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Hydromagnetic flow between two horizontal plates in a rotating system, where the lower plate is a stretching sheet and the upper is a porous solid plate, is analyzed. Heat transfer in an electrically conducting fluid bonded by two parallel plates is studied in the presence of viscous dissipation. The equations of conservation of mass and momentum and energy are reduced to a nonlinear ordinary differential equations system. Homotopy perturbation method is used to get complete analytic solution for velocity and temperature profiles. Results show an acceptable agreement between this method results and numerical solution. Also the effects of different parameters are discussed through graphs.

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Abstract-This paper deals with Approximate Analytical Solutions to nonlinear oscillations of a conservative, non-natural, single-degreeof-freedom system with odd nonlinearity. By extending the Variational approach proposed by He, we established approximate analytical formulas for the period and periodic solution.To illustrate the applicability and accuracy of the method, two examples are presented: (i) the motion of a rigid rod rocking back and forth on the circular surface without slipping, and (ii) Cubic-Quintic Duffing Oscillators. Comparison of the result which is obtained by this method with the obtained result by the Exact solution reveals that the He's Variational approach is very effective and convenient and can be easily extended to other nonlinear systems and can therefore be found widely applicable in engineering and other sciences.

This paper presents an approach for solving accurate approximate analytical solutions for strong nonlinear oscillators called improved amplitude-frequency formulation. For illustrating the accuracy of the method, we also solved equations with He's energy balance method and compared results. New algorithms offer promising approaches, which are useful for nonlinear oscillations. We find that these attained solutions not only benefit from a high degree of accuracy, but are also uniformly valid in the whole solution domain which is so simple to do and effective. The studied equations are the general motion equation and the non-dimensional nonlinear differential equation of motion for the relativistic oscillator, which their solution can be useful for researchers to extend this ability into their other works.