## 3 Numerical Method  
Equations (19)-(21) are the system of nonlinear, 3 rd order in \(f\), 2 nd order in \(\theta\) and 2 nd order in \(\phi\) respectively. First of all these non-linear ODE's are reduce into a system of first order ODE's and then solved by using shooting method. The Equations (19)-(21) can be written as:  
\[f^{\prime\prime\prime}=-\bigg{[}\frac{-d\theta^{\prime}f^{\prime \prime}+ff^{\prime\prime}-\frac{2m}{m+1}f^{\prime}-\frac{2}{m+1}Mf^{\prime}}{ a+d-d\theta-Std+H-0.5(m+)H\lambda(f^{\prime\prime})^{2}}\bigg{]}, \tag{33}\] \[\theta^{\prime\prime}=-\frac{Pr}{a_{1}+\varepsilon\theta}\bigg{[} -\frac{m-1}{m+1}\theta f^{\prime}-St\frac{m-1}{m+1}f^{\prime}-f\theta^{\prime} +\frac{\varepsilon}{Pr}\phi^{\prime 2}+Nb\phi^{\prime}\theta^{\prime}+Nt(\theta^{ \prime})^{2}\bigg{]},\] (34) \[\phi^{\prime\prime}=-LePr\bigg{[}\frac{m-1}{m+1}\phi f^{\prime}-Sc \frac{m-1}{m+1}f^{\prime}-f\phi^{\prime}-Kr((1+\delta\theta)^{n}\exp(\frac{- E}{(1+\delta\theta)})\phi)\bigg{]}-\frac{N_{b}}{N_{t}}(\theta^{\prime\prime}) \tag{35}\]  
To convert these higher order nonlinear ODE's into system of first order ODE's, let  
\[f=u_{1},\ f^{\prime}=u_{2},\ \ f^{\prime\prime}=u_{3}\ \text{and}\ f^{ \prime\prime\prime}=u_{3}^{\prime}, \tag{36}\] \[\theta=u_{4},\ \ \theta^{\prime}=u_{5}\ \ \ \text{and}\ \ \ \theta^{\prime\prime}=u_{5}^{\prime}\] (37) \[\phi=u_{6},\ \ \phi^{\prime}=u_{7}\ \ \ \text{and}\ \ \ \phi^{\prime\prime}=u_{7}^{\prime} \tag{38}\]  
The nonlinear coupled ODE's are converted into a system o first order simultaneous algebraic form, which can be defined as form as \[u_{1}^{\prime} =u_{2}, \tag{39}\] \[u_{2}^{\prime} =u_{3},\] (40) \[u_{3}^{\prime} =-\bigg{[}\frac{-du_{5}u_{3}+u_{1}u_{3}-\frac{2m}{m+1}u_{2}^{2}- \frac{2}{m+1}Mu_{2}}{a+d-du_{4}-Std+H-0.5(m+)H\lambda(u_{3})^{2}}\bigg{]},\] (41) \[u_{4}^{\prime} =u_{5},\] (42) \[u_{5}^{\prime} =-\frac{Pr}{a_{1}+\varepsilon u_{4}}\bigg{[}-\frac{m-1}{m+1}u_{4 }u_{2}+\frac{\varepsilon}{Pr}u_{5}^{2}-St\frac{m-1}{m+1}u_{2}+u_{1}u_{5}+Nbu_{ 7}u_{5}+Nt(u_{5})^{2}\bigg{]}\] (43) \[u_{6}^{\prime} =u_{7},\] (44) \[u_{7}^{\prime} =-LePr\bigg{[}u_{7}u_{1}-\frac{m-1}{m+1}u_{6}u_{2}-Sc\frac{m-1}{m +1}u_{2}-Kr((1+\delta u_{4})^{n}\exp(\frac{-E}{(1+\delta u_{4})})u_{6})\bigg{]} -\frac{N_{b}}{N_{t}}(u_{5}^{\prime}).\]  
boundary conditions are  
\[u_{1}(0)+\alpha\frac{m-1}{m+1}u_{2}(0)=0,\ u_{2}(0)=1,\ u_{4}(0)= 1+Ltu_{4}(0),\ Nbu_{6}(0)+Ntu_{4}(0)=0, \tag{45}\] \[u_{2}(\infty)=0,\ u_{4}(\infty)=0,\ u_{6}(\infty)=0 \tag{46}\]  
To determine the solution of system of seven ODE's (39)-(45) by using shooting method, seven initial assumptions are required, but in system (46) two initial guesses are given in \(f\) one in \(\theta\) and one in \(\phi\) and the other three conditions are defined as \(\eta\to\infty\). These three conditions generate result in three unknowns. The subsequent and foremost step of this method is choosing the estimated values of \(\eta\) at \(\infty\), solution process is initiated with certain initial guesses and finding out the solution of (BVP) including governing model. The method of solution with a new values of \(\eta\) at \(\eta\to\infty\) and the method is repeated until two consecutive values of \(f^{\prime\prime}(0)\), \(\theta^{\prime}(0)\) and \(\phi^{\prime}(0)\) are different only after the significant digits. Thus final values of \(\eta\) is considered as \(\eta\to\infty\).