## 3 Basic Idea of Adomian Decomposition Method (ADM)
### The ADM Solution of the Problem  
The inverse operator \(L_{y}^{-1}=\iint(. )dydy\) is applied on the second order differential Equations (14) and (15) and, using the standard form of ADM, we get:  
\[u(y,t)=g(y,t)+ML_{y}^{-1}u(y,t)+L_{y}^{-1}\Bigg{[}\frac{\partial u(y,t)}{ \partial t}\Bigg{]}-W_{e}L_{y}^{-1}\Bigg{[}\frac{\partial}{\partial y}\left( \frac{\partial u(y,t)}{\partial y}\right)^{2}\Bigg{]}, \tag{53}\]  
\[\Theta(y,t)=h(y,t)+\Pr L_{y}^{-1}\Bigg{[}\frac{\partial\Theta(y,t)}{\partial t }\Bigg{]}-EcL_{y}^{-1}\Bigg{[}\left(\frac{\partial u(y,t)}{\partial y}\right)^ {2}+W_{e}\Bigg{(}\frac{\partial u(y,t)}{\partial y}\Bigg{)}^{3}\Bigg{]}, \tag{54}\]  
To obtain the series solutions of Equations (53) and (54), summation is used as:  
\[\sum_{n=0}^{\infty}u_{n}=g(y,t)+ML_{y}^{-1}\Bigg{[}\sum_{n=0}^{\infty}u_{n}(y,t)\Bigg{]}+L_{y}^{-1}\Bigg{[}\frac{\partial}{\partial t}\sum_{n=0}^{\infty}u _{n}(y,t)\Bigg{]}-W_{e}L_{y}^{-1}\Bigg{[}\sum_{n=0}^{\infty}A_{n}(y,t)\Bigg{]}, \tag{55}\]  
\[ \begin{array}{c}\sum_{n=0}^{\infty}\Theta_{n}(y,t)=h(y,t)+\Pr L_{y}^{-1} \Big{[}\frac{\partial}{\partial t}\sum_{n=0}^{\infty}\Theta_{n}(y,t)\Big{]}- \\ EcL_{y}^{-1}[\sum_{n=0}^{\infty}B_{n}(y,t)]-EcW_{e}L_{y}^{-1}[\sum_{n=0}^{ \infty}C_{n}(y,t)],\end{array}  \tag{56}\]  
For \(n\geq 0\) the Adomian polynomials \(A_{n}(y,t),B_{n}(y,t)\) and \(C_{n}(y,t)\) from Equations (53) and (54) are defined as:  
\[ \begin{array}{c}\sum_{n=0}^{\infty}A_{n}(y,t)=\sum_{n=0}^{\infty}\frac{ \partial}{\partial y}\left(\frac{\partial u_{n}(y,t)}{\partial y^{n}}\right)^ {2},\sum_{n=0}^{\infty}B_{n}(y,t)=\sum_{n=0}^{\infty}\Big{(}\frac{\partial u_ {n}(y,t)}{\partial y^{n}}\Big{)}^{2},\\ \sum_{n=0}^{\infty}C_{n}(y,t)=\sum_{n=0}^{\infty}\Big{(}\frac{\partial u_{n}(y,t)}{\partial y^{n}}\Big{)}^{3},\end{array}  \tag{57}\]In components form Equations (55) and (56) are derived as:  
\[ \begin{array}{l}u_{0}(y,t)+u_{1}(y,t)+u_{2}(y,t)+....=g(y,t)+ML_{y}^{-1}(u_{0}( y,t)+u_{1}(y,t)+u_{2}(y,t)+...)+\\ L_{y}^{-1}\Big{[}\frac{\partial}{\partial t}(u_{0}(y,t)+u_{1}(y,t)+u_{2}(y,t)+...)\Big{]}-W_{e}L_{y}^{-1}(A_{0}(y,t)+A_{1}(y,t)+A_{2}(y,t)+...),\\ \\ \Theta_{0}(y,t)+\Theta_{1}(y,t)+\Theta_{2}(y,t)+...=h(y,t)+\Pr L_{y}^{-1}\Big{[} \frac{\partial}{\partial t}(\Theta_{0}(y,t)+\Theta_{1}(y,t)+\Theta_{2}(y,t)+... )\Big{]}-\\ EcL_{y}^{-1}[(B_{0}(y,t)+B_{1}(y,t)+B_{2}(y,t)+...)-EcW_{e}L_{y}^{-1}[(C_{0}(y, t)+C_{1}(y,t)+C_{2}(y,t)+...),\\ \end{array}  \tag{58}\]  
Expanding Equations (58) and (59) and comparing both sides for the velocity and temperature fields components, we get:  
\[u_{0}(y,t)=g(y,t)=L_{y}^{-1}\bigg{(}\frac{\partial^{2}u_{0}}{\partial y^{2}}- m\bigg{)}, \tag{60}\]  
\[\Theta_{0}(y,t)=h(y,t)=L_{y}^{-1}\bigg{(}\frac{\partial^{2}\Theta_{0}}{ \partial y^{2}}\bigg{)}, \tag{61}\]  
\[u_{1}(y,t)=ML_{y}^{-1}[u_{0}(y,t)]+L_{y}^{-1}\bigg{(}\frac{\partial u_{0}(y,t) }{\partial t}\bigg{)}-W_{e}L_{y}^{-1}[A_{0}(y,t)], \tag{62}\]  
\[\Theta_{1}(y,t)=\Pr L_{y}^{-1}\bigg{(}\frac{\partial\Theta_{0}(y,t)}{\partial t }\bigg{)}-EcL_{y}^{-1}[B_{0}(y,t)]-W_{e}EcL_{y}^{-1}[C_{0}(y,t)], \tag{63}\]  
\[u_{2}(y,t)=ML_{y}^{-1}[u_{1}(y,t)]+L_{y}^{-1}\bigg{(}\frac{\partial u_{1}(y,t) }{\partial t}\bigg{)}-W_{e}L_{y}^{-1}[A_{1}(y,t)], \tag{64}\]  
\[\Theta_{2}(y,t)=\Pr L_{y}^{-1}\bigg{(}\frac{\partial\Theta_{1}(y,t)}{\partial t }\bigg{)}-EcL_{y}^{-1}[B_{1}(y,t)]-W_{e}EcL_{y}^{-1}[C_{1}(y,t)], \tag{65}\]  
Using the physical conditions from Equations (16) and (17) in the Equations (60)-(65), the zeroth and first component solution are obtained as:  
\[u_{0}(y,t)=1+\text{Cos}(\omega t)-my+\Big{(}\frac{m}{2}\Big{)}y^{2}, \tag{66}\]  
\[\Theta_{0}(y,t)=y, \tag{67}\]  
\[ \begin{array}{l}u_{1}(y,t)=\Big{(}m^{2}W_{e}+\frac{mM}{3}+\omega Sin[\omega t ]-Cos[\omega t]-1\Big{)}y+\\ \frac{1}{2}\big{(}1+Cos[\omega t]-\omega Sin[\omega t]+2m^{2}W_{e}\big{)}y^{2} -\frac{1}{6}\big{(}mM+2m^{2}W_{e}\big{)}y^{3}+\Big{(}\frac{mM}{24}\Big{)}y^{4},\\ \end{array}