## 3 Solution by Homotopy Analysis Method
\begin{array}{l}\left.f(\eta;\Re)\right|_{\eta\ =\ 0}\ =\ 0,\ \left.\frac{ \partial f(\eta;\Re)}{\partial\eta}\right|_{\eta\ =\ 0}\ =\ 0,\ \left.\frac{ \partial f(\eta;\Re)}{\partial\eta}\right|_{\eta\to\infty}\ =\ 0,\\ \left.g(\eta;\Re)\right|_{\eta\ =\ 0}\ =\ c,\ \left.\frac{ \partial g(\eta;\Re)}{\partial\eta}\right|_{\eta\ =\ 0}\ =\ 0,\ \left.g(\eta;\Re)\right|_{\eta\to\infty}\ =\ 0,\\ \left.\frac{\partial\theta(\eta;\Re)}{\partial\eta}\right|_{\eta\ =\ 0}\ =\ -Bi_{1}(1-\theta),\ \left.\theta(\eta;\Re)\right|_{\eta\to\infty}\ =\ 0,\\ \left.\phi(\eta;\Re)\right|_{\eta\ =\ 0}\ =\ -Bi_{2}(1-\phi),\ \left.\phi(\eta;\Re) \right|_{\eta\to\infty}\ =\ 0.\end{array}  \tag{34}\]  
where \(\Re\in[0,1]\) is the implanting parameter, \(\hslash_{f},\ \hslash_{\theta},\hslash_{\phi}\) are the parameters which were used to regulate the convergence of the solution. When \(\Re\ =\ 0\) and \(\Re\ =\ 1\), we have  
\[ \begin{array}{l}\left.f(\eta;0)\ =\ \left.f_{0}(\eta),\ f(\eta;1)\ =\ \left.f(\eta ),\right.\right.\\ \left.g(\eta;0)\ =\ \left.g_{0}(\eta),\ g(\eta;1)\ =\ \left.g(\eta ),\right.\right.\\ \left.\theta(\eta;0)\ =\ \left.\theta_{0}(\eta),\ \theta(\eta;1)\ =\ \theta(\eta ),\right.\\ \left.\phi(\eta;0)\ =\ \phi_{0}(\eta),\ \phi(\eta;1)\ =\ \phi(\eta).\right.\end{array}  \tag{35}\]  
Expanding \(f(\eta;\Re),g(\eta;\Re),\theta(\eta;\Re)\) and \(\phi(\eta;\Re)\) by [PERSON]'s series  
\[ \begin{array}{l}\left.f(\eta;\Re)\ =\ \left.f_{0}(\eta)+\sum\limits_{q\ =\ 1}^{\infty}f_{q}(\eta)\Re^{q}, \right.\\ \left.g(\eta;\Re)\ =\ \left.g_{0}(\eta)+\sum\limits_{q\ =\ 1}^{\infty}g_{q}(\eta )\Re^{q},\right.\\ \left.\phi(\eta;\Re)\ =\ \left.\phi_{0}(\eta)+\sum\limits_{q\ =\ 1}^{\infty} \phi_{q}(\eta)\Re^{q}.\right.\end{array}  \tag{36}\]  
where  
\[ \begin{array}{l}\left.f_{q}(\eta)\ =\ \frac{1}{\vartheta!}\frac{ \partial f(\eta;\Re)}{\partial\eta}\right|_{\Re\ =\ 0},\mathbb{S}_{q}(\eta)\ =\ \left.\frac{1}{ \vartheta!}\frac{\partial g(\eta;\Re)}{\partial\eta}\right|_{\Re\ =\ 0},\\ \left.\theta_{q}(\eta)\ =\ \frac{1}{\vartheta!}\frac{\partial\theta(\eta;\Re)}{ \partial\eta}\right|_{\Re\ =\ 0},\end{array}  \tag{37}\]  
The secondary constraints \(\hslash_{f},\hslash_{g},\hslash_{\theta}\) and \(\hslash_{\phi}\) are nominated in such a way that the Series (35) converges at \(\Re\ =\ 1\). Changing \(\Re\ =\ 1\) in (35), we get  
\[ \begin{array}{l}\left.f(\eta)\ =\ \left.f_{0}(\eta)+\sum\limits_{q\ =\ 1}^{\infty}f_{q}(\eta), \right.\\ \left.g(\eta)\ =\ \left.g_{0}(\eta)+\sum\limits_{q\ =\ 1}^{\infty}g_{q}(\eta), \right.\\ \left.\theta(\eta)\ =\ \theta_{0}(\eta)+\sum\limits_{q\ =\ 1}^{\infty}\theta_{q}(\eta), \right.\\ \left.\phi(\eta)\ =\ \phi_{0}(\eta)+\sum\limits_{q\ =\ 1}^{\infty}\phi_{q}(\eta). \end{array}  \tag{38}\]The \(q^{th}\)-order problem satiates the following  
\[ \begin{array}{rcl}L_{f}\big{[}f_{q}(\eta)-\chi_{q}f_{q-1}(\eta)\big{]}&=&\hslash _{f}U_{q}^{f}(\eta),\\ L_{g}\big{[}g_{q}(\eta)-\chi_{q}g_{q-1}(\eta)\big{]}&=&\hslash_{g}U_{q}^{g}( \eta),\\ L_{\theta}\big{[}\theta_{q}(\eta)-\chi_{q}\theta_{q-1}(\eta)\big{]}&=&\hslash_ {\theta}U_{q}^{\theta}(\eta),\\ L_{\phi}\big{[}\phi_{q}(\eta)-\chi_{q}\phi_{q-1}(\eta)\big{]}&=&\hslash_{\phi} U_{q}^{\phi}(\eta),\end{array}  \tag{39}\]  
The boundary conditions are  
\[ \begin{array}{rcl}f(0)&=&f^{\prime}(0)&=&f^{\prime}(\infty)&=&0,\\ g^{\prime}(0)-c&=&g^{\prime}(0)&=&g^{\prime}(\infty)&=&0,\\ \theta^{\prime}(0)+Bi_{1}(1-\theta(0))&=&\theta(\infty)&=&0,\\ \phi^{\prime}(0)+Bi_{2}(1-\phi(0))&=&\phi(\infty)&=&0.\end{array}