## 2 Formulation of the Problem
The initial guesses are selected as follows  
\[\widetilde{f}_{0}(\eta)=1+\widetilde{f}_{w}+e^{-\eta},\,\,\widetilde{\theta}_ {0}(\eta)=e^{-\eta}\,\,\widetilde{\phi}_{0}(\eta)=e^{-\eta}. \tag{25}\]  
The linear operators in use are \(L_{\varphi},L_{f}\) and \(L_{\theta}\):  
\[L_{f}(f)=f_{\eta\eta\eta}-f_{\eta},\,\,L_{\theta}(\theta)=\theta_{\eta\eta}- \theta\,,\,\,L_{\varphi}(\phi)=\phi_{\eta\eta}-\phi, \tag{26}\]which exhibit the following properties:  
\[L_{f}(F_{1}+F_{2}e^{-\eta}+F_{3}e^{\eta})=0,\ L_{\theta}(F_{4}e^{-\eta}+F_{5}e^{ \eta})=0\,\ L_{\phi}(F_{6}e^{-\eta}+F_{7}e^{\eta})=0, \tag{27}\]  
where \(F_{i}(i=1\)-\(7)\) are the constants in this wide-ranging solution. The subsequent non-linear operatives \(N_{\theta},N_{\phi}\) and \(N_{f}\) are specified as:  
\[ \begin{array}{l}N_{f}[\widetilde{f}(\eta;\zeta)]=\frac{\partial^{3}\widetilde {f}(\eta;\zeta)}{\partial\eta^{3}}+\widetilde{f}(\eta;\zeta)\frac{\partial^{2 }\widetilde{f}(\eta;\zeta)}{\partial\eta^{2}}-\left(\frac{\partial\widetilde{ f}(\eta;\zeta)}{\partial\eta}\right)^{2}-M\frac{\partial\widetilde{f}(\eta; \zeta)}{\partial\eta}-MEI,\\ -\lambda\left(\frac{\partial\widetilde{f}(\eta;\zeta)}{\partial\eta}+\frac{ \eta}{2}\frac{\partial^{2}\widetilde{f}(\eta;\zeta)}{\partial\eta^{2}}\right) -\beta\left(\begin{array}{c}\left(\frac{\partial\widetilde{f}(\eta;\zeta)}{ \partial\eta}\right)^{2}\frac{\partial^{3}\widetilde{f}(\eta;\zeta)}{\partial \eta^{3}}-\\ 2\widetilde{f}(\eta;\zeta)\frac{\partial\widetilde{f}(\eta;\zeta)}{\partial \eta}\frac{\partial^{3}\widetilde{f}(\eta;\zeta)}{\partial\eta^{3}}\end{array}\right) \end{array}  \tag{28}\]  
\[ \begin{array}{l}N_{\theta}\left[\widetilde{f}(\eta;\zeta),\widetilde{\theta} (\eta;\zeta)\right]=\left(1+\frac{4}{3}Rd\right)\frac{\partial^{2}\widetilde{ \theta}(\eta;\zeta)}{\partial\eta^{2}}-\\ \Pr\left[\widetilde{f}(\eta;\zeta)\frac{\partial\widetilde{\theta}(\eta; \zeta)}{\partial\eta}-\frac{\partial\widetilde{f}(\eta;\zeta)}{\partial\eta} \widetilde{\theta}(\eta;\zeta)-\lambda\left(\widetilde{\theta}(\eta;\zeta)+ \frac{\eta}{2}\frac{\partial\widetilde{\theta}(\eta;\zeta)}{\partial\eta} \right)\right]\\ +A\frac{\partial\widetilde{f}(\eta;\zeta)}{\partial\eta}+B\widetilde{\theta}( \eta;\zeta),\\ \\ N_{\phi}\left[\widetilde{f}(\eta;\zeta),\widetilde{\theta}(\eta;\zeta), \widetilde{\phi}(\eta;\zeta)\right]=\frac{\partial^{2}\widetilde{\phi}(\eta; \zeta)}{\partial\eta^{2}}+\\ Sc\left[\widetilde{f}(\eta;\zeta)\frac{\partial\widetilde{\theta}(\eta; \zeta)}{\partial\eta}-\frac{\partial\widetilde{f}(\eta;\zeta)}{\partial\eta }\widetilde{\phi}(\eta;\zeta)-\lambda\left(3\widetilde{\phi}(\eta;\zeta)+ \frac{\eta}{2}\frac{\partial\widetilde{\phi}(\eta;\zeta)}{\partial\eta} \right)\right]\\ -Nt\left(\frac{\partial\widetilde{\theta}(\eta;\zeta)}{\partial\eta}\frac{ \partial\widetilde{\phi}(\eta;\zeta)}{\partial\eta}+\frac{\partial^{2} \widetilde{\theta}(\eta;\zeta)}{\partial\eta^{2}}\widetilde{\phi}(\eta;\zeta) \right),\end{array}  \tag{30}\]  
The zero-order problems are  
\[ \begin{array}{l}(1-\zeta)L_{f}\left[\widetilde{f}(\eta;\zeta)-\widetilde{f} _{0}(\eta)\right]=\xi\dot{h}_{f}N_{f}\left[\widetilde{f}(\eta;\zeta)\right], \\ (1-\zeta)L_{\theta}\left[\widetilde{\theta}(\eta;\zeta)-\widetilde{\theta}_{0}( \eta)\right]=\zeta\dot{h}_{\theta}N_{\theta}\left[\widetilde{f}(\eta;\zeta), \widetilde{\theta}(\eta;\zeta)\right],\\ (1-\zeta)L_{\phi}\left[\widetilde{\phi}(\eta;\zeta)-\widetilde{\phi}_{0}( \eta)\right]=\xi\dot{h}_{\phi}N_{\phi}\left[\widetilde{f}(\eta;\zeta), \widetilde{\theta}(\eta;\zeta),\widetilde{\phi}(\eta;\zeta)\right].\end{array}  \tag{31}\]  
The equivalent boundary conditions are:  
\[