## 2 Mathematical Modeling
### Solution by HAM  
Homotopy analysis method was introduced by [PERSON] [30; 31; 32] for the first time. He used one of the elementary ideas of topology, called homotopy, to derive this method. He used two homotopic functions in the derivation of this technique. The functions are called homotopic function when one of them can be continuously distorted into another. Assume that \(Z_{1},Z_{2}\) are two functions which are continuous, and \(X,Y\) are two topological spaces where \(Z_{1}\)\(\&\)\(Z_{2}\) map from \(X\) to \(Y\), then \(Z_{1}\) is said to be homotopic to \(Z_{2}\) if there is a continuous function \(\Re\),  
\[\Re:X\times[0,\;1]\to Y. \tag{17}\]  
Thus, \(x\in X\), then  
\[\Re[x,\;0]=Z_{1}(x)\text{ and }\Re[x,\;1]=Z_{2}(x). \tag{18}\]  
The mapping \(\Re\) is called homotopic. In order to solve Equations (9) and (10) with the boundary conditions Equation (11), we use HAM [38; 39; 40; 41] with the succeeding process. The initial suppositions are chosen as follows:  
\[f_{0}(\varphi)=1-e^{-\varphi}\sin\varphi,\;g_{0}(\varphi)=e^{-\varphi}. \tag{19}\]  
The linear operators are taken as \(L_{f}\) and \(\text{L}_{g}\):  
\[L_{f}(f)=f^{\prime\prime}-f^{\prime},\;\text{L}_{g}(g)=g^{\prime\prime}-g, \tag{20}\]  
which have the following succeeding properties:  
\[L_{f}(n_{1}+n_{2}e^{-\varphi}+n_{3}e^{\varphi})=0,\;\text{L}_{g}(n_{4}e^{- \varphi}+n_{5}e^{\varphi})=0, \tag{21}\]  
where \(n_{i}(i=1-5)\) are constants. The resultant non-linear operators, \(NL_{f}\) and \(NL_{g}\), are specified as\[ \begin{array}{l}\mathrm{NL}_{f}[f(\varphi;\Omega)]=(1+P)\frac{\partial^{3}f( \varphi;\Omega)}{\partial\varphi^{3}}-D\frac{\partial^{2}f(\varphi;\Omega)}{ \partial\varphi^{2}}-\left(\frac{\partial f(\varphi;\Omega)}{\partial\varphi} \right)^{2}\\ +f(\varphi;\Omega)\frac{\partial^{2}f(\varphi;\Omega)}{\partial\varphi^{2}}- \chi P\left(\frac{\partial^{2}f(\varphi;\Omega)}{\partial\varphi^{2}}\right) ^{2}-(\beta+M)\frac{\partial f(\varphi;\Omega)}{\partial\varphi}\end{array}  \tag{22}\]  
\[ \begin{array}{l}NL_{g}[g(\varphi;\Omega)]=\left(1+\frac{3}{4}K_{r}\right) \frac{\partial^{2}g(\varphi;\Omega)}{\partial\varphi^{2}}\\ +\mathrm{Pr}\left(f(\varphi;\Omega)\frac{\partial g(\varphi;\Omega)}{ \partial\varphi}-D\frac{\partial g(\varphi;\Omega)}{\partial\varphi}\right) -\varepsilon g(\varphi;\Omega).\end{array}  \tag{23}\]  
The zero-order problems from Equations (9) and (10) are  
\[ \begin{array}{l}(1-\Omega)\mathrm{L}_{f}[f(\varphi;\Omega)-f_{0}(\varphi)] =\Omega h_{f}\mathrm{NL}_{f}[f(\varphi;\Omega)],\\ (1-\Omega)\mathrm{L}_{g}[g(\varphi;\Omega)-g_{0}(\varphi)]=\Omega h_{g} \mathrm{NL}_{g}[f(\varphi;\Omega),g(\varphi;\Omega)]\end{array}  \tag{24}\]  
The equivalent boundary conditions are  
\[ \begin{array}{l}f(\varphi;\Omega)\big{|}_{\varphi=0}=0,\left.\frac{\partial f (\varphi;\Omega)}{\partial\varphi}\right|_{\varphi=0}=\sin\tau,\left.\frac{ \partial f(\varphi;\Omega)}{\partial\varphi}\right|_{\varphi\to\infty}=0. \\ g(\varphi;\Omega)\big{|}_{\varphi=0}=1,\left.g(\varphi;\Omega)\right|_{ \varphi\to\infty}=0,\end{array}  \tag{25}\]  
When \(\Omega=0\) and \(\Omega=1\), we have  
\[f(\varphi;1)=f(\varphi)\text{ and }g(\varphi;1)=g(\varphi). \tag{26}\]  
Expanding \(f(\varphi;\Omega)\) and \(g(\varphi;\Omega)\) by Taylor series,  
\[ \begin{array}{l}f(\varphi;\Omega)=f_{0}(\varphi)+\sum\limits_{q=1}^{\infty} f_{q}(\varphi)\Omega^{q}\\ g(\varphi;\Omega)=g_{0}(\varphi)+\sum\limits_{q=1}^{\infty}g_{q}(\varphi) \Omega^{q}\end{array}  \tag{27}\]  
where  
\[f_{q}(\varphi)=\left.\frac{1}{q!}\frac{\partial f(\varphi;\Omega)}{\partial \varphi}\right|_{\Omega=0}\text{ and }g_{q}(\varphi)=\left.\frac{1}{q!}\frac{ \partial g(\varphi;\Omega)}{\partial\varphi}\right|_{\Omega=0}. \tag{28}\]  
Setting \(\Omega=1\) in (28), we obtain  
\[ \begin{array}{l}f(\varphi)=f_{0}(\varphi)+\sum\limits_{q=1}^{\infty}f_{q}( \varphi),\\ g(\varphi)=g_{0}(\varphi)+\sum\limits_{q=1}^{\infty}g_{q}(\varphi).\end{array}  \tag{29}\]  
The \(q\)th-order problem satisfies the following:  
\[