*2.4. Solution by HAM*

For solution process optimal approach is used. Equations (8)–(10) with boundary conditions (11 and 12) are solved by HAM. Basic derivations of the model equations through HAM are specified in details below.

$$L\_{\widehat{f}}(\widehat{f}) = \widehat{f}^{\prime\prime\prime}, \mathcal{L}\_{\widehat{0}}(\widehat{\theta}) = \widehat{\theta}^{\prime\prime}, \mathcal{L}\_{\widehat{\phi}}(\widehat{\phi}) = \widehat{\phi}^{\prime\prime}, \tag{23}$$

Linear operators *L f* , L <sup>θ</sup> *and* <sup>L</sup> <sup>φ</sup> are signified as

$$L\_{\widehat{f}}(\mathfrak{e}\_1 + \mathfrak{e}\_2\eta + \mathfrak{e}\_3\eta^2) = 0,\\ \mathrm{L}\_{\widehat{\widehat{\phi}}}(\mathfrak{e}\_4 + \mathfrak{e}\_5\eta) = 0,\\ \mathrm{L}\_{\widehat{\phi}}(\mathfrak{e}\_6 + \mathfrak{e}\_7\eta) = 0 \,,\tag{24}$$

The consistent non-linear operators are reasonably selected as N *f* , N <sup>θ</sup> *and* <sup>N</sup> <sup>φ</sup> and identify as:

$$\begin{aligned} \mathop{\rm N}\_{\widehat{f}} \begin{bmatrix} \widehat{f} \left( \eta; \zeta \right), \widehat{\partial} \left( \eta; \zeta \right) \\ \widehat{f} \left( \eta; \zeta \right), \widehat{\partial} \left( \eta; \zeta \right) \end{bmatrix} &= A\_1 \Big( \left( 1 + 2\eta\eta \right) \widehat{f}\_{\eta\eta\eta} + 2\gamma \widehat{f}\_{\eta\eta} \Big) + A\_2 \Big( \widehat{\partial} \left\{ \widehat{f} \right\}\_{\eta\eta} - \widehat{\widehat{f}}\_{\eta\eta} \left( 1 - F\mathbf{x} \right) \Big) \\ - \widehat{f}\_{\eta} \left( M + k\_1 A\_2 \right) + A\_3 \lambda \widehat{\partial} \ \boldsymbol{\zeta} \end{aligned} \tag{25}$$

$$\mathrm{N}\_{\widehat{\boldsymbol{\theta}}} \Big[ \widehat{f} \,(\boldsymbol{\eta}; \boldsymbol{\zeta}), \widehat{\boldsymbol{\theta}} \,(\boldsymbol{\eta}; \boldsymbol{\zeta}), \widehat{\boldsymbol{\phi}} \,(\boldsymbol{\eta}; \boldsymbol{\zeta}) \Big] = (1 + 2\eta \boldsymbol{\eta}) \widehat{\boldsymbol{\theta}}\_{\eta \eta} + \boldsymbol{\eta} \, \widehat{\boldsymbol{\theta}}\_{\eta} + \mathrm{Pr} (1 + 2\eta \boldsymbol{\eta}) (\mathrm{N}\_{b} \widehat{\boldsymbol{\theta}}\_{\eta} \widehat{\boldsymbol{\phi}}\_{\eta} + \mathrm{N}\_{l} \widehat{\boldsymbol{\theta}}\_{\eta}) + \mathrm{Pr} \widehat{f} \, \widehat{\boldsymbol{\theta}}\_{\eta} \tag{26}$$

$$\mathrm{N}\_{\widehat{\phi}}[\widehat{\phi}(\eta;\zeta),\widehat{f}(\eta;\zeta),\widehat{\theta}(\eta;\zeta)]=(1+2\eta\gamma)\widehat{\phi}\_{\eta\eta}+\gamma\widehat{\phi}\_{\eta}+\frac{N\_{\mathrm{I}}}{N\_{\mathrm{b}}}\{(1+2\eta\gamma)\widehat{\theta}\_{\eta\eta}+\gamma\widehat{\theta}\_{\eta}\}+\mathrm{Sc}\widehat{f}\widehat{\phi}\_{\eta}.\tag{27}$$

For Equations (8)–(10) the 0th-order system is shown as

$$\mathbb{P}(1-\mathbb{Q})L\_{\widehat{f}}\left[\widehat{f}\left(\eta;\mathbb{Q}\right)-\widehat{f}\_{0}\left(\eta\right)\right] = p\hbar\_{\widehat{f}}\mathop{\mathrm{N}}\_{\widehat{f}}\left[\widehat{f}\left(\eta;\mathbb{Q}\right),\widehat{\theta}\left(\eta;\mathbb{Q}\right)\right] \tag{28}$$

$$\left[ (1 - \zeta) \, \mathrm{L}\_{\widehat{\,}} \widehat{\left[ \theta \left( \eta; \zeta \right) - \widehat{\theta}\_{0} (\eta) \right]} \right] = p \hbar \widehat{\,}\_{\widehat{\,}} \mathrm{N}\_{\widehat{\,}} \left[ \widehat{\,} \widehat{f} \left( \eta; \zeta \right), \widehat{\theta} \left( \eta; \zeta \right), \widehat{\phi} \left( \eta; \zeta \right) \right] \tag{29}$$

$$\mathbb{L}\left(1-\mathbb{L}\right)L\_{\widehat{\phi}}\left[\widehat{\phi}\left(\eta;\mathbb{L}\right)-\widehat{\phi}\_{0}\left(\eta\right)\right]=p\hbar\mathop{\frac{1}{\widehat{\phi}}}\mathop{\mathrm{N}}\_{\widehat{\phi}}\left[\widehat{\phi}\left(\eta;\mathbb{L}\right),\widehat{f}\left(\eta;\mathbb{L}\right),\widehat{\theta}\left(\eta;\mathbb{L}\right)\right] \tag{30}$$

Whereas, BCs are

$$\begin{aligned} \left. \widehat{f} \left( \eta; \zeta \right) \right|\_{\eta=0} &= 0, \left. \left. \frac{\partial \widehat{f} \left( \eta; \zeta \right)}{\partial \eta} \right|\_{\eta=0} = \left. \widehat{\theta} \left( \eta; \zeta \right) \right|\_{\eta=0} = 1 \\ \left. \widehat{f} \left( \eta; \zeta \right) \right|\_{\eta=\infty} &= \left. \widehat{\theta} \left( \eta; \zeta \right) \right|\_{\eta=\infty} = \widehat{\phi} \left( \eta; \zeta \right) \Big|\_{\eta=\infty} = 0, \end{aligned} \tag{31}$$

While the embedding constraint is <sup>ζ</sup> <sup>∈</sup> [0, 1], to regulate for the solution convergence - *f* , - <sup>θ</sup> and - φ are used. When ζ = 0 and ζ = 1, we have:

$$
\widehat{f}\left(\eta;1\right) = \widehat{f}\left(\eta\right), \widehat{\theta}\left(\eta;1\right) = \widehat{\theta}\left(\eta\right), \widehat{\phi}\left(\eta;1\right) = \widehat{\phi}\left(\eta\right), \tag{32}
$$

Expand the *f* (η; ζ), <sup>θ</sup>(η; <sup>ζ</sup>) and φ(η; ζ) through Taylor's series for ζ = 0

$$\begin{array}{l}\widehat{f}(\eta;\zeta) = \widehat{f}\_{0}(\eta) + \Sigma\_{n=1}^{\infty} \widehat{f}\_{n}(\eta)\zeta^{n} \\\widehat{\theta}(\eta;\zeta) = \widehat{\theta}\_{0}(\eta) + \Sigma\_{n=1}^{\infty} \widehat{\theta}\_{n}(\eta)\zeta^{n} \\\widehat{\phi}(\eta;\zeta) = \widehat{\phi}\_{0}(\eta) + \Sigma\_{n=1}^{\infty} \widehat{\phi}\_{n}(\eta)\zeta^{n} \end{array} \tag{33}$$

$$\widehat{f}\_n(\eta) = \frac{1}{n!} \frac{\widehat{\partial f}(\eta; \zeta)}{\partial \eta} \bigg|\_{\mathfrak{p}=0}, \widehat{\partial\_{\mathfrak{n}}(\eta)} = \frac{1}{n!} \frac{\widehat{\partial \partial}(\eta; \zeta)}{\partial \eta} \bigg|\_{\mathfrak{p}=0}, \widehat{\phi}\_n(\eta) = \frac{1}{n!} \frac{\widehat{\partial \phi}(\eta; \zeta)}{\partial \eta} \bigg|\_{\mathfrak{p}=0}. \tag{34}$$

Whereas, BCs are:

$$\begin{aligned} \widehat{f}'(0) &= 0, f'(0) = \widehat{\theta'}(0) = 1\\ \widehat{f}'(\infty) &= \widehat{\theta}(\infty) = \widehat{\phi}(\infty) = 0. \end{aligned} \tag{35}$$

Now

$$\begin{split} \widehat{\mathfrak{R}}\_{n}^{\widehat{f}}(\eta) &= A\_{1} \big( (1 + 2\eta\gamma) \widehat{f}\_{n-1}^{\prime\prime\prime} + 2\gamma \widehat{f}\_{n-1}^{\prime\prime} \big) - \left( \sum\_{j=0}^{w-1} \widehat{f}\_{w-1-j} \widehat{f}\_{j}^{\prime\prime} - f\_{n-1}^{\prime\prime 2} (1 - \text{Fx}) \right) \\ &- \widehat{f}\_{n-1}^{\prime\prime}(M + k\_{1}A\_{2}) + A\_{3} \lambda \widehat{\theta}\_{n-1} \big) . \end{split} \tag{36}$$

$$\hat{\mathcal{R}}\_{n}^{\widehat{\boldsymbol{\theta}}}(\boldsymbol{\eta}) = (1 + 2\eta\boldsymbol{\gamma})\hat{\boldsymbol{\theta}}\_{n-1}^{\boldsymbol{\prime}\prime} + \hat{\boldsymbol{\gamma}}\hat{\boldsymbol{\theta}}\_{n-1}^{\boldsymbol{\prime}} + \text{Pr}(1 + 2\eta\boldsymbol{\gamma})|\mathcal{N}\_{b}\sum\_{j=0}^{w-1} \hat{\boldsymbol{\theta}}\_{w-1-j}^{\prime} \hat{\boldsymbol{\phi}}\_{j} + \text{N}\_{t}\hat{\boldsymbol{\theta}}\_{n-1}^{\prime 2}|\_{+} + \text{Pr}\sum\_{j=0}^{w-1} \hat{\boldsymbol{\theta}}\_{w-1-j}^{\prime} \hat{\boldsymbol{f}}\_{j},\tag{37}$$

$$\hat{\mathfrak{R}}\_{n}^{\hat{\phi}}(\eta) = (1 + 2\eta\mathcal{V})\hat{\phi}\_{n-1}^{\prime\prime} + \frac{N\_{t}}{N\_{b}}((1 + 2\eta\mathcal{V})\hat{\phi}\_{n-1}^{\prime\prime} + \gamma\widehat{\phi}\_{n-1}^{\prime}) + \gamma\widehat{\phi}\_{n-1}^{\prime} + \text{Sc}\sum\_{j=0}^{w-1} \widehat{f}\_{w-1-j} \widehat{\phi}\_{j}^{\prime} \,. \tag{38}$$

While

$$\chi\_n = \begin{cases} 0, \text{ if } \zeta \le 1 \\ 1, \text{ if } \zeta > 1. \end{cases} \tag{39}$$
