*3.2. mth Order Deformation Problems*

Considering Equations (48) and (52) for homotopy at *m*th order as

$$\mathcal{L}\_f[f\_m(\zeta) - \chi\_m f\_{m-1}(\zeta)] = \hbar\_f \mathcal{R}\_m^f(\zeta),\tag{68}$$

$$f\_m(0) = 0, \quad f\_m(1) = 0, \quad f\_m'(0) = 0, \quad f\_m'(1) = 0,\tag{69}$$

$$\mathcal{R}\_{m}^{f}(\zeta) = \mathcal{B}\_{1}f\_{m-1}^{\prime\prime\prime} + \text{Re}\left[\sum\_{k=o}^{m-1} f\_{m-1-k} f\_{k}^{\prime\prime\prime} + 2g\_{m-1-k} g\_{k}^{\prime} - \text{MB}\_{2} f\_{m-1}^{\prime\prime}\right] - k\_{2}\text{Re}\mathcal{B}\_{1}f\_{m-1}^{\prime\prime} - \text{k}\mathcal{B}\_{1}f\_{m}^{\prime\prime}.\tag{70}$$

$$2k\_{3}\text{Re}\frac{1}{\rho\_{\text{lnf}}}\sum\_{k=o}^{m-1} f\_{m-1-k}^{\prime\prime} f\_{k}^{\prime\prime\prime}.\tag{70}$$

Considering Equations (49) and (53) for homotopy at *m*th order as

$$\mathbf{L}\_{\mathcal{S}}[\mathbf{g}\_{m}(\boldsymbol{\zeta}) - \chi\_{m}\mathbf{g}\_{m-1}(\boldsymbol{\zeta})] = \boldsymbol{\hbar}\_{\mathcal{S}}\mathbf{R}\_{m}^{\mathcal{S}}(\boldsymbol{\zeta}),\tag{71}$$

$$\mathcal{g}\_{\mathfrak{m}}(0) = 0, \quad \mathcal{g}\_{\mathfrak{m}}(1) = 0,\tag{72}$$

$$R\_{m}^{\mathcal{S}}(\zeta) = B\_{1}g\_{m-1}^{\prime\prime} + \text{Re}\left[\sum\_{k=o}^{m-1} \Im f\_{m-1-k} g\_{k}^{\prime} - M \mathcal{B}\_{2} g\_{m-1}^{\prime}\right] - k\_{2} \mathcal{B}\_{1} g\_{m-1} - k\_{3} \frac{1}{\rho\_{\ln f}} \sum\_{k=o}^{m-1} g\_{m-1-k} g\_{k}.\tag{73}$$

Considering Equations (50) and (54) for homotopy at *m*th order as

$$\mathbf{L}\_{\theta}[\theta\_{m}(\zeta) - \chi\_{m}\theta\_{m-1}(\zeta)] = \hbar\_{\theta}\mathcal{R}\_{m}^{\theta}(\zeta),\tag{74}$$

$$
\theta\_m(0) = 0, \quad \theta\_m(1) = 0,\tag{75}
$$

$$R\_m^{\theta}(\zeta) = B\_3 \frac{k\_{mf}}{k\_f} \theta\_{m-1}^{\prime\prime} + \frac{1}{Rd} Pr \text{Re} \left[ 2 \sum\_{k=o}^{m-1} f\_{m-1-k} \theta\_k^{\prime} + MB\_4 \text{Ec} \left[ \sum\_{k=o}^{m-1} f\_{m-1-k}^{\prime} f\_k^{\prime} + \sum\_{k=o}^{m-1} g\_{m-1-k} \xi\_k \right] \right]. \tag{76}$$

Considering Equations (51) and (55) for homotopy at *m*th order as

$$\mathbf{L}\_{\varphi}[\varphi\_{m}(\zeta) - \chi\_{m}\varphi\_{m-1}(\zeta)] = \hbar\_{\varphi}\mathcal{R}\_{m}^{\varphi}(\zeta),\tag{77}$$

$$
\varphi\_m'(0) = 0, \qquad \varphi\_m(1) = 0,\tag{78}
$$

$$R\_m^{\mathcal{P}}(\zeta) = \boldsymbol{\varrho}\_{m-1}^{\prime\prime} + \mathrm{ReSc} \left[ 2 \sum\_{k=0}^{m-1} f\_{m-1-k} \boldsymbol{\varrho}\_k^{\prime} + k\_4 \left[ \boldsymbol{\varrho}\_{m-1} + \boldsymbol{\varrho}\_{m-1-k} \sum\_{l=0}^{k} \boldsymbol{\varrho}\_{k-l} \boldsymbol{\varrho}\_{l-l} \boldsymbol{\varrho}\_l - 2 \sum\_{k=0}^{m-1} \boldsymbol{\varrho}\_{m-1-k} \boldsymbol{\varrho}\_k \right] \right],\tag{79}$$

$$\chi\_m = \begin{cases} \ 0, & m \le 1 \\ \ 1, & m > 1. \end{cases} \tag{80}$$

Adding the particular solutions *f* ∗ *<sup>m</sup>*(*ζ*), *g* ∗ *<sup>m</sup>*(*ζ*), *θ* ∗ *<sup>m</sup>*(*ζ*) and *ϕ* ∗ *<sup>m</sup>*(*ζ*), Equations (68), (71), (74) and (77) yield the general solutions as

$$f\_m(\zeta) = f\_m^\*(\zeta) + E\_1 + E\_2\zeta + E\_3\zeta^2 + E\_4\zeta^3,\tag{81}$$

$$\mathbf{g}\_{\mathfrak{m}}(\zeta) = \mathbf{g}\_{\mathfrak{m}}^\*(\zeta) + \mathbf{E}\_5 + \mathbf{E}\_6 \zeta\_\prime \tag{82}$$

$$
\theta\_m(\mathcal{J}) = \theta\_m^\*(\mathcal{J}) + E\_7 + E\_8 \mathcal{J}\_\prime \tag{83}
$$

$$
\varphi\_m(\zeta) = \varphi\_m^\*(\zeta) + E\_9 + E\_{10}\zeta. \tag{84}
$$
