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In this paper the Homotopy Analysis Method (HAM), is applied to find the approximate solution of Hirota-Satsuma coupled (KdV) equations, which don't need a small parameter for solution. The results obtained by HAM is compared with exact solution, the results divulge that the Homotopy Analysis Method are most accurate, closed and suitable to exact solution of the equation, as compare to Homotopy Perturbation Method. It is predicated that the HAM can be found usually.

This paper presents the analysis of the process of blade coating through a computational intensive method for an incompressible Newtonian fluid along with Magnetohydrodynamics (MHD). The slip between the substrate and the fluid is also taken into account. The nature of the existing steady solutions has been investigated with the help of exact and numerical methods. Those obtained exact solutions include the solutions for the velocity profiles, volumetric flow rate and pressure gradient. The important engineering quantities like maximum pressure, pressure distribution and load are also computed. It is assumed that the relative velocity between the plate and the fluid is proportional to the shear rate at the plate. An external magnetic field is applied normal to the plates. It is observed and concluded that both slip parameter and the magnetic field parameter serve as the controlling parameters in the industrial blade coating process.

The present paper presents the comparison of analytical techniques. We establish the existence of the phenomena of the noise terms in the perturbation series solution and find the exact solution of the nonlinear problems. If the noise terms exist, the Homotopy Analysis method gives the same series solution as in Adomian Decomposition Method as well as homotopy Perturbation Method (Wahab et al, 2015) and we get the exact solution using the initial guess in Homotopy Analysis Method using the results obtained by Adomian Decomposition Method.

This paper deals with the approximate analytical solution of the Navier-Stokes equations in cylindrical coordinates. The homotopy perturbation method is used to get the analytical approximation. Depending upon different available choices for the linear operator, we also have the advantage to choose different initial approximations to start our analysis. The analysis is done without calculating the Adomian's polynomials.