The selection criteria play an important role in the portfolio optimization using any ratio model. In this paper, the authors have considered the mean return as profit and variance of return as risk on the asset return as selection criteria, as the first stage to optimize the selected portfolio. Furthermore, the sharp ratio (SR) has been considered to be the optimization ratio model. In this regard, the historical data taken from Shanghai Stock Exchange (SSE) has been considered. A metaheuristic technique has been developed, with financial tool box available in MATLAB and the particle swarm optimization (PSO) algorithm. Hence, called as the hybrid particle swarm optimization (HPSO) or can also be called as financial tool box particle swarm optimization (FTB-PSO). In this model, the budgets as constraint, where as two different models i.e. with and without short sale, have been considered. The obtained results have been compared with the existing literature and the proposed technique is found to be optimum and better in terms of profit.

The pressure gradient term plays a vital role in convective heat transfer in the boundary layer flow of a Maxwell fluid over a stretching sheet. The importance of the effects of the term can be monitored by developing Maxwell’s equation of momentum and energy with the pressure gradient term. To achieve this goal, an approximation technique, i.e. Homotopy Perturbation Method (HPM) is employed with an application of algorithms of Adams Method (AM) and Gear Method (GM). With this approximation method we can study the effects of the pressure gradient (m), Deborah number (β), the ratio of the free stream velocity parameter to the stretching sheet parameter (ɛ) and Prandtl number (Pr) on both the momentum and thermal boundary layer thicknesses. The results have been compared in the absence and presence of the pressure gradient term m . It has an impact of thinning of the momentum and boundary layer thickness for non-zero values of the pressure gradient. The convergence of the system has been taken into account for the stretching sheet parameter ɛ. The result of the system indicates the significant thinning of the momentum and thermal boundary layer thickness in velocity and temperature profiles. On the other hand, some results show negative values of f '(η) and θ (η) which indicates the case of fluid cooling.

Boundary layer flow of convective heat transfer with pressure gradient over a flat plate is solved with an application of algorithms of Adams Method (AM) and Gear Method (GM) using Homotopy Perturbation Method (HPM). The distributions of temperature and velocity in the boundary layer are examined, particularly on the influences due to Prandtl number (Pr) and pressure gradient (m). Consequently, the equations of momentum and energy are resolved concurrently. These HPM outcomes have been compared with the previous published work in the literature; and these are found to be in good agreement with the results obtained from numerical methods.

In this article, Maxwell fluid over a flat plate for convective boundary layer flow with pressure gradient parameter is considered. The aim of this study is to compare and analyze the effects of the presence and absence of λ (relaxation time), and also the effects of m (pressure gradient parameter) and Pr (Prandtl number) on the momentum and thermal boundary layer thicknesses. An approximation technique namely Homotopy Perturbation Method (HPM) has been used with an implementation of Adam and Gear Method’s algorithms. The obtained results have been compared for zero relaxation time and also pressure gradient parameter with the published work of Fathizadeh and Rashidi. The current outcomes are found to be in good agreement with the published results. Physical interpretations have been given for the effects of the m, Pr and β (Deborah number) with λ. This study will play an important role in industrial and engineering applications.

Exponential growths are observed in several natural phenomena, including cosmological expansion of the universe, nuclear fission, population growth, global viral spread, and financial markets fluctuations. Several deterministic and stochastic methods have been used for the modeling of dynamics of these phenomena. Global reliable data are essential to parameterise a defined model so as to enhance confidence in model outputs. Four-parameter logistic model comes out to be best forecast model for the total Covid-19 infected people in Pakistan and India. Five-parameter logistic model is best forecast model for Bangladesh. Simulation results revel that maximum total number of cases will be about 1.0, 0.6 and 0.25 million in India, Pakistan and Bangladesh respectively. Covid-19 infected rate will be closed to 11000, 8000 and 2000 cases per day during peak period in these countries.