Optimal Homotopy Analysis Method

Optimal Homotopy Analysis Method

DOI: 10.4018/978-1-5225-2713-8.ch008
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Optimal homotopy analysis method is a powerful tool for nonlinear differential equations. In this method, the convergence of the series solutions is controlled by one or more parameters which can be determined by minimizing a certain function. There are several approaches to determine the optimal values of these parameters, which can be divided into two categories, i.e., global optimization approach and step-by-step optimization approach. In the global optimization approach, all the parameters are optimized simultaneously at the last order of approximation. However, this process leads to a system of coupled, nonlinear algebraic equations in multiple variables which are very difficult to solve. In the step-by-step approach, the optimal values of these parameters are determined sequentially, that is, they are determined one by one at different orders of approximation. In this way, the computational efficiency is significantly improved, especially when high order of approximation is needed. In this chapter, we provide extensive examples in heat transfer equations.
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Deburge and Han (1972) studied a problem concerning heat transfer in channel flow. Natural convection of a non-Newtonian copper-water nanofluid was investigated by Domairry et al. (2012). They conclude that as the nanoparticle volume fraction increases, the momentum boundary layer thickness increases, whereas the thermal boundary layer thickness decreases. Sheikholeslami et al. (2013) studied the problem of natural convection between a circular enclosure and a sinusoidal cylinder. They con- cluded that stream lines, isotherms, and the number, size and formation of the cells inside the enclosure strongly depend on the Rayleigh number, values of amplitude and the number of undulations of the enclosure. Sheikholeslami et al. (2013) performed a numerical study to investigate natural convection in a square cavity with curve boundaries filled with Cu-water nanofluid. Their results proved that the change of inclination angle has a significant impact on the thermal and hydrodynamic flow fields. Recently several authors investigated about natural convection heat transfer (Sheikholeslami, Gorji-Bandpy & Soleiman, 2013; Sheikholeslami, Gorji-Bandpy & Ganji, 2013a, 2013b, 2012; Sheikholeslami et al., 2014; Sheikholeslami, Hatami & Ganji, 2013, 2014; Sheikholeslami et al., 2013a, 2013b, 2014; Sheikholeslami, Gorji-Bandpy & Domairry, 2013; Sheikholeslami et al., 2012; Soleimani et al., 2012; Ganji et al., 2013; Bhattrai & Tang, 2013; Elhefny & Liang, 2013; Reyhani et al., 2013; Sheikholeslami et al., 2014).

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