tools of non-linear analysis. The author’s thoroughly discussed the local and global asymptotical stabilities of underling model upon the disease free, endemic equilibrium and reproductive number. We used the techniques of Laplace Adomian decomposition method for the approximate solution of consider system. Furthermore, author’s interpret the dynamics of proposed system graphically via Mathematica, from which we observed that disease can be either controlled to a large extent or eliminate, if transmission rate is reduced and increase the rate of treatment.

In this paper, we obtain the results of controllability for first order impulsive non–linear time varying delay dynamic systems and Hammerstein impulsive system on time scale, using the theory of fixed points such as Banach fixed point theorem combined with Lipchitz conditions and non linear functional analysis. We also provide examples to support our theoretical results.