In this paper, we are putting forward the periodic solution of non-linear oscillators by means of variational iterative method (VIM) using Laplace transform. Here, we present a comparative study of the new technique based on Laplace transform and the previous techniques of maximum minimum approach (MMA) and amplitude frequency formulation (AFF) for the analytical results. For the non-linear oscillators, MMA, AFF and VIM by Laplace transform give the same analytical results. Comparison of analytical results of VIM by Laplace transform with numerical results by fourth-order Runge–Kutta (RK) method conforms the soundness of the method for solving non-linear oscillators as well as for the time and boundary conditions of the non-linear oscillators.

In this work, the combined effects of velocity slip and convective heat boundary conditions on a hybrid nano-fluid over a nonlinear curved stretching surface were considered. Two kinds of fluids, namely, hybrid nano-fluid and aluminum oxide (Al2O3)- and iron oxide (Fe3O4)-based nano-fluid, were also taken into account. We transformed the governing model into a nonlinear system of ordinary differential equations (ODEs). For this we used the similarity transformation method. The solution of the transformed ODE system was computed via a higher-order numerical approximation scheme known as the shooting method with the Runge–Kutta method of order four (RK-4). It is noticed that the fluid velocity was reduced for the magnetic parameter, curvature parameter, and slip parameters, while the temperature declined with higher values of the magnetic parameter, Prandtl number, and convective heat transfer. Furthermore, the physical quantities of engineering interest, i.e., the behavior of the skin fraction and the Nusselt number, are presented. These behaviors are also illustrated graphically along with the numerical values in a comparison with previous work in numerical tabular form.

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<abstract><p>In this paper we consider fractional-order mathematical model describing the spread of the smoking model in the sense of Caputo operator with tobacco in the form of snuffing. The threshold quantity $ \mathcal{R}_0 $ and equilibria of the model are determined. We prove the existence of the solution via fixed-point theory and further examine the uniqueness of of the solution of the considered model. The new version of numerical approximation's framework for the approximation of Caputo operator is used. Finally, the numerical results are presented to justify the significance of the arbitrary fractional order derivative. The analysis shows fractional-order model of tobacco smoking in Caputo sense gives useful information as compared to the classical integer order tobacco smoking model.</p></abstract>