If Equation (8) is written in the compact form of (13), we get

$$\begin{array}{}{\displaystyle F(\mathbf{u})=0,}\end{array}$$(22)

where **u** = (*f*, *f*′, *f*″, *f*″′, *θ*, *ϕ*). Upon applying Quasi-linearization on (22) we get

$$\begin{array}{}{\displaystyle {a}_{0,r}{f}_{r+1}+{a}_{1,r}{f}_{r+1}^{\prime}+{a}_{2,r}{f}_{r+1}^{\u2033}+{a}_{3,r}{f}_{r+1}^{\u2034}}\\ {\displaystyle +{a}_{4,r}{f}_{r+1}^{(iv)}+{a}_{5,r}{\theta}_{r+1}+{a}_{6,r}{\varphi}_{r+1}={R}_{r}^{(1)},}\end{array}$$(23)

where

$$\begin{array}{}{\displaystyle {a}_{0,r}:={F}_{f}({\mathbf{u}}_{r})=\beta ({f}_{r}^{\u2033}-\frac{1}{2}{f}_{r}^{(iv)})+(1+{\lambda}_{1}){f}_{r}^{\u2033},}\\ {\displaystyle {a}_{1,r}:={F}_{{f}^{\prime}}({\mathbf{u}}_{r})=(1+{\lambda}_{1})(-{M}^{2}-4{f}_{r}^{\prime}))}\\ {\displaystyle {a}_{2,r}:={F}_{{f}^{\u2033}}({\mathbf{u}}_{r})=\beta ({f}_{r}+3{f}_{r}^{\u2033})+(1+{\lambda}_{1}){f}_{r},}\\ {\displaystyle {a}_{3,r}:={F}_{{f}^{\u2034}}({\mathbf{u}}_{r})=1,}\\ {\displaystyle {a}_{4,r}:={F}_{{f}^{(iv)}}({\mathbf{u}}_{r})=-\frac{\beta}{2}{f}_{r}\phantom{\rule{thinmathspace}{0ex}}}\\ {\displaystyle {a}_{5,r}:={F}_{\theta}({\mathbf{u}}_{r})=2(1+{\lambda}_{1}){\lambda}_{3},}\\ {\displaystyle {a}_{6,r}:={F}_{\varphi}({\mathbf{u}}_{r})=2(1+{\lambda}_{1}){\lambda}_{4}}\\ {\displaystyle {R}_{r}^{(1)}={f}_{r}{f}_{r}^{\u2033}+\frac{3}{2}\beta ({f}_{r}^{\u2033}{)}^{2}-2({f}_{r}^{\prime}{)}^{2}{\lambda}_{1}-2({f}_{r}^{\prime}{)}^{2}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}-\frac{1}{2}\beta {f}_{r}{f}_{r}^{(iv)}+\beta {f}_{r}{f}_{r}^{\u2033}+{\lambda}_{1}{f}_{r}{f}_{r}^{\u2033}}\end{array}$$

Similarly, Quasi-linearization transforms (9) to

$$\begin{array}{}{\displaystyle \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle {b}_{0,r}{f}_{r+1}+{b}_{1,r}{f}_{r+1}^{\prime}+{b}_{2,r}{\theta}_{r+1}}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle +{b}_{3,r}{\theta}_{r+1}^{\prime}+{b}_{4,r}{\theta}^{\u2033}={R}_{r}^{(2)},}\end{array}$$(24)

where

$$\begin{array}{}{\displaystyle {b}_{0,r}:=\frac{1}{2}{\theta}_{r}^{\prime},\phantom{\rule{thinmathspace}{0ex}}{b}_{1,r}:=-{\theta}_{r},\phantom{\rule{thinmathspace}{0ex}}{b}_{2,r}:=-{f}_{r}^{\prime},\phantom{\rule{thinmathspace}{0ex}}{b}_{3,r}:=\frac{1}{2}{f}_{r},}\\ {\displaystyle {b}_{4,r}:=\frac{4+3R}{6Pr\phantom{\rule{thinmathspace}{0ex}}R},\phantom{\rule{thinmathspace}{0ex}}{R}^{(2)}=\frac{1}{2}{f}_{r}{\theta}_{r}^{\prime}-{f}_{r}^{\prime}{\theta}_{r}.}\end{array}$$

Proceeding in a similar manner also transforms (10) to

$$\begin{array}{}{\displaystyle {c}_{0,r}{f}_{r+1}+{c}_{1,r}{f}_{r+1}^{\prime}+{c}_{2,r}{\varphi}_{r+1}}\\ {\displaystyle +{c}_{3,r}{\varphi}_{r+1}^{\prime},+{c}_{4,r}{\varphi}^{\u2033}={R}_{r}^{(3)}}\end{array}$$(25)

where

$$\begin{array}{}{\displaystyle {c}_{0,r}:=\frac{1}{2}{\varphi}_{r}^{\prime},\phantom{\rule{thinmathspace}{0ex}}{c}_{1,r}:=-{\varphi}_{r},\phantom{\rule{thinmathspace}{0ex}}{c}_{2,r}:=-{f}_{r}^{\prime}-\gamma ,}\\ {\displaystyle {c}_{3,r}:=\frac{1}{2}{f}_{r},{c}_{4,r}:=\frac{1}{2Sc},{R}^{(2)}=\frac{1}{2}{f}_{r}{\varphi}_{r}^{\prime}-{f}_{r}^{\prime}{\varphi}_{r}}\end{array}$$

Equations (23), (24) and (25) are subject to boundary conditions

$$\begin{array}{}{\displaystyle {f}_{r+1}(0)=0,\phantom{\rule{thinmathspace}{0ex}}{f}_{r+1}^{\prime}(0)=1,\phantom{\rule{thinmathspace}{0ex}}{f}_{r+1}^{\prime}(\mathrm{\infty})=0,\phantom{\rule{thinmathspace}{0ex}}{f}_{r+1}^{\u2033}(\mathrm{\infty})=1,}\end{array}$$(26)

$$\begin{array}{}{\displaystyle {\theta}_{r+1}(0)=1,\phantom{\rule{thinmathspace}{0ex}}{\theta}_{r+1}(\mathrm{\infty})=0,}\end{array}$$(27)

$$\begin{array}{}{\displaystyle {\varphi}_{r+1}(0)=1,\phantom{\rule{thinmathspace}{0ex}}{\varphi}_{r+1}(\mathrm{\infty})=0,}\end{array}$$(28)

respectively. Chebyshev differentiation replaces the differential equations (23), (24) and (25) with a
single linear system

$$\begin{array}{}{\displaystyle A{\mathbf{U}}_{r+1}={\mathbf{R}}_{r},}\end{array}$$(29)

where

$$\begin{array}{}{\displaystyle A=\left(\begin{array}{ccc}{A}^{(11)}& {A}^{(12)}& {A}^{(13)}\\ {A}^{(21)}& {A}^{(22)}& O\\ {A}^{(31)}& O& {A}^{(33)}\end{array}\right),\phantom{\rule{1em}{0ex}}{\mathbf{U}}_{r+1}=\left(\begin{array}{c}{\mathbf{F}}_{r+1}\\ {\mathit{\Theta}}_{r+1}\\ {\mathit{\Phi}}_{r+1}\end{array}\right),}\\ {\mathbf{R}}_{r}=\left(\begin{array}{c}{\mathbf{R}}_{r}^{(1)}\\ {\mathbf{R}}_{r}^{(2)}\\ {\mathbf{R}}_{r}^{(3)}\end{array}\right)\end{array}$$(30)

$$\begin{array}{}{\displaystyle {A}^{(11)}=\text{diag}\left\{-\frac{\beta}{2}{\mathbf{F}}_{r}\right\}{\hat{D}}^{4}+{\hat{D}}^{3}+\text{diag}\{\beta ({\mathbf{F}}_{r}+3{\mathbf{F}}_{r}^{\u2033})}\\ {\displaystyle +(1+{\lambda}_{1}){\mathbf{F}}_{r}\}{\hat{D}}^{2}+[-(1+{\lambda}_{1}){M}^{2}I-4(1+{\lambda}_{1}){\mathbf{F}}_{r}^{\prime}]\hat{D}+}\\ {\displaystyle \text{diag}\left\{\beta ({\mathbf{F}}_{r}^{\u2033}-\frac{1}{2}{\mathbf{F}}_{r}^{(iv)})+(1+\lambda 1){\mathbf{F}}_{r}^{\u2033}\right\},}\end{array}$$

$$\begin{array}{}{\displaystyle {A}^{(12)}=2(1+{\lambda}_{1}){\lambda}_{3}I,\phantom{\rule{thinmathspace}{0ex}}{A}^{(13)}=2(1+{\lambda}_{1}){\lambda}_{4}I}\\ {\displaystyle {\mathbf{R}}_{r}^{(1)}={\mathbf{F}}_{r}\circ {\mathbf{F}}_{r}^{\u2033}+\frac{3}{2}\beta {\mathbf{F}}_{r}^{\u2033}\circ {\mathbf{F}}_{r}^{\u2033}-2{\lambda}_{1}{\mathbf{F}}_{r}^{\prime}\circ {\mathbf{F}}_{r}^{\prime}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}-2{\mathbf{F}}_{r}^{\prime}\circ {\mathbf{F}}_{r}^{\prime}-\frac{\beta}{2}{\mathbf{F}}_{r}\circ {\mathbf{F}}_{r}^{(iv)}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}+\beta {\mathbf{F}}_{r}\circ {\mathbf{F}}_{r}^{\u2033}+{\lambda}_{1}{\mathbf{F}}_{r}\circ {\mathbf{F}}_{r}^{\u2033},}\\ {\displaystyle {A}^{(21)}=\text{diag}\{-{\mathit{\Theta}}_{r}\}\hat{D}+\text{diag}\left\{\frac{1}{2}{\mathit{\Theta}}_{r}^{\prime}\right\},}\\ {\displaystyle {A}^{(22)}=\frac{4+3R}{6\phantom{\rule{thinmathspace}{0ex}}Pr\phantom{\rule{thinmathspace}{0ex}}R}{\hat{D}}^{2}+\text{diag}\left\{\frac{1}{2}{\mathbf{F}}_{r}\right\}\hat{D}+\text{diag}-{\mathbf{F}}_{r}^{\prime},}\\ {\displaystyle {\mathbf{R}}_{r}^{(2)}=\frac{1}{2}{\mathbf{F}}_{r}\circ {\mathit{\Theta}}_{r}^{\prime}-{\mathbf{F}}_{r}^{\prime}\circ {\mathit{\Theta}}_{r},}\\ {\displaystyle {A}^{(31)}=\text{diag}\{-{\mathit{\Phi}}_{r}\}\hat{D}+\text{diag}\left\{\frac{1}{2}{\mathit{\Phi}}_{r}^{\prime}\right\},}\\ {\displaystyle {A}^{(33)}=\frac{1}{Sc}{\hat{D}}^{2}+\text{diag}\left\{\frac{1}{2}{\mathbf{F}}_{r}\right\}\hat{D}+\text{diag}\{-{\mathbf{F}}_{r}^{\prime}\}-\gamma I,}\\ {\displaystyle {\mathbf{R}}_{r}^{(2)}=\frac{1}{2}{\mathbf{F}}_{r}\circ {\mathit{\Phi}}_{r}^{\prime}-{\mathbf{F}}_{r}^{\prime}\circ {\mathit{\Phi}}_{r},}\\ {\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathbf{F}}_{r}=[{f}_{r}({\xi}_{0})\phantom{\rule{thinmathspace}{0ex}}{f}_{r}({\xi}_{1})\phantom{\rule{thinmathspace}{0ex}}\dots \phantom{\rule{thinmathspace}{0ex}}{f}_{r}({\xi}_{N}){]}^{T},}\\ {\displaystyle \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\mathit{\Theta}}_{r}=[{\theta}_{r}({\xi}_{0})\phantom{\rule{thinmathspace}{0ex}}{\theta}_{r}({\xi}_{1})\phantom{\rule{thinmathspace}{0ex}}\dots \phantom{\rule{thinmathspace}{0ex}}{\theta}_{r}({\xi}_{N}){]}^{T}.}\end{array}$$

*Φ*_{r} = [*Φ*_{r}(*ξ*_{0}),*Φ*_{r}(*ξ*_{1}) … *Φ*_{r}(*ξ*_{N})]^{T}, *I* is the (*N* + 1) × (*N* + 1) identity matrix and

$$\begin{array}{}{\displaystyle \text{diag}\{{u}_{0},{u}_{1},\dots ,{u}_{N}\}=\left(\begin{array}{c}{u}_{0}\\ & {u}_{1}\\ & & \ddots \\ & & & {u}_{N}\end{array}\right)}\end{array}$$

is an (*N* + 1) × (*N* + 1) diagonal matrix. The linear system (29) is subject to boundary conditions

$$\begin{array}{}{\displaystyle {f}_{r+1}({\xi}_{N})=0,\phantom{\rule{thinmathspace}{0ex}}\sum _{k=0}^{N}{\hat{D}}_{Nk}f({\xi}_{k})=1,}\\ {\displaystyle \sum _{k=0}^{N}{\hat{D}}_{0k}f({\xi}_{k})=0,}\end{array}$$(31)

$$\begin{array}{}{\displaystyle \sum _{k=0}^{N}[{\hat{D}}^{2}{]}_{0k}f({\xi}_{k})=0,}\end{array}$$(32)

$$\begin{array}{}{\displaystyle {\theta}_{r+1}({\xi}_{N})=1,\phantom{\rule{thinmathspace}{0ex}}{\theta}_{r+1}({\xi}_{0})=0,}\end{array}$$(33)

$$\begin{array}{}{\displaystyle {\varphi}_{r+1}({\xi}_{N})=1,\phantom{\rule{thinmathspace}{0ex}}{\varphi}_{r+1}({\xi}_{0})=0.}\end{array}$$(34)

Solution of the linear system (29) is preceded by the following:

We include boundary conditions (32) through (34) in a manner similar to that as in paper by Motsa *et al*. [21].

We choose initial approximations as

$$\begin{array}{}{\displaystyle {f}_{0}(\eta )=1-{e}^{-\eta},{\theta}_{0}(\eta )={e}^{-\eta},{\varphi}_{0}(\eta )={e}^{-\eta},}\end{array}$$

so that we satisfy boundary conditions (26) through (28).

Since **U**_{0} is now known, if we solve Equation (29) for each *r* = 0, 1, ⋯ we get subsequent approximations **U**_{1}, **U**_{2}, ⋯.

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