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This paper investigates a new approach called Homotopy Analysis Decomposition Method (HADM) for solving nonlinear differential equations, the method was developed by incorporating Adomian polynomial into Homotopy Analysis Method. The Adomian polynomial was used to decompose the nonlinear term in the equation then apply the scheme of homotopy analysis method. The accuracy and efficiency of the proposed method was validated by considering algebraically decaying viscous boundary layer  flow due to a moving sheet. Diagonal Pade approximation was used to get the skin friction. The obtained results were presented along with other methods in the literature in tabular form to show the computational efficiency of the new approach. The results were found to agree with those in literature. Owing to its small size of computation, the method is not aected by discretization error as the results are presented in form of polynomials.

Crude oil is an essential source of energy. Without access to energy, output growth is impossible. As a result of this link, volatility in oil prices has the ability to induce fluctuations in the output of both developed and developing economies. Moreover, factors such as business cycles and policy changes often introduce nonlinearity into the transmission mechanism of oil price shocks. This study therefore examines not only the interconnectedness of oil price volatility and output growth, but also the nonlinear, asymmetric impact of oil price volatility on output growth in the countries making up the Group of Seven. To this end, monthly data on West Texas Intermediate oil price and industrial production indices of the Group of Seven countries over the period 1990:01 to 2019:08 is used for empirical analysis. The study employs the DCC and cDCC-GARCH techniques for symmetric empirical analysis. The asymmetric empirical analysis is also conducted via GJR-GARCH, FIEGARCH, HYGARCH and cDCC-GARCH techniques. The findings reveal disparities in the magnitudes of the positive and negative (asymmetric) effects of oil price shocks on output growth. The results also reveal that past news and lagged volatility have a significant impact on the current conditional volatility of the output growth of the Group of Seven countries. The study concludes that the impact of oil price volatility on output growth in the selected economies is asymmetric, the volatility is highly persistent and clustered, and the asymmetric GARCH models outperform the symmetric GARCH models.

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A model for the interaction of HIV with the T CD  4 cells was examined in which an optimal treatment parameter was introduced to control the infectivity termin the HIV dynamic model. The control class was chosen to be a measurable function defined with an objective functional which maximizes the T cell count and minimizes the systemic cost based on the percentage effect of the antiretroviral therapy drug. Optimal control was characterized by applying pontryagin's maximum principle. The values of the objective function at the optimal control shows that the greatest effect do occur when treatment is initiated earliest. Also, results of the numerical simulations indicate that the rate of uninfected T CD  4 increased and virus population decreased due to treatment parameter.