5. wth order deformation problem

By taking w times derivatives of Eqs. (28)–(32) and then dividing by w! as well as substituting p ¼ 0, we obtained the following equations:

$$L\_f \left[ f\_w(\eta) - \chi\_w f\_{w-1}(\eta) \right] = \hbar\_f \mathfrak{R}\_w^f(\eta), \tag{43}$$

$$L\_{\theta}[\Theta\_{w}(\eta) - \chi\_{w}\Theta\_{w-1}(\eta)] = \hbar\_{\theta}\,\mathfrak{R}\_{w}^{\ell}(\eta),\tag{44}$$

$$L\_{\phi} \left[ \phi\_w(\eta) - \chi\_w \phi\_{w-1}(\eta) \right] = \hbar\_{\phi} \, \Re\_w^f(\eta) \,\tag{45}$$

where

$$\chi\_{w} = \begin{cases} 0, & \text{if} \quad p \le 1 \\ 1, & \text{if} \quad p > 1 \end{cases}.$$

$$\mathfrak{R}\_{w}^{f}(\eta) = \left( 1 + \frac{1}{\beta\_{1}} \right) \left( \eta \check{f}\_{w-1} + f\_{w-1}'' \right) + \text{Re} \sum\_{j=0}^{w-1} \left( f\_{w-1-j} f\_{j}'' - f\_{w-1-j}' f\_{j}' \right) + \left( \text{Gr}\_{l} \theta\_{w-1} + \text{Gr}\_{l} \phi\_{w-1} \right), \tag{46}$$

$$\mathfrak{R}\_{w}^{\theta}(\eta) = \eta \theta\_{w-1}^{\eta} + \theta\_{w-1}^{\prime} + \text{Pr.Re} \sum\_{j=0}^{w-1} \left( f\_{w-1-j} \theta\_{j}^{\prime} - 2f\_{w-1-j}^{\prime} \theta\_{j} \right) + \eta \theta\_{w-1}^{\prime} \sum\_{j=0}^{w-1} \left( \mathcal{N}\_{l} \theta\_{w-1}^{\prime} + \mathcal{N}\_{l} \phi\_{w-1}^{\prime} \right), \tag{47}$$

$$\mathfrak{R}\_{w}^{\phi}(\eta) = \eta \phi\_{w-1}^{\eta} + \phi\_{w-1}^{\prime} + L e. \text{Re} \sum\_{j=0}^{w-1} \left( f\_{w-1-j} \phi\_j^{\prime} - 2f\_{w-1-j}^{\prime} \phi\_j \right) + \frac{N\_t}{N\_b} \left( \eta \theta\_{w-1}^{\eta} + \theta\_{w-1}^{\prime} \right). \tag{48}$$

The related boundary conditions are

$$\begin{aligned} f\_w(1) = f\_w'(1) = \theta\_w(1) = \phi\_w(1) = 1, \\ f\_w''(\beta) = \theta\_w'(\beta) = \phi\_w'(\beta) = 0. \end{aligned} \tag{49}$$

The general solution of Eqs. (42)–(44) is given by

$$\begin{aligned} f\_w(\eta) &= \varepsilon\_1 + \varepsilon\_2 \eta + \varepsilon\_3 \eta^2 + f\_w^\*(\eta), \\ \Theta\_w(\eta) &= \varepsilon\_4 + \varepsilon\_5 \eta + \Theta\_w^\*(\eta), \\ \phi\_w(\eta) &= \varepsilon\_6 + \varepsilon\_7 \eta + \phi\_w^\*(\eta). \end{aligned} \tag{50}$$

Here <sup>∗</sup> f <sup>∗</sup> ð Þ <sup>ξ</sup> , g ð Þ <sup>ξ</sup> and <sup>θ</sup><sup>∗</sup> ð Þ <sup>ξ</sup> represent the particular solutions, and the constant Aið<sup>i</sup> <sup>¼</sup> <sup>1</sup> � <sup>8</sup><sup>Þ</sup> w w <sup>w</sup> are determined from boundary conditions (49).
